24th Workshop on Logic, Language, Information and Computation (WoLLIC 2017)

24th Workshop on Logic, Language, Information and Computation (WoLLIC 2017) WoLLIC 2017 was held in London, UK, 18–21 July 2017, in the Department of Computer Science, University College London (UCL). WoLLIC (http://wollic.org) is a series of workshops, which started in 1994 with the aim of fostering interdisciplinary research in pure and applied logic. The idea is to have a forum that is large enough in the number of possible interactions between logic and the sciences related to information and computation, and yet is small enough to allow for concrete and useful interaction among participants. Previous versions were held at Recife (Pernambuco, Brazil) in 1994 and 1995; Salvador (Bahia, Brazil) in 1996; Fortaleza (Ceará, Brazil) in 1997; São Paulo (Brazil) in 1998; Itatiaia (Rio de Janeiro, Brazil) in 1999; Natal (Rio Grande do Norte, Brazil) in 2000; Brasília (Distrito Federal, Brazil) in 2001; Rio de Janeiro (Brazil) in 2002; Ouro Preto (Minas Gerais, Brazil) in 2003; Fontainebleau (France) in 2004; Florianópolis (Santa Catarina, Brazil) in 2005; Stanford (California, USA) in 2006; Rio de Janeiro in 2007; Edinburgh in 2008; Tokyo in 2009; Brasília in 2010; Philadelphia in 2011; Buenos Aires in 2012; Darmstadt in 2013; Valparaiso in 2014; Bloomington (IN) in 2015; Puebla in 2016. It is planned that the meeting will take place in the following years and locations: 2018 in Bogotá, 2019 in Utrecht, 2020 in Arequipa, 2021 in Chennai (tbc). WoLLIC’s scientific sponsors include Association for Symbolic Logic (ASL), Interest Group in Pure and Applied Logics (IGPL), Association for Logic, Language and Information (FoLLI), European Association for Theoretical Computer Science (EATCS), European Association for Computer Science Logic (EACSL), Sociedade Brasileira de Computação (SBC) and Sociedade Brasileira de Lógica (SBL). WoLLIC 2017 once again carried on the tradition of promoting interdisciplinary and cross-disciplinary research, topics ranging from mathematical proof theory to formal semantics of natural languages. The Program Committee consisted of M. Baaz (University of Technology, Vienna, Austria), J. Baldwin (University of Illinois at Chicago, USA), D. Bartozova (Universidade de São Paulo, Brazil), A. Ciabattoni (University of Technology, Vienna, Austria), W. Dean (University of Warwick, UK), E. Grädel (RWTH Aachen, Germany), V. Halbach (University of Oxford, UK), J. Kennedy (Helsinki University, Finland (Chair)), D. Kozen (Cornell University, USA), J. Makowsky (Technion - Israel Institute of Technology, Israel), L. Moss (Indiana University, USA), A. Palmigiano (Delft University, the Netherlands), M. Sadrzadeh (Queen Mary, UK), S. Smets (Amsterdam University, the Netherlands), A. Tornquist (Kobenhavns Universitet, Denmark), R. Verbrugge (University of Groningen, the Netherlands), A. Villaveces (Universidad Nacional, Colombia) and Ph. Welch (University of Bristol, UK). The Organizing Committee members were J. Brotherston (UCL), P. Oliva (Queen Mary), A. G. de Oliveira (U Fed Pernambuco), R. de Queiroz (U Fed Pernambuco, co-chair), A. Silva (UCL, Local co-chair), M. Sadrzadeh (Queen Mary, Local co-chair). As in previous instances, WoLLIC 2017 included invited talks as well as contributed papers. The Program Committee received 61 submissions. A thorough review process by the programme committee, assisted by a number of external reviewers, has led to the acceptance of 23 papers for presentation at the meeting and inclusion in the Proceedings, which was published as volume 10388 of Springer’s Lecture Notes in Computer Science (FoLLI subseries). The conference programme also included two tutorial lectures and 8 invited talks by 8 prominent invited speakers, who have graciously accepted the programme committee’s invitation: Hazel Brickhill (Bristol) (University of Bristol), Michael Detlefsen (University of Notre Dame), Alexander Kurz (University of Leicester), Frederike Moltmann (New York University), David Pym (University College London), Nicole Schweikardt (Humboldt Universität), Fan Yang (Delft University), Boris Zilber (University of Oxford). There were also four tutorials given by Michael Detlefsen, Alexander Kurz, Frederike Moltmann and Nicole Schweikardt. A special issue of Archive for Mathematical Logic (Springer), guest edited by Juliette Kennedy and Ruy de Queiroz, will include peer-reviewed full versions of a selection of the conference presentations. Special Session: Screening of Films about Mathematicians. On the evening of Thursday, July 20th, as a tribute to a recent project which focuses on the cross-cultural connections that are made through mathematics and the impact that Navajo Math Circles can have on a community, there was a screening of George Csicsery’s Navajo Math Circles (2016), a one-hour film, documenting the process of a two-year period in which hundreds of Navajo children in recent years have found themselves at the centre of a lively collaboration with mathematicians from around the world. Juliette Kennedy (Program Chair) Ruy de Queiroz (Org Comm Co-Chair) Alexandra Silva (Local Co-Chair) Mehrnoosh Sadrzadeh (Local Co-Chair). Tutorials Coalgebraic Logic Alexander Kurz Department of Informatics, University of Leicester, University Road, Leicester, LE1 7RH, UK. E-mail: ak155@le.ac.uk In the tutorial we have the time to look at some of the category theory behind the theory sketched in the lecture. Natural Language Metaphysics Friederike Moltmann Research Director (DR1), Centre Nationale de la Recherche Scientifique (CNRS), France. E-mail: fmoltmann@univ-paris1.fr This tutorial gives an introduction to the branch of metaphysics that can be called ‘natural language metaphysics’. It discusses various sorts of appeals philosophers have made to natural language, of ways in analyzes in linguistic semantics involve metaphysical notions, of the ways natural language reflects ontological notions and structures, and of cases of discrepancies between the ontology implicit in natural language and the reflective ontology of philosophers or non-philosophers. Recommended reading: ‘Natural Language Ontology’. Oxford Research Encyclopedia of Linguistics. Oxford UP, New York, 2017 (online). A Tutorial on Database Theory Nicole Schweikardt Humboldt-Universität zu Berlin, Institut für Informatik, Unter den Linden 6, D-10099 Berlin, Germany. E-mail: schweikn@informatik.hu-berlin.de In this tutorial I want to give a brief introduction to database theory, including basic definitions, typical research questions and some fundamental results. Invited Talks A New Way to Measure Subsets of Ordinals: Generalised Closed Unbounded and Stationary Sets Hazel Brickhill University of Helsinki, P.O. Box 33 (Yliopistonkatu 4), 00014, Finland. E-mail: hazel.brickhill@helsinki.fi The notions of closed unbounded and stationary set are central to Set Theory, and provide a way to measure how ‘large’ or ‘thick’ a subset of an ordinal is. In this talk I will introduce a generalization of these notions, based on the generalization of stationarity defined in [1]. Surprisingly for a new concept is set theory, generalized closed unbounded and stationary sets are very simple to define and accessible. They provide a finer calibration measuring the sizes of sets between ‘stationary’ and ‘containing a closed unbounded set’, and are closely related to the phenomena of stationary reflection and indescribability. I will give combinatorial definitions for these notions and also describe how they can be characterized in terms of derived topologies [2]. These notions are being used to answer questions about provability logic (see [2]), and promise a range of further applications. References [1] J. Bagaria , M. Magidor and H. Sakai . Reflection and indescribability in the constructible universe . Israel Journal of Mathematics , 208 , 1 – 11 , 2015 . Google Scholar CrossRef Search ADS [2] J. Bagaria . Derived topologies on ordinals and stationary reflection . https://www.newton.ac.uk/files/preprints/ni16031.pdf, 2016 . Formalism: Historical and Conceptual Background Michael Detlefsen Department of Philosophy, University of Notre Dame, 100 Malloy Hall Notre Dame, IN 46556, Notre Dame, IN 46556 USA. E-mail: mdetlef1@nd.edu The aim will of this talk is to survey and clarify the historical sources of and motives for formalism. Central here will be the clarification of certain distinctive views concerning non-semantical uses of language in reasoning, and of the conditions of its usefulness and reliability. Formalism: Hilbert’s Proposals Michael Detlefsen Department of Philosophy, University of Notre Dame, 100 Malloy Hall Notre Dame, IN 46556, Notre Dame, IN 46556 USA. E-mail: mdetlef1@nd.edu In a late statement of his formalist programme, Hilbert described a proper approach to consistency problems as being based on observation. The aim will be to clarify this much overlooked idea and to briefly explore its significance. Coalgebraic Logic Alexander Kurz Department of Informatics, University of Leicester, University Road, Leicester, LE1 7RH, UK. E-mail: ak155@le.ac.uk We will start with examples and remarks on induction and coinduction and then argue that the appropriate general framework to elucidate them is given by algebras and coalgebras for a functor on a category. Moreover, coalgebras allow us to develop a general theory of dynamic systems. To better understand the generality offered by the notion of functor on a category, we will look at presentations of functors by operations and equations. As an application, we then show how to set up a general theory of logics for coalgebras (=dynamic systems), parametric in a functor. Natural Language Metaphysics Friederike Moltmann Research Director (DR1), Centre Nationale de la Recherche Scientifique (CNRS), France. E-mail: fmoltmann@univ-paris1.fr Metaphysics in the past was considered mainly a pursuit of philosophers, asking questions about being in most general terms. While some philosophers made appeal to natural language in order to argue for a ontological category or metaphysical notion, others have rejected such an appeal arguing that the ontology reflected in language diverges significantly from what there really is, from any philosophically accepted ontology. Whatever one’s view may be of what a philosopher should pursue, it has become clear, especially with the development of natural language semantics (and syntax), that the ontology reflected in natural language is an important object of study in itself, as the subject matter of natural language ontology or more generally natural language metaphysics. This talk discusses a range of issues that arise for natural language metaphysics, such as the sorts of data that it should and that it should not take into account, the ways natural language reflects ontological notions and structures, why there are discrepancies between the ontology implicit in natural language and the reflective ontology of philosophers or non-philosophers and how the relation of natural language metaphysics can be conceived with respect to other projects in metaphysics. Resource Semantics: Logic as a Modelling Technology David Pym University College London, Department of Computer Science, Gower Street, London WC1E 6BT, UK. E-mail: D.Pym@cs.ucl.ac.uk The development of Bunched Implication (BI), the logic of bunched implications, together with its resource semantics, led to the formulation of Separation Logic. However, this rather succesful story sits within a broader, quite systematic logical context. I will review the (family of) logics that are supported by resource semantics, explaining their more-or-less uniform meta-theoretic basis and illustrating their uses in a range of modelling applications. Database Query Answering under Updates Nicole Schweikardt Humboldt-Universität zu Berlin, Institut für Informatik, Unter den Linden 6, D-10099 Berlin, Germany. E-mail: schweikn@informatik.hu-berlin.de Query evaluation is one of the most fundamental tasks in databases, and a vast amount of literature is devoted to the complexity of this problem. This talk will focus on query evaluation in the ‘dynamic setting’, where the database may be updated by inserting or deleting tuples. In this setting, an evaluation algorithm receives a query Q and an initial database D and starts with a preprocessing phase that computes a suitable data structure to represent the result of evaluating Q on D. After every database update, the data structure is updated so that it represents the result of evaluating Q on the updated database. The data structure shall be designed in such a way that it quickly provides the query result, preferably in constant time (i.e. independent of the database size). We focus on the following flavours of query evaluation: (1) Testing: decide whether a given tuple t is contained in Q(D). (2) Counting: compute the number of tuples that belong to Q(D). (3) Enumeration: enumerate Q(D) with a bounded delay between the output tuples. Here, as usual, Q(D) denotes the k-ary relation obtained by evaluating a k-ary query Q on a relational database D. For Boolean queries, all three tasks boil down to (4) Answering: decide if Q(D) is non-empty. Compared to the dynamic descriptive complexity framework introduced by Patnaik and Immerman (1997), which focuses on the expressive power of first-order logic on dynamic databases and has led to a rich body of literature, we are interested in the computational complexity of query evaluation. We say that a query evaluation algorithm is efficient if the update time is either constant or at most polylogarithmic in the size of the database. In this talk I want to give an overview of recent results in this area. Characterizing dependencies in logic and sciences Fan Yang Department of Mathematics and Statistics, University of Helsinki, Finland. E-mail: fan.yang.c@gmail.com In this talk, we discuss theory and applications of dependence and independence logic (DIL), which was introduced by Väänänen [5] and by Grädel and Väänänen [1] with the specific aim of expressing and reasoning about dependence and independence relations. This framework extends first-order logic with new atomic formulas, called dependence and independence atoms, that explicitly specify the dependence relations. In contrast to the usual Tarskian semantics, where formulas are evaluated under single assignments, formulas of DIL are evaluated under sets of assignments (called teams) instead. Such a semantics is call team semantics, introduced by Hodges [2, 3]. The first part of the talk consists of a very brief tutorial on DIL. We will cover the team semantics, expressive power and axiomatization problem of DIL and its notable variants. The second part of the talk discusses (potential) applications of DIL in social and natural sciences, such as database theory, social choice theory, quantum theory and economics. In particular, we will formulate Arrow’s Impossibility Theorem in social choice theory as a type of dependence strengthening theorem in the framework of DIL, and thereby argue that DIL offers an interesting new perspective on social choice theory. This part of the talk is based on a joint work with Eric Pacuit [4]. References [1] E. Grädel and J. Väänänen . Dependence and independence . Studia Logica , 101 , 399 – 410 , 2013 . Google Scholar CrossRef Search ADS [2] W. Hodges . Compositional semantics for a language of imperfect information . Logic Journal of the IGPL , 5 , 539 – 563 , 1997 . Google Scholar CrossRef Search ADS [3] W. Hodges . Some strange quantifiers . In Structures in Logic and Computer Science: A Selection of Essays in Honor of A. Ehrenfeucht , J. Mycielski , G. Rozenberg and A. Salomaa , eds., pp. 51 – 65 . Vol. 1261 of Lecture Notes in Computer Science . London : Springer , 1997 . Google Scholar CrossRef Search ADS [4] E. Pacuit and F. Yang . Dependence and independence in social choice: Arrow’s theorem . In Dependence Logic: Theory and Application , H. V. S. Abramsky , J. Kontinen and J. Väänänen , eds., pp. 235 – 260 . Progress in Computer Science and Applied Logic , Birkhauser , 2016 . Google Scholar CrossRef Search ADS [5] J. Väänänen . Dependence Logic: A New Approach to Independence Friendly Logic . Cambridge : Cambridge University Press , 2007 . Google Scholar CrossRef Search ADS Positive model theory and approximation by finite structures Boris Zilber Mathematical Institute, University of Oxford, and Merton College, 24-29 St Giles, Oxford, OX1 3LB, UK. E-mail: zilber@maths.ox.ac.uk I will introduce a notion of structural approximation and will discuss non-trivial examples of approximations by finite structures of real manifolds and a model of quantum mechanics. Contributed Talks Graph Turing Machines Nathanael L. Ackerman and Cameron E. Freer Department of Mathematics, Harvard University, Cambridge, USA. E-mail: cameron@remine.com We consider graph Turing machines, a model of parallel computation on a graph, which provides a natural generalization of several standard computational models, including ordinary Turing machines and cellular automata. In this extended abstract, we give bounds on the computational strength of functions that graph Turing machines can compute. We also begin the study of the relationship between the computational power of a graph Turing machine and structural properties of its underlying graph. (Acknowledgements: The authors would like to thank Tomislav Petrović, Linda Brown Westrick and the anonymous referees of earlier versions for the helpful comments.) Independence-Friendly Logic without Henkin Quantification Fausto Barbero, Lauri Hella and Raine Rönnholm University of Helsinki, Philosophy, Faculty of Arts, Finland. E-mail: fausto.barbero@helsinki.fi and University of Tampere, Mathematics, Faculty of Natural Sciences, Finland. E-mail: lauri.hella,raine.ronnholm@uta.fi We analyze from a global point of view the expressive resources of IF logic that do not stem from Henkin (partially ordered) quantification. When one restricts attention to regular IF sentences, this amounts to the study of the fragment of IF logic which is individuated by the game-theoretical property of Action Recall. We prove that the fragment of Action Recall can express all existential second-order (ESO) properties. This can be accomplished already by the prenex fragment of Action Recall, whose only second-order source of expressiveness are the so-called signalling patterns. The proof shows that a complete set of Henkin prefixes is explicitly definable in the fragment of Action Recall. In the more general case, in which also irregular IF sentences are allowed, we show that full ESO expressive power can be achieved using neither Henkin nor signalling patterns. Total Search Problems in Bounded Arithmetic and Improved Witnessing Arnold Beckmann and Jean-José Razafindrakotoe Department of Computer Science, College of Science, Swansea University, UK. E-mail: a.beckmann@swansea.ac.uk, jjrazaf@icloud.com We define a new class of total search problems as a subclass of Megiddo and Papadimitriou’s class of total $$\textsf{NP}$$ search problems, in which solutions are verifiable in $$\textsf{AC}^0$$. We denote this class $$\forall \,\exists \textsf{AC}^0$$. We show that all total $$\textsf{NP}$$ search problems are equivalent, w.r.t. $$\textsf{AC}^0$$-many-one reductions, to search problems in $$\forall \,\exists \textsf{AC}^0$$. Furthermore, we show that $$\forall \,\exists \textsf{AC}^0$$ contains well-known problems such as the Stable Marriage and the Maximal Independent Set problems. We introduce the class of Inflationary Iteration problems in $$\forall \,\exists \textsf{AC}^0$$, and show that it characterizes the provably total $$\textsf{NP}$$ search problems of the bounded arithmetic theory corresponding to polynomial time. Cook and Nguyen introduced a generic way of defining a bounded arithmetic theory $$\textsf{VC}$$ for complexity classes $$\textsf{C}$$ which can be obtained using a complete problem. For such C we will define a new class $$\textsf{KPT[C]}$$ of $$\forall \,\exists \textsf{AC}^0$$ search problems based on Student–Teacher games in which the student has computing power limited to $$\textsf{AC}^0$$. We prove that $$\textsf{KPT[C]}$$ characterizes the provably total $$\textsf{NP}$$ search problems of the bounded arithmetic theory corresponding to $$\textsf{C}$$. All our characterizations are obtained via ‘new-style’ witnessing theorems, where reductions are provable in a theory corresponding to $$\textsf{AC}^0$$. On the Reflection Calculus with Partial Conservativity Operators Lev D. Beklemishev Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia. E-mail: bekl@mi.ras.ru Strictly positive logics recently attracted attention both in the description logic and in the provability logic communities for their combination of efficiency and sufficient expressivity. The language of Reflection Calculus (RC) consists of implications between formulas built up from propositional variables and the constant ‘true’ using only conjunction and the diamond modalities which are interpreted in Peano arithmetic as restricted uniform reflection principles. We extend the language of RC by another series of modalities representing the operators associating with a given arithmetical theory T its fragment axiomatized by all theorems of T of arithmetical complexity $$\varPi ^0_n$$, for all n > 0. We note that such operators, in a precise sense, cannot be represented in the full language of modal logic. We formulate a formal system extending RC that is sound and, as we conjecture, complete under this interpretation. We show that in this system one is able to express iterations of reflection principles up to any ordinal $$<\varepsilon _0$$. On the other hand, we provide normal forms for its variable-free fragment. Thereby, the variable-free fragment is shown to be algorithmically decidable and complete w.r.t. its natural arithmetical semantics. (Acknowledgements: This work is supported by the Russian Science Foundation under grant 16-11-10252.) On the Length of Medial-Switch-Mix Derivations Paola Bruscoli and Lutz Straßburger University of Bath, Bath, UK. Inria Palaiseau, France. E-mail: P.Bruscoli@Bath.ac.uk Switch and medial are two inference rules that play a central role in many deep inference proof systems. In specific proof systems, the mix rule may also be present. In this paper we show that the maximal length of a derivation using only the inference rules for switch, medial and mix, modulo associativity and commutativity of the two binary connectives involved, is quadratic in the size of the formula at the conclusion of the derivation. This shows, at the same time, the termination of the rewrite system. Generalized Relations in Linguistics and Cognition Bob Coecke, Fabrizio Genovese, Martha Lewis and Dan Marsden Mathematical Institute, University of Bern, Switzerland. E-mail: daniel.marsden@cs.ox.ac.uk Categorical compositional models of natural language exploit grammatical structure to calculate the meaning of sentences from the meanings of individual words. This approach outperforms conventional techniques for some standard NLP tasks. More recently, similar compositional techniques have been applied to conceptual space models of cognition. Compact closed categories, particularly the category of finite-dimensional vector spaces, have been the most common setting for categorical compositional models. When addressing a new problem domain, such as conceptual space models of meaning, a key problem is finding a compact closed category that captures the features of interest. We propose categories of generalized relations as source of new, practical models for cognition and NLP. We demonstrate using detailed examples that phenomena such as fuzziness, metrics, convexity, semantic ambiguity and meaning that varies with context can all be described by relational models. Crucially, by exploiting a technical framework described in previous work of the authors, we also show how we can combine multiple features into a single model, providing a flexible family of new categories for categorical compositional modelling. Proof Theory and Ordered Groups Almudena Colacito and George Metcalfe Department of Computer Science, University of Oxford, Oxford, UK. E-mail: {almudena.colacito, george.metcalfe}@math.unibe.ch Ordering theorems, characterizing when partial orders of a group extend to total orders, are used to generate hypersequent calculi for varieties of lattice-ordered groups (ℓ-groups). These calculi are then used to provide new proofs of theorems arising in the theory of ordered groups. More precisely, an analytic calculus for abelian ℓ-groups is generated using an ordering theorem for abelian groups; a calculus is generated for ℓ-groups and new decidability proofs are obtained for the equational theory of this variety and extending finite subsets of free groups to right orders; and a calculus for representable ℓ-groups is generated and a new proof is obtained that free groups are orderable. (Acknowledgements: The second author is supported by Swiss National Science Foundation grant 200021_146748 and the EU Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 689176.) Constructive canonicity for lattice-based fixed point logics Willem Conradie, Andrew Craig, Alessandra Palmigiano and Zhiguang Zhao Department of Pure and Applied Mathematics, University of Johannesburg, South Africa and Delft University of Technology, Delft, the Netherlands. In the present paper, we prove canonicity results for lattice-based fixed-point logics in a constructive meta-theory. Specifically, we prove two types of canonicity results, depending on how the fixed-point binders are interpreted. These results smoothly unify the constructive canonicity results for inductive inequalities, proved in a general lattice setting, with the canonicity results for fixed-point logics on a bi-intuitionistic base, proven in a non-constructive setting. (Acknowledgements: The research of the first author has been funded by the National Research Foundation of South Africa, Grant number 81309. The research of the third and fourth author has been funded by the NWO Vidi grant 016.138.314, the NWO Aspasia grant 015.008.054 and a Delft Technology Fellowship awarded in 2013.) Non-commutative Logic for Compositional Distributional Semantics Karin Cvetko-Vah, Mehrnoosh Sadrzadeh, Dimitri Kartsaklis and Benjamin Blundell Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia. School of Electronic Engineering and Computer Science, Queen Mary, University of London, London, UK. ITS Research, Queen Mary University of London, London, UK. Distributional models of natural language use vectors to provide a contextual foundation for meaning representation. These models rely on large quantities of real data, such as corpora of documents, and have found applications in natural language tasks, such as word similarity, disambiguation, indexing, and search. Compositional distributional models extend the distributional ones from words to phrases and sentences. Logical operators are usually treated as noise by these models and no systematic treatment is provided so far. In this paper, we show how skew lattices and their encoding in upper triangular matrices provide a logical foundation for compositional distributional models. In this setting, one can model commutative as well as non-commutative logical operations of conjunction and disjunction. We provide theoretical foundations, a case study, and experimental results for an entailment task on real data. On Fragments of Higher Order Logics that on Finite Structures Collapse to Second Order Flavio Ferrarotti, Senén González and José Turull-Torres Software Competence Center Hagenberg, Austria and Depto. de Ingeniería e Investigaciones Tecnológicas, Universidad Nacional de La Matanza, Argentina, and Massey University, New Zealand. E-mail: {Flavio.Ferrarotti, Senen.Gonzalez}@scch.at jmturull1952@gmail.com We define new fragments of higher-order logics of order three and above, and investigate their expressive power over finite models. The key unifying property of these fragments is that they all admit inexpensive algorithmic translations of their formulae to equivalent second-order logic formulae. That is, within these fragments we can make use of third- and higher-order quantification without paying the extremely high complexity price associated with them. Although theoretical in nature, the results reported here are more significant from a practical perspective. It turns out that there are many examples of properties of finite models (queries from the perspective of relational databases) that can be simply and elegantly defined by formulae of the higher-order fragments studied in this work. For many of those properties, the equivalent second-order formulae can be very complicated and unintuitive. In particular when they concern properties of complex objects, such as hypergraphs, and the equivalent second-order expressions require the encoding of those objects into plain relations. (Acknowledgements: Work supported by Austrian Science Fund (FWF):[I2420-N31]. Project: Higher-Order Logics and Structures. Initiated during a project sponsored visit of Prof. José María Turull-Torres. The research reported in this paper has been partly supported by the Austrian Ministry for Transport, Innovation and Technology, the Federal Ministry of Science, Research and Economy, and the Province of Upper Austria in the frame of the COMET center SCCH.) Computable Quotient Presentations of Models of Arithmetic and Set Theory Michał Tomasz Godziszewski and Joel David Hamkins Logic Department, Institute of Philosophy, University of Warsaw, Warszawa, Poland. Mathematics, Philosophy, Computer Science, The Graduate Center of The City University of New York,New York, USA. Mathematics, College of Staten Island of CUNY, Staten Island, USA. We prove various extensions of the Tennenbaum phenomenon to the case of computable quotient presentations of models of arithmetic and set theory. Specifically, no non-standard model of arithmetic has a computable quotient presentation by a c.e. equivalence relation. No $$\varSigma _1$$-sound non-standard model of arithmetic has a computable quotient presentation by a co-c.e. equivalence relation. No non-standard model of arithmetic in the language $$\{+,\cdot ,\leq \}$$ has a computably enumerable quotient presentation by any equivalence relation of any complexity. No model of ZFC or even much weaker set theories has a computable quotient presentation by any equivalence relation of any complexity. And similarly no non-standard model of finite set theory has a computable quotient presentation. (Acknowledgements: This article is a preliminary report of results following up research initiated at the conference Mathematical Logic and its Applications, held in memory of Professor Yuzuru Kakuda of Kobe University in September 2016 at the Research Institute for Mathematical Sciences (RIMS) in Kyoto. The second author is grateful for the chance twenty years ago to be a part of Kakuda-sensei’s logic group in Kobe, a deeply formative experience that he is pleased to see growing into a lifelong connection with Japan. He is grateful to the organizer Makoto Kikuchi and his other Japanese hosts for supporting this particular research visit, as well as to Bakhadyr Khoussainov for insightful conversations. The first author has been supported by the National Science Centre (Poland) research grant NCN PRELUDIUM UMO-2014/13/N/HS1/02058. He also thanks the Mathematics Program of the CUNY Graduate Center in New York for his research visit as a Fulbright Visiting Scholar between September 2016 and April 2017. Commentary concerning this paper can be made at http://jdh.hamkins.org/computable-quotient-presentations.) Lattice Logic Properly Displayed Giuseppe Greco and Alessandra Palmigiano Delft University of Technology, Delft, the Netherlands. University of Johannesburg, Johannesburg, South Africa. We introduce a proper display calculus for (non-distributive) Lattice Logic which is sound, complete, conservative, and enjoys cut-elimination and subformula property. Properness (i.e. closure under uniform substitution of all parametric parts in rules) is the main interest and added value of the present proposal, and allows for the smoothest Belnap-style proof of cut-elimination, and for the most comprehensive account of axiomatic extensions and expansions of Lattice Logic in a single overarching framework. Our proposal builds on an algebraic and order-theoretic analysis of the semantic environment of lattice logic, and applies the guidelines of the multi-type methodology in the design of display calculi. (Acknowledgements: This research has been funded by the NWO Vidi grant 016.138.314, the NWO Aspasia grant 015.008.054 and a Delft Technology Fellowship awarded to the second author in 2013.) Multi-type Display Calculus for Semi De Morgan Logic Giuseppe Greco, Fei Liang, M. Andrew Moshier and Alessandra Palmigiano Delft University of Technology, Delft, the Netherlands. Chapman University, Orange, USA. Institute of Logic and Cognition Sun Yat-sen University, Guangzhou, China. University of Johannesburg, Johannesburg, South Africa. E-mail: liangf25@mail2.sysu.edu.cn We introduce a proper multi-type display calculus for semi De Morgan logic which is sound, complete, conservative, and enjoys cut-elimination and subformula property. Our proposal builds on an algebraic analysis of semi De Morgan algebras and applies the guidelines of the multi-type methodology in the design of display calculi. (Acknowledgements: This research is supported by the NWO Vidi grant 016.138.314, the NWO Aspasia grant 015.008.054 and a Delft Technology Fellowship awarded to the second author in 2013.) Shift Registers Fool Finite Automata Bjorn Kjos-Hanssen University of Hawai‘i at Manoa Honolulu, USA. E-mail: bjoernkh@hawaii.edu Let x be an m-sequence, a maximal length sequence produced by a linear feedback shift register. We show that x has maximal subword complexity function in the sense of Allouche and Shallit. We show that this implies that the nondeterministic automatic complexity $$A_N(x)$$ is close to maximal: $$n/2-A_N(x)=O(\log ^2 n)$$, where n is the length of x. In contrast, Hyde has shown $$A_N(y)\leq n/2+1$$ for all sequences y of length n. The Lambek Calculus with Iteration: Two Variants Stepan Kuznetsov University of Hawai‘i at Manoa Honolulu, USA. E-mail: bjoernkh@hawaii.edu Formulae of the Lambek calculus are constructed using three binary connectives, multiplication and two divisions. We extend it using a unary connective, positive Kleene iteration. For this new operation, following its natural interpretation, we present two lines of calculi. The first one is a fragment of infinitary action logic and includes an omega-rule for introducing iteration to the antecedent. We also consider a version with infinite (but finitely branching) derivations and prove equivalence of these two versions. In Kleene algebras, this line of calculi corresponds to the *-continuous case. For the second line, we restrict our infinite derivations to cyclic (regular) ones. We show that this system is equivalent to a variant of action logic that corresponds to general residuated Kleene algebras, not necessarily *-continuous. Finally, we show that, in contrast with the case without division operations (considered by Kozen), the first system is strictly stronger than the second one. To prove this, we use a complexity argument. Namely, we show, using methods of Buszkowski and Palka, that the first system is $$\varPi ^0_n$$-hard, and therefore is not recursively enumerable and cannot be described by a calculus with finite derivations. (Acknowledgements: This work is supported by the Russian Science Foundation under grant 16-11-10252.) Dependent Event Types Zhaohui Luo and Sergei Soloviev Steklov Mathematical Institute of RAS, Moscow, Russia. E-mail: sk@mi.ras.ru This paper studies how dependent types can be employed for a refined treatment of event types, offering a nice improvement to Davidson’s event semantics. We consider dependent event types indexed by thematic roles and illustrate how, in the presence of refined event types, subtyping plays an essential role in semantic interpretations. We consider two extensions with dependent event types: first, the extension of Church’s simple type theory as employed in Montague semantics that is familiar with many linguistic semanticists and, secondly, the extension of a modern type theory as employed in MTT-semantics. The former uses subsumptive subtyping, while the latter uses coercive subtyping, to capture the subtyping relationships between dependent event types. Both of these extensions have nice meta-theoretic properties such as normalization and logical consistency; in particular, we shall show that the former can be faithfully embedded into the latter and hence has expected meta-theoretic properties. As an example of applications, it is shown that dependent event types give a natural solution to the incompatibility problem (sometimes called the event quantification problem) in combining event semantics with the traditional compositional semantics, both in the Montagovian setting with the simple type theory and in the setting of MTT-semantics. (Acknowledgements: The first author is partially supported by EU COST Action CA15123 and CAS/SAFEA International Partnership Program. The second author is an associated researcher at ITMO University, St. Petersburg, Russia. Partially supported by EU COST Action CA15123 and Russian Federation Grant 074-U01.) A Geometry of Interaction Machine for Gödel’s System T Ian Mackie Department of Informatics, University of Sussex, Brighton, UK. Gödel’s System T is the simply typed lambda calculus extended with numbers and an iterator. The higher-order nature of the language gives it enormous expressive power—the language can represent all the primitive recursive functions and beyond, for instance Ackermann’s function. In this paper we use System T as a minimalistic functional language. We give an interpretation using a data flow model that incorporates ideas from the geometry of interaction and game semantics. The contribution is a reversible model of higher-order computation which can also serve as a novel compilation technique. Disjoint Fibring of Non-deterministic Matrices Sérgio Marcelino and Carlos CaleiroSQIG - Instituto de Telecomunicações, Departamento de Matemática - Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal. E-mail: smarcel@math.tecnico.ulisboa.pt In this paper we give a first definitive step towards endowing the general mechanism for combining logics known as fibring with a meaningful and useful semantics given by non-deterministic logical matrices (Nmatrices). We present and study the properties of two semantical operations: a unary operation of $$\omega $$-power of a given Nmatrix, and a binary operation of strict product of Nmatrices with disjoint similarity types (signatures). We show that, together, these operations can be used to characterize the disjoint fibring of propositional logics, when each of these logics is presented by a single Nmatrix. As an outcome, we also provide a decidability and complexity result about the resulting fibred logic. We illustrate the constructions with a few meaningful examples. (Acknowledgements: Work done under the scope of Project UID/EEA/50008/2013 of Instituto de Telecomunicações, financed by the applicable framework (FCT/MEC through national funds and co-funded by FEDER-PT2020). The first author also acknowledges the FCT postdoctoral grant SFRH/BPD/76513/2011. This research is part of the MoSH initiative of SQIG at Instituto de Telecomunicações.) Concrete Mathematics. Finitistic Approach to Foundations Marcin Mostowski and Marek Czarnecki Institute of Philosophy, Department of Logic, Jagiellonian University, Cracow, Poland. Institute of Philosophy, Department of Logic, Warsaw University, Warsaw, Poland. E-mail: marcin.mostowski@uj.edu.pl, m.czarnecki2@uw.edu.pl We discuss the idea of concrete mathematics inspired by Hilbert’s idea of finitistic mathematics as the part of mathematics not engaged into actual infinity. We explicate it as the part of mathematics based on $$\Delta ^0_2$$ arithmetical concepts. The explication is justified by equivalence of $$\Delta ^0_2$$ definability with algorithmic learnability (an epistemic argument) and with FM-representability (representability in finite models, an ontological argument). We show that the essential part of classical mathematics can be interpreted in the concrete framework. We claim that current mathematics is a social game of proving theorems on some axiomatic set theoretic background. On the other hand, concrete mathematics is the reality on which our mathematical experience is based. This is what makes the game intersubjective. Nevertheless, this game is one of the most efficient methods of building our mathematical knowledge. (Acknowledgements: This work was funded by the Polish National Science Centre grant number 2013/11/B/HS1/04168.) Solovay’s Completeness Without Fixed Points Fedor Pakhomov Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia. E-mail: pakhfn@mi.ras.ru In this paper we present a new proof of Solovay’s theorem on arithmetical completeness of Gödel–Löb provability logic GL. Originally, completeness of GL with respect to interpretation of $$\Box $$ as provability in Peano Arithmetic (PA) was proved by Solovay in 1976. The key part of Solovay’s proof was his construction of an arithmetical evaluation for a given modal formula that made the formula unprovable in PA if it were unprovable in GL. The arithmetical sentences for the evaluations were constructed using certain arithmetical fixed points. The method developed by Solovay have been used for establishing similar semantics for many other logics. In our proof we develop new more explicit construction of required evaluations that doesn’t use any fixed points in their definitions. To our knowledge, it is the first alternative proof of the theorem that is essentially different from Solovay’s proof in this key part. (Acknowledgements: This work is supported by the Russian Science Foundation under grant 14-50-00005.) An Epistemic Generalization of Rationalizability Rohit Parikh City University of New York, New York, USA. E-mail: rparikh@gc.cuny.edu Rationalizability, originally proposed by Bernheim and Pearce, generalizes the notion of Nash equilibrium. Nash equilibrium requires common knowledge of strategies. Rationalizability only requires common knowledge of rationality. However, their original notion assumes that the payoffs are common knowledge. That is agents do know what world they are in, but may be ignorant of what other agents are playing. We generalize the original notion of rationalizability to consider situations where agents do not know what world they are in, or where some know but others do not know. Agents who know something about the world can take advantage of their superior knowledge. It may also happen that both Ann and Bob know about the world but Ann does not know that Bob knows. How might they act? We will show how a notion of rationalizability in the context of partial knowledge, represented by a Kripke structure, can be developed. On Two Concepts of Ultrafilter Extensions of First-Order Models and Their Generalizations Nikolai L. Poliakov and Denis I. Saveliev Financial University, Moscow, Russia. and Institute for Information Transmission Problems of the Russian Academy of Sciences, Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia. E-mail: d.i.saveliev@gmail.com There exist two known concepts of ultrafilter extensions of first-order models, both in a certain sense canonical. One of them [1] comes from modal logic and universal algebra, and in fact goes back to [2]. Another one [3, 4] comes from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups [5] as its main precursor. By a classical fact, the space of ultrafilters over a discrete space is its largest compactification. The main result of [3, 4], which confirms a canonicity of this extension, generalizes this fact to discrete spaces endowed with a first-order structure. An analogous result for the former type of ultrafilter extensions was obtained in [6]. Here we offer a uniform approach to both types of extensions. It is based on the idea to extend the extension procedure itself. We propose a generalization of the standard concept of first-order models in which functional and relational symbols are interpreted rather by ultrafilters over sets of functions and relations than by functions and relations themselves. We provide two specific operations which turn generalized models into ordinary ones, and establish necessary and sufficient conditions under which the latter are the two canonical ultrafilter extensions of some models. (Acknowledgements: The second author was supported by Grant 16-01-00615 of the Russian Foundation for Basic Research.) Knowledge Is a Diamond Vít Punčochár Institute of Philosophy, Czech Academy of Sciences, Prague, Czech Republic. E-mail: vit.puncochar@centrum.cz In the standard epistemic logic, the knowledge operator is represented as a box operator, a universal quantifier over a set of possible worlds. There is an alternative approach to the semantics of knowledge, according to which an agent a knows $$\alpha $$ iff a has a reliable (e.g. sensory) evidence that supports $$\alpha $$. In this interpretation, knowledge is viewed rather as an existential, i.e. a diamond modality. In this paper, we will propose a formal semantics for substructural logics that allows to model knowledge on the basis of this intuition. The framework is strongly motivated by a similar semantics introduced in [3]. However, as we will argue, our framework overcomes some unintuitive features of the semantics from [3]. Most importantly, knowledge does not distribute over disjunction in our logic. (Acknowledgements: The work on this paper was supported by grant no. 16-07954J of the Czech Science Foundation.) Cut-Elimination for the Modal Grzegorczyk Logic via Non-well-founded Proofs Yury Savateev and Daniyar Shamkanov National Research University Higher School of Economics, Moskva, Russia and Steklov Mathematical Institute of the Russian Academy of Sciences, Moskva. Russia. E-mail: yury.savateev@gmail.com We present a sequent calculus for the modal Grzegorczyk logic Grz allowing non-well-founded proofs and obtain the cut-elimination theorem for it by constructing a continuous cut-elimination mapping acting on these proofs. (Acknowledgements: The article was prepared within the framework of the Basic Research Program at the National Research University Higher School of Economics (HSE) and supported within the framework of a subsidy by the Russian Academic Excellence Project ‘5-100’. Both authors also acknowledge support from the Russian Foundation for Basic Research (grant no. 15-01-09218a).) Global Neighbourhood Completeness of the Gödel–Löb Provability Logic Daniyar Shamkanov Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia. E-mail: daniyar.shamkanov@gmail.com The Gdel-Lb provability logic GL is strongly neighbourhood complete in the case of the so-called local semantic consequence relation. In the given paper, we consider Hilbert-style non-well-founded derivations in GL and establish that GL with the obtained derivability relation is strongly neighbourhood complete in the case of the global semantic consequence relation. (Acknowledgements: This work was supported by the Russian Science Foundation (grant no. 14-50-00005).) Substructural Logics with a Reflexive Transitive Closure Modality Igor Sedlár Institute of Philosophy, Czech Academy of Sciences, Prague, Czech Republic. E-mail: sedlar@flu.cas.cz Reflexive transitive closure modalities represent a number of important notions, such as common knowledge in a group of agents or non-deterministic iteration of actions. Normal modal logics with such modalities are well explored but weaker logics are not. We add a reflexive transitive closure box modality to the modal non-associative commutative full Lambek calculus with a simple negation. Decidability and weak completeness of the resulting system are established and extensions of the results to stronger substructural logics are discussed. As a special case, we obtain decidability and weak completeness for intuitionistic modal logic with the reflexive transitive closure box. (Acknowledgements: This work has been supported by the joint project of the German Science Foundation (DFG) and the Czech Science Foundation (GA CR) number 16-07954J (SEGA: From shared evidence to group attitudes). The author would like to thank the anonymous reviewers for a number of suggestions, and to Adam Přenosil for reading a draft of the paper. A preliminary version of the paper was presented at the 8th International Workshop on Logic and Cognition in Guangzhou, China; the author is indebted to the audience for valuable feedback.) Coherent Diagrammatic Reasoning in Compositional Distributional Semantics Gijs Jasper Wijnholds Queen Mary University of London, London, UK. E-mail: g.j.wijnholds@qmul.ac.uk The framework of Categorical Compositional Distributional models of meaning [3], inspired by category theory, allows one to compute the meaning of natural language phrases, given basic meaning entities assigned to words. Composing word meanings is the result of a functorial passage from syntax to semantics. To keep one from drowning in technical details, diagrammatic reasoning is used to represent the information flow of sentences that exists independently of the concrete instantiation of the model. Not only does this serve the purpose of clarification, it moreover offers computational benefits as complex diagrams can be transformed into simpler ones, which under coherence can simplify computation on the semantic side. Until now, diagrams for compact closed categories and monoidal closed categories have been used (see [2, 3]). These correspond to the use of pregroup grammar [12] and the Lambek calculus [9] for syntactic structure, respectively. Unfortunately, the diagrammatic language of Baez and Stay [1] has not been proven coherent. In this paper, we develop a graphical language for the (categorical formulation of) the non-associative Lambek calculus [10]. This has the benefit of modularity where extension of the system are easily incorporated in the graphical language. Moreover, we show the language is coherent with monoidal closed categories without associativity, in the style of Selinger’s survey paper [17]. Algorithmic Sahlqvist Preservation for Modal Compact Hausdorff Spaces Zhiguang Zhao Delft University of Technology, Delft, the Netherlands. E-mail: zhaozhiguang23@gmail.com In this paper, we use the algorithm ALBA to reformulate the proof in [1, 2] that over modal compact Hausdorff spaces, the validity of Sahlqvist sequents are preserved from open assignments to arbitrary assignments. In particular, we prove an adapted version of the topological Ackermann lemma based on the Esakia-type lemmas proved in [1, 2]. Received 28 March 2018 © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Logic Journal of the IGPL Oxford University Press

24th Workshop on Logic, Language, Information and Computation (WoLLIC 2017)

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Abstract

WoLLIC 2017 was held in London, UK, 18–21 July 2017, in the Department of Computer Science, University College London (UCL). WoLLIC (http://wollic.org) is a series of workshops, which started in 1994 with the aim of fostering interdisciplinary research in pure and applied logic. The idea is to have a forum that is large enough in the number of possible interactions between logic and the sciences related to information and computation, and yet is small enough to allow for concrete and useful interaction among participants. Previous versions were held at Recife (Pernambuco, Brazil) in 1994 and 1995; Salvador (Bahia, Brazil) in 1996; Fortaleza (Ceará, Brazil) in 1997; São Paulo (Brazil) in 1998; Itatiaia (Rio de Janeiro, Brazil) in 1999; Natal (Rio Grande do Norte, Brazil) in 2000; Brasília (Distrito Federal, Brazil) in 2001; Rio de Janeiro (Brazil) in 2002; Ouro Preto (Minas Gerais, Brazil) in 2003; Fontainebleau (France) in 2004; Florianópolis (Santa Catarina, Brazil) in 2005; Stanford (California, USA) in 2006; Rio de Janeiro in 2007; Edinburgh in 2008; Tokyo in 2009; Brasília in 2010; Philadelphia in 2011; Buenos Aires in 2012; Darmstadt in 2013; Valparaiso in 2014; Bloomington (IN) in 2015; Puebla in 2016. It is planned that the meeting will take place in the following years and locations: 2018 in Bogotá, 2019 in Utrecht, 2020 in Arequipa, 2021 in Chennai (tbc). WoLLIC’s scientific sponsors include Association for Symbolic Logic (ASL), Interest Group in Pure and Applied Logics (IGPL), Association for Logic, Language and Information (FoLLI), European Association for Theoretical Computer Science (EATCS), European Association for Computer Science Logic (EACSL), Sociedade Brasileira de Computação (SBC) and Sociedade Brasileira de Lógica (SBL). WoLLIC 2017 once again carried on the tradition of promoting interdisciplinary and cross-disciplinary research, topics ranging from mathematical proof theory to formal semantics of natural languages. The Program Committee consisted of M. Baaz (University of Technology, Vienna, Austria), J. Baldwin (University of Illinois at Chicago, USA), D. Bartozova (Universidade de São Paulo, Brazil), A. Ciabattoni (University of Technology, Vienna, Austria), W. Dean (University of Warwick, UK), E. Grädel (RWTH Aachen, Germany), V. Halbach (University of Oxford, UK), J. Kennedy (Helsinki University, Finland (Chair)), D. Kozen (Cornell University, USA), J. Makowsky (Technion - Israel Institute of Technology, Israel), L. Moss (Indiana University, USA), A. Palmigiano (Delft University, the Netherlands), M. Sadrzadeh (Queen Mary, UK), S. Smets (Amsterdam University, the Netherlands), A. Tornquist (Kobenhavns Universitet, Denmark), R. Verbrugge (University of Groningen, the Netherlands), A. Villaveces (Universidad Nacional, Colombia) and Ph. Welch (University of Bristol, UK). The Organizing Committee members were J. Brotherston (UCL), P. Oliva (Queen Mary), A. G. de Oliveira (U Fed Pernambuco), R. de Queiroz (U Fed Pernambuco, co-chair), A. Silva (UCL, Local co-chair), M. Sadrzadeh (Queen Mary, Local co-chair). As in previous instances, WoLLIC 2017 included invited talks as well as contributed papers. The Program Committee received 61 submissions. A thorough review process by the programme committee, assisted by a number of external reviewers, has led to the acceptance of 23 papers for presentation at the meeting and inclusion in the Proceedings, which was published as volume 10388 of Springer’s Lecture Notes in Computer Science (FoLLI subseries). The conference programme also included two tutorial lectures and 8 invited talks by 8 prominent invited speakers, who have graciously accepted the programme committee’s invitation: Hazel Brickhill (Bristol) (University of Bristol), Michael Detlefsen (University of Notre Dame), Alexander Kurz (University of Leicester), Frederike Moltmann (New York University), David Pym (University College London), Nicole Schweikardt (Humboldt Universität), Fan Yang (Delft University), Boris Zilber (University of Oxford). There were also four tutorials given by Michael Detlefsen, Alexander Kurz, Frederike Moltmann and Nicole Schweikardt. A special issue of Archive for Mathematical Logic (Springer), guest edited by Juliette Kennedy and Ruy de Queiroz, will include peer-reviewed full versions of a selection of the conference presentations. Special Session: Screening of Films about Mathematicians. On the evening of Thursday, July 20th, as a tribute to a recent project which focuses on the cross-cultural connections that are made through mathematics and the impact that Navajo Math Circles can have on a community, there was a screening of George Csicsery’s Navajo Math Circles (2016), a one-hour film, documenting the process of a two-year period in which hundreds of Navajo children in recent years have found themselves at the centre of a lively collaboration with mathematicians from around the world. Juliette Kennedy (Program Chair) Ruy de Queiroz (Org Comm Co-Chair) Alexandra Silva (Local Co-Chair) Mehrnoosh Sadrzadeh (Local Co-Chair). Tutorials Coalgebraic Logic Alexander Kurz Department of Informatics, University of Leicester, University Road, Leicester, LE1 7RH, UK. E-mail: ak155@le.ac.uk In the tutorial we have the time to look at some of the category theory behind the theory sketched in the lecture. Natural Language Metaphysics Friederike Moltmann Research Director (DR1), Centre Nationale de la Recherche Scientifique (CNRS), France. E-mail: fmoltmann@univ-paris1.fr This tutorial gives an introduction to the branch of metaphysics that can be called ‘natural language metaphysics’. It discusses various sorts of appeals philosophers have made to natural language, of ways in analyzes in linguistic semantics involve metaphysical notions, of the ways natural language reflects ontological notions and structures, and of cases of discrepancies between the ontology implicit in natural language and the reflective ontology of philosophers or non-philosophers. Recommended reading: ‘Natural Language Ontology’. Oxford Research Encyclopedia of Linguistics. Oxford UP, New York, 2017 (online). A Tutorial on Database Theory Nicole Schweikardt Humboldt-Universität zu Berlin, Institut für Informatik, Unter den Linden 6, D-10099 Berlin, Germany. E-mail: schweikn@informatik.hu-berlin.de In this tutorial I want to give a brief introduction to database theory, including basic definitions, typical research questions and some fundamental results. Invited Talks A New Way to Measure Subsets of Ordinals: Generalised Closed Unbounded and Stationary Sets Hazel Brickhill University of Helsinki, P.O. Box 33 (Yliopistonkatu 4), 00014, Finland. E-mail: hazel.brickhill@helsinki.fi The notions of closed unbounded and stationary set are central to Set Theory, and provide a way to measure how ‘large’ or ‘thick’ a subset of an ordinal is. In this talk I will introduce a generalization of these notions, based on the generalization of stationarity defined in [1]. Surprisingly for a new concept is set theory, generalized closed unbounded and stationary sets are very simple to define and accessible. They provide a finer calibration measuring the sizes of sets between ‘stationary’ and ‘containing a closed unbounded set’, and are closely related to the phenomena of stationary reflection and indescribability. I will give combinatorial definitions for these notions and also describe how they can be characterized in terms of derived topologies [2]. These notions are being used to answer questions about provability logic (see [2]), and promise a range of further applications. References [1] J. Bagaria , M. Magidor and H. Sakai . Reflection and indescribability in the constructible universe . Israel Journal of Mathematics , 208 , 1 – 11 , 2015 . Google Scholar CrossRef Search ADS [2] J. Bagaria . Derived topologies on ordinals and stationary reflection . https://www.newton.ac.uk/files/preprints/ni16031.pdf, 2016 . Formalism: Historical and Conceptual Background Michael Detlefsen Department of Philosophy, University of Notre Dame, 100 Malloy Hall Notre Dame, IN 46556, Notre Dame, IN 46556 USA. E-mail: mdetlef1@nd.edu The aim will of this talk is to survey and clarify the historical sources of and motives for formalism. Central here will be the clarification of certain distinctive views concerning non-semantical uses of language in reasoning, and of the conditions of its usefulness and reliability. Formalism: Hilbert’s Proposals Michael Detlefsen Department of Philosophy, University of Notre Dame, 100 Malloy Hall Notre Dame, IN 46556, Notre Dame, IN 46556 USA. E-mail: mdetlef1@nd.edu In a late statement of his formalist programme, Hilbert described a proper approach to consistency problems as being based on observation. The aim will be to clarify this much overlooked idea and to briefly explore its significance. Coalgebraic Logic Alexander Kurz Department of Informatics, University of Leicester, University Road, Leicester, LE1 7RH, UK. E-mail: ak155@le.ac.uk We will start with examples and remarks on induction and coinduction and then argue that the appropriate general framework to elucidate them is given by algebras and coalgebras for a functor on a category. Moreover, coalgebras allow us to develop a general theory of dynamic systems. To better understand the generality offered by the notion of functor on a category, we will look at presentations of functors by operations and equations. As an application, we then show how to set up a general theory of logics for coalgebras (=dynamic systems), parametric in a functor. Natural Language Metaphysics Friederike Moltmann Research Director (DR1), Centre Nationale de la Recherche Scientifique (CNRS), France. E-mail: fmoltmann@univ-paris1.fr Metaphysics in the past was considered mainly a pursuit of philosophers, asking questions about being in most general terms. While some philosophers made appeal to natural language in order to argue for a ontological category or metaphysical notion, others have rejected such an appeal arguing that the ontology reflected in language diverges significantly from what there really is, from any philosophically accepted ontology. Whatever one’s view may be of what a philosopher should pursue, it has become clear, especially with the development of natural language semantics (and syntax), that the ontology reflected in natural language is an important object of study in itself, as the subject matter of natural language ontology or more generally natural language metaphysics. This talk discusses a range of issues that arise for natural language metaphysics, such as the sorts of data that it should and that it should not take into account, the ways natural language reflects ontological notions and structures, why there are discrepancies between the ontology implicit in natural language and the reflective ontology of philosophers or non-philosophers and how the relation of natural language metaphysics can be conceived with respect to other projects in metaphysics. Resource Semantics: Logic as a Modelling Technology David Pym University College London, Department of Computer Science, Gower Street, London WC1E 6BT, UK. E-mail: D.Pym@cs.ucl.ac.uk The development of Bunched Implication (BI), the logic of bunched implications, together with its resource semantics, led to the formulation of Separation Logic. However, this rather succesful story sits within a broader, quite systematic logical context. I will review the (family of) logics that are supported by resource semantics, explaining their more-or-less uniform meta-theoretic basis and illustrating their uses in a range of modelling applications. Database Query Answering under Updates Nicole Schweikardt Humboldt-Universität zu Berlin, Institut für Informatik, Unter den Linden 6, D-10099 Berlin, Germany. E-mail: schweikn@informatik.hu-berlin.de Query evaluation is one of the most fundamental tasks in databases, and a vast amount of literature is devoted to the complexity of this problem. This talk will focus on query evaluation in the ‘dynamic setting’, where the database may be updated by inserting or deleting tuples. In this setting, an evaluation algorithm receives a query Q and an initial database D and starts with a preprocessing phase that computes a suitable data structure to represent the result of evaluating Q on D. After every database update, the data structure is updated so that it represents the result of evaluating Q on the updated database. The data structure shall be designed in such a way that it quickly provides the query result, preferably in constant time (i.e. independent of the database size). We focus on the following flavours of query evaluation: (1) Testing: decide whether a given tuple t is contained in Q(D). (2) Counting: compute the number of tuples that belong to Q(D). (3) Enumeration: enumerate Q(D) with a bounded delay between the output tuples. Here, as usual, Q(D) denotes the k-ary relation obtained by evaluating a k-ary query Q on a relational database D. For Boolean queries, all three tasks boil down to (4) Answering: decide if Q(D) is non-empty. Compared to the dynamic descriptive complexity framework introduced by Patnaik and Immerman (1997), which focuses on the expressive power of first-order logic on dynamic databases and has led to a rich body of literature, we are interested in the computational complexity of query evaluation. We say that a query evaluation algorithm is efficient if the update time is either constant or at most polylogarithmic in the size of the database. In this talk I want to give an overview of recent results in this area. Characterizing dependencies in logic and sciences Fan Yang Department of Mathematics and Statistics, University of Helsinki, Finland. E-mail: fan.yang.c@gmail.com In this talk, we discuss theory and applications of dependence and independence logic (DIL), which was introduced by Väänänen [5] and by Grädel and Väänänen [1] with the specific aim of expressing and reasoning about dependence and independence relations. This framework extends first-order logic with new atomic formulas, called dependence and independence atoms, that explicitly specify the dependence relations. In contrast to the usual Tarskian semantics, where formulas are evaluated under single assignments, formulas of DIL are evaluated under sets of assignments (called teams) instead. Such a semantics is call team semantics, introduced by Hodges [2, 3]. The first part of the talk consists of a very brief tutorial on DIL. We will cover the team semantics, expressive power and axiomatization problem of DIL and its notable variants. The second part of the talk discusses (potential) applications of DIL in social and natural sciences, such as database theory, social choice theory, quantum theory and economics. In particular, we will formulate Arrow’s Impossibility Theorem in social choice theory as a type of dependence strengthening theorem in the framework of DIL, and thereby argue that DIL offers an interesting new perspective on social choice theory. This part of the talk is based on a joint work with Eric Pacuit [4]. References [1] E. Grädel and J. Väänänen . Dependence and independence . Studia Logica , 101 , 399 – 410 , 2013 . Google Scholar CrossRef Search ADS [2] W. Hodges . Compositional semantics for a language of imperfect information . Logic Journal of the IGPL , 5 , 539 – 563 , 1997 . Google Scholar CrossRef Search ADS [3] W. Hodges . Some strange quantifiers . In Structures in Logic and Computer Science: A Selection of Essays in Honor of A. Ehrenfeucht , J. Mycielski , G. Rozenberg and A. Salomaa , eds., pp. 51 – 65 . Vol. 1261 of Lecture Notes in Computer Science . London : Springer , 1997 . Google Scholar CrossRef Search ADS [4] E. Pacuit and F. Yang . Dependence and independence in social choice: Arrow’s theorem . In Dependence Logic: Theory and Application , H. V. S. Abramsky , J. Kontinen and J. Väänänen , eds., pp. 235 – 260 . Progress in Computer Science and Applied Logic , Birkhauser , 2016 . Google Scholar CrossRef Search ADS [5] J. Väänänen . Dependence Logic: A New Approach to Independence Friendly Logic . Cambridge : Cambridge University Press , 2007 . Google Scholar CrossRef Search ADS Positive model theory and approximation by finite structures Boris Zilber Mathematical Institute, University of Oxford, and Merton College, 24-29 St Giles, Oxford, OX1 3LB, UK. E-mail: zilber@maths.ox.ac.uk I will introduce a notion of structural approximation and will discuss non-trivial examples of approximations by finite structures of real manifolds and a model of quantum mechanics. Contributed Talks Graph Turing Machines Nathanael L. Ackerman and Cameron E. Freer Department of Mathematics, Harvard University, Cambridge, USA. E-mail: cameron@remine.com We consider graph Turing machines, a model of parallel computation on a graph, which provides a natural generalization of several standard computational models, including ordinary Turing machines and cellular automata. In this extended abstract, we give bounds on the computational strength of functions that graph Turing machines can compute. We also begin the study of the relationship between the computational power of a graph Turing machine and structural properties of its underlying graph. (Acknowledgements: The authors would like to thank Tomislav Petrović, Linda Brown Westrick and the anonymous referees of earlier versions for the helpful comments.) Independence-Friendly Logic without Henkin Quantification Fausto Barbero, Lauri Hella and Raine Rönnholm University of Helsinki, Philosophy, Faculty of Arts, Finland. E-mail: fausto.barbero@helsinki.fi and University of Tampere, Mathematics, Faculty of Natural Sciences, Finland. E-mail: lauri.hella,raine.ronnholm@uta.fi We analyze from a global point of view the expressive resources of IF logic that do not stem from Henkin (partially ordered) quantification. When one restricts attention to regular IF sentences, this amounts to the study of the fragment of IF logic which is individuated by the game-theoretical property of Action Recall. We prove that the fragment of Action Recall can express all existential second-order (ESO) properties. This can be accomplished already by the prenex fragment of Action Recall, whose only second-order source of expressiveness are the so-called signalling patterns. The proof shows that a complete set of Henkin prefixes is explicitly definable in the fragment of Action Recall. In the more general case, in which also irregular IF sentences are allowed, we show that full ESO expressive power can be achieved using neither Henkin nor signalling patterns. Total Search Problems in Bounded Arithmetic and Improved Witnessing Arnold Beckmann and Jean-José Razafindrakotoe Department of Computer Science, College of Science, Swansea University, UK. E-mail: a.beckmann@swansea.ac.uk, jjrazaf@icloud.com We define a new class of total search problems as a subclass of Megiddo and Papadimitriou’s class of total $$\textsf{NP}$$ search problems, in which solutions are verifiable in $$\textsf{AC}^0$$. We denote this class $$\forall \,\exists \textsf{AC}^0$$. We show that all total $$\textsf{NP}$$ search problems are equivalent, w.r.t. $$\textsf{AC}^0$$-many-one reductions, to search problems in $$\forall \,\exists \textsf{AC}^0$$. Furthermore, we show that $$\forall \,\exists \textsf{AC}^0$$ contains well-known problems such as the Stable Marriage and the Maximal Independent Set problems. We introduce the class of Inflationary Iteration problems in $$\forall \,\exists \textsf{AC}^0$$, and show that it characterizes the provably total $$\textsf{NP}$$ search problems of the bounded arithmetic theory corresponding to polynomial time. Cook and Nguyen introduced a generic way of defining a bounded arithmetic theory $$\textsf{VC}$$ for complexity classes $$\textsf{C}$$ which can be obtained using a complete problem. For such C we will define a new class $$\textsf{KPT[C]}$$ of $$\forall \,\exists \textsf{AC}^0$$ search problems based on Student–Teacher games in which the student has computing power limited to $$\textsf{AC}^0$$. We prove that $$\textsf{KPT[C]}$$ characterizes the provably total $$\textsf{NP}$$ search problems of the bounded arithmetic theory corresponding to $$\textsf{C}$$. All our characterizations are obtained via ‘new-style’ witnessing theorems, where reductions are provable in a theory corresponding to $$\textsf{AC}^0$$. On the Reflection Calculus with Partial Conservativity Operators Lev D. Beklemishev Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia. E-mail: bekl@mi.ras.ru Strictly positive logics recently attracted attention both in the description logic and in the provability logic communities for their combination of efficiency and sufficient expressivity. The language of Reflection Calculus (RC) consists of implications between formulas built up from propositional variables and the constant ‘true’ using only conjunction and the diamond modalities which are interpreted in Peano arithmetic as restricted uniform reflection principles. We extend the language of RC by another series of modalities representing the operators associating with a given arithmetical theory T its fragment axiomatized by all theorems of T of arithmetical complexity $$\varPi ^0_n$$, for all n > 0. We note that such operators, in a precise sense, cannot be represented in the full language of modal logic. We formulate a formal system extending RC that is sound and, as we conjecture, complete under this interpretation. We show that in this system one is able to express iterations of reflection principles up to any ordinal $$<\varepsilon _0$$. On the other hand, we provide normal forms for its variable-free fragment. Thereby, the variable-free fragment is shown to be algorithmically decidable and complete w.r.t. its natural arithmetical semantics. (Acknowledgements: This work is supported by the Russian Science Foundation under grant 16-11-10252.) On the Length of Medial-Switch-Mix Derivations Paola Bruscoli and Lutz Straßburger University of Bath, Bath, UK. Inria Palaiseau, France. E-mail: P.Bruscoli@Bath.ac.uk Switch and medial are two inference rules that play a central role in many deep inference proof systems. In specific proof systems, the mix rule may also be present. In this paper we show that the maximal length of a derivation using only the inference rules for switch, medial and mix, modulo associativity and commutativity of the two binary connectives involved, is quadratic in the size of the formula at the conclusion of the derivation. This shows, at the same time, the termination of the rewrite system. Generalized Relations in Linguistics and Cognition Bob Coecke, Fabrizio Genovese, Martha Lewis and Dan Marsden Mathematical Institute, University of Bern, Switzerland. E-mail: daniel.marsden@cs.ox.ac.uk Categorical compositional models of natural language exploit grammatical structure to calculate the meaning of sentences from the meanings of individual words. This approach outperforms conventional techniques for some standard NLP tasks. More recently, similar compositional techniques have been applied to conceptual space models of cognition. Compact closed categories, particularly the category of finite-dimensional vector spaces, have been the most common setting for categorical compositional models. When addressing a new problem domain, such as conceptual space models of meaning, a key problem is finding a compact closed category that captures the features of interest. We propose categories of generalized relations as source of new, practical models for cognition and NLP. We demonstrate using detailed examples that phenomena such as fuzziness, metrics, convexity, semantic ambiguity and meaning that varies with context can all be described by relational models. Crucially, by exploiting a technical framework described in previous work of the authors, we also show how we can combine multiple features into a single model, providing a flexible family of new categories for categorical compositional modelling. Proof Theory and Ordered Groups Almudena Colacito and George Metcalfe Department of Computer Science, University of Oxford, Oxford, UK. E-mail: {almudena.colacito, george.metcalfe}@math.unibe.ch Ordering theorems, characterizing when partial orders of a group extend to total orders, are used to generate hypersequent calculi for varieties of lattice-ordered groups (ℓ-groups). These calculi are then used to provide new proofs of theorems arising in the theory of ordered groups. More precisely, an analytic calculus for abelian ℓ-groups is generated using an ordering theorem for abelian groups; a calculus is generated for ℓ-groups and new decidability proofs are obtained for the equational theory of this variety and extending finite subsets of free groups to right orders; and a calculus for representable ℓ-groups is generated and a new proof is obtained that free groups are orderable. (Acknowledgements: The second author is supported by Swiss National Science Foundation grant 200021_146748 and the EU Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 689176.) Constructive canonicity for lattice-based fixed point logics Willem Conradie, Andrew Craig, Alessandra Palmigiano and Zhiguang Zhao Department of Pure and Applied Mathematics, University of Johannesburg, South Africa and Delft University of Technology, Delft, the Netherlands. In the present paper, we prove canonicity results for lattice-based fixed-point logics in a constructive meta-theory. Specifically, we prove two types of canonicity results, depending on how the fixed-point binders are interpreted. These results smoothly unify the constructive canonicity results for inductive inequalities, proved in a general lattice setting, with the canonicity results for fixed-point logics on a bi-intuitionistic base, proven in a non-constructive setting. (Acknowledgements: The research of the first author has been funded by the National Research Foundation of South Africa, Grant number 81309. The research of the third and fourth author has been funded by the NWO Vidi grant 016.138.314, the NWO Aspasia grant 015.008.054 and a Delft Technology Fellowship awarded in 2013.) Non-commutative Logic for Compositional Distributional Semantics Karin Cvetko-Vah, Mehrnoosh Sadrzadeh, Dimitri Kartsaklis and Benjamin Blundell Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia. School of Electronic Engineering and Computer Science, Queen Mary, University of London, London, UK. ITS Research, Queen Mary University of London, London, UK. Distributional models of natural language use vectors to provide a contextual foundation for meaning representation. These models rely on large quantities of real data, such as corpora of documents, and have found applications in natural language tasks, such as word similarity, disambiguation, indexing, and search. Compositional distributional models extend the distributional ones from words to phrases and sentences. Logical operators are usually treated as noise by these models and no systematic treatment is provided so far. In this paper, we show how skew lattices and their encoding in upper triangular matrices provide a logical foundation for compositional distributional models. In this setting, one can model commutative as well as non-commutative logical operations of conjunction and disjunction. We provide theoretical foundations, a case study, and experimental results for an entailment task on real data. On Fragments of Higher Order Logics that on Finite Structures Collapse to Second Order Flavio Ferrarotti, Senén González and José Turull-Torres Software Competence Center Hagenberg, Austria and Depto. de Ingeniería e Investigaciones Tecnológicas, Universidad Nacional de La Matanza, Argentina, and Massey University, New Zealand. E-mail: {Flavio.Ferrarotti, Senen.Gonzalez}@scch.at jmturull1952@gmail.com We define new fragments of higher-order logics of order three and above, and investigate their expressive power over finite models. The key unifying property of these fragments is that they all admit inexpensive algorithmic translations of their formulae to equivalent second-order logic formulae. That is, within these fragments we can make use of third- and higher-order quantification without paying the extremely high complexity price associated with them. Although theoretical in nature, the results reported here are more significant from a practical perspective. It turns out that there are many examples of properties of finite models (queries from the perspective of relational databases) that can be simply and elegantly defined by formulae of the higher-order fragments studied in this work. For many of those properties, the equivalent second-order formulae can be very complicated and unintuitive. In particular when they concern properties of complex objects, such as hypergraphs, and the equivalent second-order expressions require the encoding of those objects into plain relations. (Acknowledgements: Work supported by Austrian Science Fund (FWF):[I2420-N31]. Project: Higher-Order Logics and Structures. Initiated during a project sponsored visit of Prof. José María Turull-Torres. The research reported in this paper has been partly supported by the Austrian Ministry for Transport, Innovation and Technology, the Federal Ministry of Science, Research and Economy, and the Province of Upper Austria in the frame of the COMET center SCCH.) Computable Quotient Presentations of Models of Arithmetic and Set Theory Michał Tomasz Godziszewski and Joel David Hamkins Logic Department, Institute of Philosophy, University of Warsaw, Warszawa, Poland. Mathematics, Philosophy, Computer Science, The Graduate Center of The City University of New York,New York, USA. Mathematics, College of Staten Island of CUNY, Staten Island, USA. We prove various extensions of the Tennenbaum phenomenon to the case of computable quotient presentations of models of arithmetic and set theory. Specifically, no non-standard model of arithmetic has a computable quotient presentation by a c.e. equivalence relation. No $$\varSigma _1$$-sound non-standard model of arithmetic has a computable quotient presentation by a co-c.e. equivalence relation. No non-standard model of arithmetic in the language $$\{+,\cdot ,\leq \}$$ has a computably enumerable quotient presentation by any equivalence relation of any complexity. No model of ZFC or even much weaker set theories has a computable quotient presentation by any equivalence relation of any complexity. And similarly no non-standard model of finite set theory has a computable quotient presentation. (Acknowledgements: This article is a preliminary report of results following up research initiated at the conference Mathematical Logic and its Applications, held in memory of Professor Yuzuru Kakuda of Kobe University in September 2016 at the Research Institute for Mathematical Sciences (RIMS) in Kyoto. The second author is grateful for the chance twenty years ago to be a part of Kakuda-sensei’s logic group in Kobe, a deeply formative experience that he is pleased to see growing into a lifelong connection with Japan. He is grateful to the organizer Makoto Kikuchi and his other Japanese hosts for supporting this particular research visit, as well as to Bakhadyr Khoussainov for insightful conversations. The first author has been supported by the National Science Centre (Poland) research grant NCN PRELUDIUM UMO-2014/13/N/HS1/02058. He also thanks the Mathematics Program of the CUNY Graduate Center in New York for his research visit as a Fulbright Visiting Scholar between September 2016 and April 2017. Commentary concerning this paper can be made at http://jdh.hamkins.org/computable-quotient-presentations.) Lattice Logic Properly Displayed Giuseppe Greco and Alessandra Palmigiano Delft University of Technology, Delft, the Netherlands. University of Johannesburg, Johannesburg, South Africa. We introduce a proper display calculus for (non-distributive) Lattice Logic which is sound, complete, conservative, and enjoys cut-elimination and subformula property. Properness (i.e. closure under uniform substitution of all parametric parts in rules) is the main interest and added value of the present proposal, and allows for the smoothest Belnap-style proof of cut-elimination, and for the most comprehensive account of axiomatic extensions and expansions of Lattice Logic in a single overarching framework. Our proposal builds on an algebraic and order-theoretic analysis of the semantic environment of lattice logic, and applies the guidelines of the multi-type methodology in the design of display calculi. (Acknowledgements: This research has been funded by the NWO Vidi grant 016.138.314, the NWO Aspasia grant 015.008.054 and a Delft Technology Fellowship awarded to the second author in 2013.) Multi-type Display Calculus for Semi De Morgan Logic Giuseppe Greco, Fei Liang, M. Andrew Moshier and Alessandra Palmigiano Delft University of Technology, Delft, the Netherlands. Chapman University, Orange, USA. Institute of Logic and Cognition Sun Yat-sen University, Guangzhou, China. University of Johannesburg, Johannesburg, South Africa. E-mail: liangf25@mail2.sysu.edu.cn We introduce a proper multi-type display calculus for semi De Morgan logic which is sound, complete, conservative, and enjoys cut-elimination and subformula property. Our proposal builds on an algebraic analysis of semi De Morgan algebras and applies the guidelines of the multi-type methodology in the design of display calculi. (Acknowledgements: This research is supported by the NWO Vidi grant 016.138.314, the NWO Aspasia grant 015.008.054 and a Delft Technology Fellowship awarded to the second author in 2013.) Shift Registers Fool Finite Automata Bjorn Kjos-Hanssen University of Hawai‘i at Manoa Honolulu, USA. E-mail: bjoernkh@hawaii.edu Let x be an m-sequence, a maximal length sequence produced by a linear feedback shift register. We show that x has maximal subword complexity function in the sense of Allouche and Shallit. We show that this implies that the nondeterministic automatic complexity $$A_N(x)$$ is close to maximal: $$n/2-A_N(x)=O(\log ^2 n)$$, where n is the length of x. In contrast, Hyde has shown $$A_N(y)\leq n/2+1$$ for all sequences y of length n. The Lambek Calculus with Iteration: Two Variants Stepan Kuznetsov University of Hawai‘i at Manoa Honolulu, USA. E-mail: bjoernkh@hawaii.edu Formulae of the Lambek calculus are constructed using three binary connectives, multiplication and two divisions. We extend it using a unary connective, positive Kleene iteration. For this new operation, following its natural interpretation, we present two lines of calculi. The first one is a fragment of infinitary action logic and includes an omega-rule for introducing iteration to the antecedent. We also consider a version with infinite (but finitely branching) derivations and prove equivalence of these two versions. In Kleene algebras, this line of calculi corresponds to the *-continuous case. For the second line, we restrict our infinite derivations to cyclic (regular) ones. We show that this system is equivalent to a variant of action logic that corresponds to general residuated Kleene algebras, not necessarily *-continuous. Finally, we show that, in contrast with the case without division operations (considered by Kozen), the first system is strictly stronger than the second one. To prove this, we use a complexity argument. Namely, we show, using methods of Buszkowski and Palka, that the first system is $$\varPi ^0_n$$-hard, and therefore is not recursively enumerable and cannot be described by a calculus with finite derivations. (Acknowledgements: This work is supported by the Russian Science Foundation under grant 16-11-10252.) Dependent Event Types Zhaohui Luo and Sergei Soloviev Steklov Mathematical Institute of RAS, Moscow, Russia. E-mail: sk@mi.ras.ru This paper studies how dependent types can be employed for a refined treatment of event types, offering a nice improvement to Davidson’s event semantics. We consider dependent event types indexed by thematic roles and illustrate how, in the presence of refined event types, subtyping plays an essential role in semantic interpretations. We consider two extensions with dependent event types: first, the extension of Church’s simple type theory as employed in Montague semantics that is familiar with many linguistic semanticists and, secondly, the extension of a modern type theory as employed in MTT-semantics. The former uses subsumptive subtyping, while the latter uses coercive subtyping, to capture the subtyping relationships between dependent event types. Both of these extensions have nice meta-theoretic properties such as normalization and logical consistency; in particular, we shall show that the former can be faithfully embedded into the latter and hence has expected meta-theoretic properties. As an example of applications, it is shown that dependent event types give a natural solution to the incompatibility problem (sometimes called the event quantification problem) in combining event semantics with the traditional compositional semantics, both in the Montagovian setting with the simple type theory and in the setting of MTT-semantics. (Acknowledgements: The first author is partially supported by EU COST Action CA15123 and CAS/SAFEA International Partnership Program. The second author is an associated researcher at ITMO University, St. Petersburg, Russia. Partially supported by EU COST Action CA15123 and Russian Federation Grant 074-U01.) A Geometry of Interaction Machine for Gödel’s System T Ian Mackie Department of Informatics, University of Sussex, Brighton, UK. Gödel’s System T is the simply typed lambda calculus extended with numbers and an iterator. The higher-order nature of the language gives it enormous expressive power—the language can represent all the primitive recursive functions and beyond, for instance Ackermann’s function. In this paper we use System T as a minimalistic functional language. We give an interpretation using a data flow model that incorporates ideas from the geometry of interaction and game semantics. The contribution is a reversible model of higher-order computation which can also serve as a novel compilation technique. Disjoint Fibring of Non-deterministic Matrices Sérgio Marcelino and Carlos CaleiroSQIG - Instituto de Telecomunicações, Departamento de Matemática - Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal. E-mail: smarcel@math.tecnico.ulisboa.pt In this paper we give a first definitive step towards endowing the general mechanism for combining logics known as fibring with a meaningful and useful semantics given by non-deterministic logical matrices (Nmatrices). We present and study the properties of two semantical operations: a unary operation of $$\omega $$-power of a given Nmatrix, and a binary operation of strict product of Nmatrices with disjoint similarity types (signatures). We show that, together, these operations can be used to characterize the disjoint fibring of propositional logics, when each of these logics is presented by a single Nmatrix. As an outcome, we also provide a decidability and complexity result about the resulting fibred logic. We illustrate the constructions with a few meaningful examples. (Acknowledgements: Work done under the scope of Project UID/EEA/50008/2013 of Instituto de Telecomunicações, financed by the applicable framework (FCT/MEC through national funds and co-funded by FEDER-PT2020). The first author also acknowledges the FCT postdoctoral grant SFRH/BPD/76513/2011. This research is part of the MoSH initiative of SQIG at Instituto de Telecomunicações.) Concrete Mathematics. Finitistic Approach to Foundations Marcin Mostowski and Marek Czarnecki Institute of Philosophy, Department of Logic, Jagiellonian University, Cracow, Poland. Institute of Philosophy, Department of Logic, Warsaw University, Warsaw, Poland. E-mail: marcin.mostowski@uj.edu.pl, m.czarnecki2@uw.edu.pl We discuss the idea of concrete mathematics inspired by Hilbert’s idea of finitistic mathematics as the part of mathematics not engaged into actual infinity. We explicate it as the part of mathematics based on $$\Delta ^0_2$$ arithmetical concepts. The explication is justified by equivalence of $$\Delta ^0_2$$ definability with algorithmic learnability (an epistemic argument) and with FM-representability (representability in finite models, an ontological argument). We show that the essential part of classical mathematics can be interpreted in the concrete framework. We claim that current mathematics is a social game of proving theorems on some axiomatic set theoretic background. On the other hand, concrete mathematics is the reality on which our mathematical experience is based. This is what makes the game intersubjective. Nevertheless, this game is one of the most efficient methods of building our mathematical knowledge. (Acknowledgements: This work was funded by the Polish National Science Centre grant number 2013/11/B/HS1/04168.) Solovay’s Completeness Without Fixed Points Fedor Pakhomov Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia. E-mail: pakhfn@mi.ras.ru In this paper we present a new proof of Solovay’s theorem on arithmetical completeness of Gödel–Löb provability logic GL. Originally, completeness of GL with respect to interpretation of $$\Box $$ as provability in Peano Arithmetic (PA) was proved by Solovay in 1976. The key part of Solovay’s proof was his construction of an arithmetical evaluation for a given modal formula that made the formula unprovable in PA if it were unprovable in GL. The arithmetical sentences for the evaluations were constructed using certain arithmetical fixed points. The method developed by Solovay have been used for establishing similar semantics for many other logics. In our proof we develop new more explicit construction of required evaluations that doesn’t use any fixed points in their definitions. To our knowledge, it is the first alternative proof of the theorem that is essentially different from Solovay’s proof in this key part. (Acknowledgements: This work is supported by the Russian Science Foundation under grant 14-50-00005.) An Epistemic Generalization of Rationalizability Rohit Parikh City University of New York, New York, USA. E-mail: rparikh@gc.cuny.edu Rationalizability, originally proposed by Bernheim and Pearce, generalizes the notion of Nash equilibrium. Nash equilibrium requires common knowledge of strategies. Rationalizability only requires common knowledge of rationality. However, their original notion assumes that the payoffs are common knowledge. That is agents do know what world they are in, but may be ignorant of what other agents are playing. We generalize the original notion of rationalizability to consider situations where agents do not know what world they are in, or where some know but others do not know. Agents who know something about the world can take advantage of their superior knowledge. It may also happen that both Ann and Bob know about the world but Ann does not know that Bob knows. How might they act? We will show how a notion of rationalizability in the context of partial knowledge, represented by a Kripke structure, can be developed. On Two Concepts of Ultrafilter Extensions of First-Order Models and Their Generalizations Nikolai L. Poliakov and Denis I. Saveliev Financial University, Moscow, Russia. and Institute for Information Transmission Problems of the Russian Academy of Sciences, Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia. E-mail: d.i.saveliev@gmail.com There exist two known concepts of ultrafilter extensions of first-order models, both in a certain sense canonical. One of them [1] comes from modal logic and universal algebra, and in fact goes back to [2]. Another one [3, 4] comes from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups [5] as its main precursor. By a classical fact, the space of ultrafilters over a discrete space is its largest compactification. The main result of [3, 4], which confirms a canonicity of this extension, generalizes this fact to discrete spaces endowed with a first-order structure. An analogous result for the former type of ultrafilter extensions was obtained in [6]. Here we offer a uniform approach to both types of extensions. It is based on the idea to extend the extension procedure itself. We propose a generalization of the standard concept of first-order models in which functional and relational symbols are interpreted rather by ultrafilters over sets of functions and relations than by functions and relations themselves. We provide two specific operations which turn generalized models into ordinary ones, and establish necessary and sufficient conditions under which the latter are the two canonical ultrafilter extensions of some models. (Acknowledgements: The second author was supported by Grant 16-01-00615 of the Russian Foundation for Basic Research.) Knowledge Is a Diamond Vít Punčochár Institute of Philosophy, Czech Academy of Sciences, Prague, Czech Republic. E-mail: vit.puncochar@centrum.cz In the standard epistemic logic, the knowledge operator is represented as a box operator, a universal quantifier over a set of possible worlds. There is an alternative approach to the semantics of knowledge, according to which an agent a knows $$\alpha $$ iff a has a reliable (e.g. sensory) evidence that supports $$\alpha $$. In this interpretation, knowledge is viewed rather as an existential, i.e. a diamond modality. In this paper, we will propose a formal semantics for substructural logics that allows to model knowledge on the basis of this intuition. The framework is strongly motivated by a similar semantics introduced in [3]. However, as we will argue, our framework overcomes some unintuitive features of the semantics from [3]. Most importantly, knowledge does not distribute over disjunction in our logic. (Acknowledgements: The work on this paper was supported by grant no. 16-07954J of the Czech Science Foundation.) Cut-Elimination for the Modal Grzegorczyk Logic via Non-well-founded Proofs Yury Savateev and Daniyar Shamkanov National Research University Higher School of Economics, Moskva, Russia and Steklov Mathematical Institute of the Russian Academy of Sciences, Moskva. Russia. E-mail: yury.savateev@gmail.com We present a sequent calculus for the modal Grzegorczyk logic Grz allowing non-well-founded proofs and obtain the cut-elimination theorem for it by constructing a continuous cut-elimination mapping acting on these proofs. (Acknowledgements: The article was prepared within the framework of the Basic Research Program at the National Research University Higher School of Economics (HSE) and supported within the framework of a subsidy by the Russian Academic Excellence Project ‘5-100’. Both authors also acknowledge support from the Russian Foundation for Basic Research (grant no. 15-01-09218a).) Global Neighbourhood Completeness of the Gödel–Löb Provability Logic Daniyar Shamkanov Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia. E-mail: daniyar.shamkanov@gmail.com The Gdel-Lb provability logic GL is strongly neighbourhood complete in the case of the so-called local semantic consequence relation. In the given paper, we consider Hilbert-style non-well-founded derivations in GL and establish that GL with the obtained derivability relation is strongly neighbourhood complete in the case of the global semantic consequence relation. (Acknowledgements: This work was supported by the Russian Science Foundation (grant no. 14-50-00005).) Substructural Logics with a Reflexive Transitive Closure Modality Igor Sedlár Institute of Philosophy, Czech Academy of Sciences, Prague, Czech Republic. E-mail: sedlar@flu.cas.cz Reflexive transitive closure modalities represent a number of important notions, such as common knowledge in a group of agents or non-deterministic iteration of actions. Normal modal logics with such modalities are well explored but weaker logics are not. We add a reflexive transitive closure box modality to the modal non-associative commutative full Lambek calculus with a simple negation. Decidability and weak completeness of the resulting system are established and extensions of the results to stronger substructural logics are discussed. As a special case, we obtain decidability and weak completeness for intuitionistic modal logic with the reflexive transitive closure box. (Acknowledgements: This work has been supported by the joint project of the German Science Foundation (DFG) and the Czech Science Foundation (GA CR) number 16-07954J (SEGA: From shared evidence to group attitudes). The author would like to thank the anonymous reviewers for a number of suggestions, and to Adam Přenosil for reading a draft of the paper. A preliminary version of the paper was presented at the 8th International Workshop on Logic and Cognition in Guangzhou, China; the author is indebted to the audience for valuable feedback.) Coherent Diagrammatic Reasoning in Compositional Distributional Semantics Gijs Jasper Wijnholds Queen Mary University of London, London, UK. E-mail: g.j.wijnholds@qmul.ac.uk The framework of Categorical Compositional Distributional models of meaning [3], inspired by category theory, allows one to compute the meaning of natural language phrases, given basic meaning entities assigned to words. Composing word meanings is the result of a functorial passage from syntax to semantics. To keep one from drowning in technical details, diagrammatic reasoning is used to represent the information flow of sentences that exists independently of the concrete instantiation of the model. Not only does this serve the purpose of clarification, it moreover offers computational benefits as complex diagrams can be transformed into simpler ones, which under coherence can simplify computation on the semantic side. Until now, diagrams for compact closed categories and monoidal closed categories have been used (see [2, 3]). These correspond to the use of pregroup grammar [12] and the Lambek calculus [9] for syntactic structure, respectively. Unfortunately, the diagrammatic language of Baez and Stay [1] has not been proven coherent. In this paper, we develop a graphical language for the (categorical formulation of) the non-associative Lambek calculus [10]. This has the benefit of modularity where extension of the system are easily incorporated in the graphical language. Moreover, we show the language is coherent with monoidal closed categories without associativity, in the style of Selinger’s survey paper [17]. Algorithmic Sahlqvist Preservation for Modal Compact Hausdorff Spaces Zhiguang Zhao Delft University of Technology, Delft, the Netherlands. E-mail: zhaozhiguang23@gmail.com In this paper, we use the algorithm ALBA to reformulate the proof in [1, 2] that over modal compact Hausdorff spaces, the validity of Sahlqvist sequents are preserved from open assignments to arbitrary assignments. In particular, we prove an adapted version of the topological Ackermann lemma based on the Esakia-type lemmas proved in [1, 2]. Received 28 March 2018 © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. permissions@oup.com

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Logic Journal of the IGPLOxford University Press

Published: May 1, 2018

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