TY - JOUR
AU - Zhang, Yuanlin
AB - Abstract In this article, we present a new version of the language of Epistemic Specifications. The goal is to simplify and improve the intuitive and formal semantics of the language. We describe an algorithm for computing solutions of programs written in this new version of the language. The new semantics is illustrated by a number of examples, including an Epistemic Specifications-based framework for conformant planning. In addition, we introduce the notion of an epistemic logic program with sorts . This extends recent efforts to define a logic programming language that includes the means for explicitly specifying the domains of predicate parameters. An algorithm and its implementation as a solver for epistemic logic programs with sorts is also discussed. 1 Introduction We define a new version of the language of Epistemic Specifications —an extension of the language of answer set programming (ASP)—that was introduced by Gelfond in the early 1990s [ 14 , 15 ]. The language of Epistemic Specifications allows for introspective reasoning through the use of modal operators K (‘known’) and M (‘may be true’), similar to □ and ⋄ of modal logic. The original version allowed formalizing certain types of knowledge not expressible in ASP, but its semantics was not fully satisfactory. The difficulty was related to issues with the formalization of the rationality principle which states that a rational agent should believe only what he is forced to believe . The semantics did not seem to adequately capture this principle for programs with recursion through modal operators. The problem was partially addressed in [ 16 ] where some progress was made with respect to the semantics of programs with recursion through the K operator. The approach was, however, not sufficient for programs with recursion through M . The version of the language presented herein addresses that issue. Concurrent with this work was an effort by Balai et al . [ 3 , 4 ] to enhance ASP by providing a framework for specifying sorts for the parameters of predicates in a language they called SPARC . This promoted better program design, helped to eliminate certain common errors (e.g. misspellings), and encouraged development of rules specific to a particular domain without the concern for rule safety that has been associated with variable terms by popular ASP solvers (e.g. clingo [ 21 ], DLV [ 10 ]). We utilize these recent efforts on sorts in order to provide within the language of Epistemic Specifications a framework for specifying predicate parameter domains. We call a program written in this new language an epistemic logic program with sorts (ELPS) . In addition to the benefits a sorted signature provides to the programmer, it also allows for a more straightforward algorithm for computing world views. We describe such an algorithm—one that can take advantage of existing state-of-the-art ASP solvers—and give a synopsis of test results from our implementation. In the next section, we provide some background information about the language. In Section 3 , we give the syntax and semantics for the new version. We then show comparisons between the new and old versions in Section 4 . An algorithm for an epistemic logic program solver is given in Section 5 . Epistemic logic programs with sorts are presented in Section 6 . The algorithm is modified for sorts in Section 7 , and an implementation and test results are presented in Section 8 . We next look at applications in Section 9 , including a framework for a conformant planner that takes advantage of the added features of the language. Finally, we describe related work, conclusions, and future work. Appendices are given with proofs of the theorems presented in the article, plus an alternative modal reduct that offers deeper insight into the new semantics. 2 Background The first paper [ 14 ] on Epistemic Specifications illustrated the need for a new formalism by considering the following problem. E xample 1 [Scholarship Eligibility Problem] Consider the following three ASP rules (with rule labels added for later reference) used by a certain college to determine eligibility of students for scholarships, where the variable X ranges over a given set of students. A fourth rule—given here in plain text—is that a student should be interviewed by the scholarship committee if his/her eligibility cannot be determined using the prior three rules . R1 : eligible(X)←highGPA(X) . R2 : eligible(X)←minority(X),fairGPA(X) . R3 : ¬eligible(X)←¬fairGPA(X),¬highGPA(X) . A reasonable attempt to express this fourth rule in ASP is as follows: However, when used with a database containing incomplete information, the resulting program is unable to correctly determine if a student is supposed to be interviewed. This is illustrated by considering the following disjunctive rule representing information from a database that Mike's GPA is either fair or high. Note that his minority status is not given and is therefore unknown. Together, rules R1 – R5 comprise a program having the following two answer sets: Although the program correctly represents Mike's eligibility for a scholarship as indeterminate, it fails to entail that Mike should be interviewed. A general representation was unable to be found for solving this problem using ASP. R4 : interview(X)←   noteligible ( X ), not ¬eligible(X) . R5 : fairGPA(mike)or highGPA(mike) . A1={highGPA(mike),eligible(mike)} , and A2={fairGPA(mike),interview(mike)} . Gelfond's remedy to this problem was to extend ASP by introducing modal operators, providing for more powerful introspective reasoning with respect to all answer sets (the K modal operator) or some answer set (the M modal operator). The intuitive meaning of these operators was influenced by modal logics of knowledge and belief (see, e.g. [ 13 ]). Semantically, programs in the language of Epistemic Specifications define possible collections of belief sets of a rational agent—called world views —associated with the program. The world view of a program without modal operators coincides with the collection of its answer sets if taken as an ASP program. If a program has modal operators, a world view W satisfies subjective literal K ℓ if literal ℓ belongs to all the belief sets in W , and subjective literal M ℓ if ℓ belongs to at least one belief set in W . For example, {{p,r},{q,r}} is the world view of the following program: The language of Epistemic Specifications provides a natural way to deal with the problem described in Example 1. por q . r←M p . E xample 2 [Scholarship Eligibility Problem Revisited] The fourth rule can be expressed in the language of Epistemic Specifications as The epistemic logic program consisting of rule R6 together with rules R1 – R3 and R5 from Example 1 has the world view W={A1,A2} where Notice that interview(mike) is in every belief set of the world view of this program; thus, we now have a program that correctly entails that Mike should be interviewed. R6 : interview(X)←not K eligible(X) , not K ¬eligible(X) . A1={highGPA(mike),eligible(mike),interview(mike)} , and A2={fairGPA(mike),interview(mike)} . Now let us consider the following program: Under the original semantics, this program has two world views: {{}} and {{p}} . We believe a rational agent should not accept {{p}} as a world view of this program as support for p is clearly circular and belief in Kp should require stronger conviction than that given (e.g. since the one-rule program p←p does not force belief in p , it is clear that p←K p should not since Kp requires more conviction to accept than p ). p←K p . Gelfond updated the semantics in [ 16 ], fixing the problem with recursion through K such that {{}} was the only world view of p←K p ; however, the following program still had two world views: {{}} and {{p}} . We couldn't find a justification for this result compatible with the rationality principle. p←M p . 3 Syntax and semantics of epistemic specifications—new version In what follows, we will distinguish between different versions of Epistemic Specifications by the years of their introduction. The new version will be referred to as ES2014 . 3.1 Syntax of ES2014 The language of Epistemic Specifications consists of the usual first-order signature, ASP logical connectives ( not , or , ¬ , etc.), and modal operators K and M . An objective literal is the same as a literal in ASP. A subjective literal is an expression of the form K ℓ or Mℓ where ℓ is an objective literal. A default-negated literal is an expression of the form not g where g is an objective or subjective literal. An extended literal is an objective, subjective, or default-negated literal. A rule is of the form where k≥0 , n≥0 , ℓ s are objective literals, and e s are extended literals. A program in this language, called an epistemic logic program , is a set of rules. ℓ1or … or ℓk←e1,…,en , 3.2 Semantics of ES2014 Note: In the following definitions, ℓ is used to denote a ground objective literal. By an ES structure , W , we mean a non-empty collection of consistent sets of ground objective literals. A pair 〈A,W〉 is called a pointed ES structure of W where A∈W . To define the semantics of Epistemic Specifications, we first define the notion of a model of an epistemic logic program Π . W is a model of Π if for each rule R in Π , R is satisfied by every pointed ES structure of W . The notion of satisfiability is defined below. D efinition 1 [ When an Extended Literal Is Satisfied ] Let W be an ES structure and 〈A,W〉 be a pointed ES structure of W . Note that the satisfiability of an objective literal does not depend on W and the satisfiability of a subjective literal does not depend on A . For example, if 〈A,W〉⊨ℓ we can simply say that A⊨ℓ ; likewise, if 〈A,W〉⊨K ℓ we can simply say that W⊨K ℓ . 〈A,W〉⊨ℓ     if ℓ∈A . 〈A,W〉⊨not ℓ   if ℓ∉A . 〈A,W〉⊨K ℓ    if ∀S∈W:ℓ∈S . 〈A,W〉⊨not K ℓ  if ∃S∈W:ℓ∉S . 〈A,W〉⊨Mℓ    if ∃S∈W:ℓ∈S . 〈A,W〉⊨not Mℓ  if ∀S∈W:ℓ∉S . D efinition 2 [ When a Rule Is Satisfied ] Let Π be a ground epistemic logic program, W be an ES structure, 〈A,W〉 be a pointed ES structure of W , and R be a rule in Π . 〈A,W〉⊨R if where head(R) denotes the set of objective literals in the head of R , and body(R) denotes the set of extended literals in the body of R . ∃ℓ∈head(R):〈A,W〉⊨ℓ , or ∃e∈body(R):〈A,W〉⊭e D efinition 3 [ Modal Reduct ] 1 Let Π be a ground epistemic logic program and W be an ES structure. The modal reduct of Π with respect to W , denoted by ΠW , is the ASP program (with nested expressions2 ) obtained from Π by replacing/removing subjective literals and deleting rules according to the following table. s if W⊨s if W⊭s K ℓ replace K ℓ with ℓ delete the rule not K ℓ remove not K ℓ replace not K ℓ with not ℓ Mℓ remove Mℓ replace Mℓ with notnot ℓ not Mℓ replace not Mℓ with not ℓ delete the rule s if W⊨s if W⊭s K ℓ replace K ℓ with ℓ delete the rule not K ℓ remove not K ℓ replace not K ℓ with not ℓ Mℓ remove Mℓ replace Mℓ with notnot ℓ not Mℓ replace not Mℓ with not ℓ delete the rule View Large s if W⊨s if W⊭s K ℓ replace K ℓ with ℓ delete the rule not K ℓ remove not K ℓ replace not K ℓ with not ℓ Mℓ remove Mℓ replace Mℓ with notnot ℓ not Mℓ replace not Mℓ with not ℓ delete the rule s if W⊨s if W⊭s K ℓ replace K ℓ with ℓ delete the rule not K ℓ remove not K ℓ replace not K ℓ with not ℓ Mℓ remove Mℓ replace Mℓ with notnot ℓ not Mℓ replace not Mℓ with not ℓ delete the rule View Large D efinition 4 [ World View ] W is a world view of Π if W=AS(ΠW) where AS(ΠW) denotes the set of all answer sets of ΠW . Π is said to be inconsistent if Π has no world view. Not only does this new definition retain the ‘fix’ for unintended world views due to recursion through K (as provided in ES2011), it also provides a reasonable solution to the problem of unintended world views due to recursion through M . The next example demonstrates this important new feature of our new definition. E xample 3 Let Π be the following epistemic logic program. Consider the ES structure W1={{}} . ΠW1 is which has two answer sets, {} and {p} ; thus, since W1≠AS(ΠW1) , W1 is not a world view of Π . Now consider the ES structure W2={{p}} . ΠW2 is whose answer set is {p} ; thus, since W2=AS(ΠW2) , W2 is a world view of Π . p←M p . p←notnot p . p . It is worth noting that, per our original intuition, we believed W1 should have been the only world view of Π . New insights into the nature of modal operators forced us to change our view. In the next section, we give a short description of our new intuition. 3.3 Preference order for extended literals and new intuition It was observed by looking at the definitions for satisfiability that a rational agent should find it easier to accept certain extended literals over others. This is clear when we look at the fact that, e.g. given a pointed ES structure 〈A,W〉 , in order to establish K ℓ , it must be demonstrated that ℓ belongs to all belief sets in W . To establish ℓ , it must be demonstrated that ℓ belongs to a particular belief set in W , namely A . But to establish Mℓ , it is sufficient to demonstrate that ℓ belongs to some belief set in W . This implies a preference order as illustrated below. We believe this to be an important observation, one that is key to the development of an intuitive understanding of the semantics of our new definition, and for our acceptance of {{p}} as the one and only world view of p←M p . In order to formalize our intuition about the preference of ℓ with respect to Mℓ , the definition of the reduct uses a translation of Mℓ into notnot ℓ rather than ℓ when Mℓ is not satisfied. The next example demonstrates another feature of the new semantics in that replacement of Mℓ by notnot ℓ when Mℓ is not satisfied allows acceptance of a world view where none existed under any of the semantics of previous language versions. E xample 4 Let Π be the following epistemic logic program. Consider the ES structure W={{p}} . ΠW is whose answer set is {p} ; thus, since W=AS(ΠW) , {{p}} is a world view of Π . por q . p←M q . por q . p←notnot q . In the next section, we provide more examples to demonstrate the ensuing effects of the new semantics. 4 Comparison with previous language version The modal reduct in Gelfond's 1994 definition [ 15 ] (referred to henceforth as ES1994 ) simply removed all satisfied subjective literals from the bodies of rules and removed any rules containing unsatisfied subjective literals. ES1994 also allowed empty world views and world views with an inconsistent belief set. The syntax of ES2014 differs from ES1994 by changing ¬K ℓ and ¬Mℓ to not K ℓ and not Mℓ , respectively. The table below shows world views and respective modal reducts of various example programs for ES1994 and ES2014. Among the differences given between ES1994 and ES2014 semantics is that a world view is now required to be non-empty and composed of consistent sets of literals. It may be viewed that since the definition of a world view was dependent on ASP semantics that the consistency requirement for belief sets in a world view came indirectly as a result of the evolution of ASP semantics. (The non-empty world view requirement was first proposed by Truszczyński in [ 24 ].) 5 An algorithm for finding world views of an ES2014 program We propose an algorithm that takes an epistemic logic program as its input and outputs the world views (if any) of the program. We first generate candidate solutions by creating an ASP program from the input that emulates all possible modal reducts with respect to combinations of truth assignments to distinct ground subjective literals of the input program. The answer sets of the ASP program are then computed and grouped into candidate world views based on associated subjective literal truth values. Finally, candidates are verified as world views if all associated subjective literals are satisfied. The algorithm requires all rules to be safe . We use the following definition: D efinition 5 [ Safe Rule ] A rule is said to be safe if for each variable that occurs in the rule there is at least one occurrence of the variable in a positive literal of the body. For ease of presentation, k_ℓ , k0_ℓ , k1_ℓ , m_ℓ , m0_ℓ , m1_ℓ , or d_ℓ will be used to denote the fresh atom obtained from literal ℓ by prefixing it with k_ , k0_ , k1_ , m_ , m0_ , m1_ , or d_ (respectively), and with substitution of 0 for ¬ within the fresh atom implicitly understood if ℓ is a classically-negated atom. For example, if ℓ=p(X) then k0_ℓ=k0_p(X) , but if ℓ=¬p(X) then k0_ℓ=k0_0p(X) . An atom of the form k_ℓ , k0_ℓ , or k1_ℓ will be referred to as a k-atom , whereas an atom of the form m_ℓ , m0_ℓ , or m1_ℓ will be referred to as an m-atom . By a k-/m-literal we mean a k-/m-atom or its negation. If W is a collection of sets of literals, we denote by W∖KM the collection of sets from W with all k-/m-literals removed. An atom of the form d_ℓ is used strictly as a device to achieve proper grounding of rules associated with a subjective literal of the form K ℓ , ensuring rule safety when ℓ contains a variable. After grounding, atoms of the form d_ℓ are removed. Likewise, rbody+(R) —denoting the comma-delimited list of relevant positive literals from the body of rule R when R contains a subjective literal containing a variable—is used to achieve proper grounding and safety of added rules. In the following algorithm, k1_ℓ is to be intuitively understood as ‘ K ℓ is true ’ and k0_ℓ as ‘ K ℓ is false .’ Respectively, m1_ℓ ( m0_ℓ ) corresponds to ‘ Mℓ is true ( false )’. Algorithm 1. [ ELPsolver : Finds the World Views of an Epistemic Logic Program] (1) Create ASP program Π′ from the rules of Π . Rules without subjective literals are left unchanged. For each rule R containing a subjective literal, replace the subjective literal and add new rules per the following table. (2) Compute (the ground version of Π′ ) with an appropriate ASP grounder, then create from by removing each rule with an atom of the form d_ℓ in its head along with every occurrence of d_ℓ from the rest of the program and removing the rbody+(R) literals from the bodies of rules added in step 1, and then compute the answer sets of with an appropriate ASP solver. Note: If there are no answer sets (i.e. is inconsistent), then Π has no world view. If Π has no subjective literals and is consistent, then the collection of answer sets of is the only world view of Π . In either case, the algorithm is finished and no further steps are performed. (3) Group the answer sets of by common k-/m-atoms of the form k0_ℓ , k1_ℓ , m0_ℓ , and m1_ℓ . Each group W of answer sets is said to represent (i.e. if its k-/m-literals are removed) a candidate world view of Π . (4) For each W representing a candidate world view of Π , check that the following conditions are met for all its k-/m-atoms: (a) if k1_ℓ is in the sets of W , then ℓ is in every set of W ; (b) if k0_ℓ is in the sets of W , then ℓ is missing from at least one set of W ; (c) if m1_ℓ is in the sets of W , then ℓ is in at least one set of W ; and (d) if m0_ℓ is in the sets of W , then ℓ is missing from every set of W . W∖KM is a world view of Π if conditions (a)–(d) above are met. T heorem 1 [ The Algorithm Is Sound ] 3 Given as input an epistemic logic program Π with only safe rules and an equivalent finite ground representation, if W is output by Algorithm 1 then W is a world view of Π . T heorem 2 [ The Algorithm Is Complete ] Given as input an epistemic logic program Π with only safe rules and an equivalent finite ground representation, if W is a world view of Π then W is output by Algorithm 1. 6 Epistemic logic programs with sorts Under the original semantics of both ASP and Epistemic Specifications, the signature of a program is typically implied by the symbols that occur, with universal domain for all predicate parameters. Additionally, a rule containing a variable is considered a shorthand for all its ground instantiations. Thus, for the following program the last rule is simply a shorthand for the rules p(a) . p(b) . q(c) . r(X)←p(X) . r(a)←p(a) . r(b)←p(b) . r(c)←p(c) . Now consider the same initial program, but with a sorted signature explicitly given that specifies the domain for each predicate parameter per the following table: Note that the parameter domains for p and r differ. One might ask whether the last rule of the program is still valid, and if so, what the ground instantiations should be. We use the same convention as in [ 4 ] where the ground instantiations of a rule with variables are only those where each variable instantiation maintains predicate parameter values within their declared domains. Thus, the last rule is now a shorthand for only one rule since any other instantiation for the variable X would not result in values for the parameters of p and r that would be in both of their respective domains. PREDICATE SYMBOL PARAMETER DOMAIN p {a,b} q {c} r {a,c} PREDICATE SYMBOL PARAMETER DOMAIN p {a,b} q {c} r {a,c} View Large PREDICATE SYMBOL PARAMETER DOMAIN p {a,b} q {c} r {a,c} PREDICATE SYMBOL PARAMETER DOMAIN p {a,b} q {c} r {a,c} View Large r(a)←p(a) . 6.1 Syntax of an ELPS Syntactically, the difference between an epistemic logic program and an epistemic logic program with sorts is that the latter includes explicit sort definitions and predicate declarations in addition to the program rules. A sort is a (non-empty) set of ground terms intended for use in defining the domain of predicate parameters. Various means for defining sorts have been proposed (e.g. see [ 28 ]). In this article, we use the same framework as that used in SPARC [ 3 , 4 ] which uses a preface before the program rules to define the sorts and provide predicate declarations. The first section of the framework begins with the keyword sorts followed by sort definitions of the form where sort_name begins with the symbol # followed by a unique identifier starting with a lower-case letter, and sort_expression is one of the following forms where n>0 , each t is a ground term, f is a function symbol, and ⊗ denotes a set operator (union, intersection, or difference) where the set operations result in a non-empty set. sort_name=sort_expression , {t1,…,tn} f(sort_name1,…,sort_namen) sort_name1⊗⋯⊗sort_namen , The next section begins with the keyword predicates followed by predicate declarations of the form where n≥0 . Every predicate in the program must be declared uniquely in this section before being used in a program rule. In addition to explicitly providing the domain for each parameter of a predicate, this allows for static type-checking of rules to discover misspellings and arity mismatches. predicate_name(sort_name1,…,sort_namen) , The final section begins with the keyword rules followed by the rules of the program. Each rule is of the form given in Section 3.1 for epistemic logic program rules. 6.2 Semantics For a program to be considered valid, every occurrence of a ground atom in a program rule must match its predicate declaration with respect to predicate name, arity, and parameter values within the sort domain. Rules with variables are instantiated as previously described. In short, for a valid ELPS, the net effect of the sorted signature is to restrict the grounding of its rules. The grounding of an ELPS results in a set of ground rules; hence, the semantics is the same as that presented in Section 3.2 . 6.3 Example programs To illustrate the syntax and semantics of an ELPS, we provide several example programs with their ground versions, candidate world views, modal reducts answer sets of modal reducts, and world views. E xample 5 View Large View Large E xample 6 View Large View Large E xample 7 View Large View Large 7 Algorithm for finding world views of an ELPS The algorithm proposed for solving an ELPS is the same as that described in Section 5 except that step 1 is replaced by the one given on the following page, and, as neither d_l nor rbody+(R) literals are added, step 2 need only compute the answer sets of the ASP program created in step 1. Additionally, sort atoms added in step 1 should be removed from the final results. As proofs of Theorems 1 and 2 are based on the ground program produced in steps 1 and 2 of the previous algorithm, the proofs also hold for the ground version of the program produced by step 1 of this algorithm. 8 Implementation and test results The algorithm was implemented (see [ 1 ]) using the Java programming language, working off a previous implementation of SPARC. This made coding fairly easy since the same framework for a sorted signature is shared by both ELPS and SPARC. The solver is a loosely coupled system that uses Potassco's clingo for finding the answer sets of the ASP program created in step 1 of the algorithm. The choice of clingo was appropriate due to its excellent performance and direct support for doubly default-negated literals (i.e. nested expressions of the form notnot   ℓ ). Other solvers could be used with (possibly) minor modifications made for differences in syntax/supported features. Test results showed that performance is reasonably good—even for this loosely coupled proof-of-concept solver—provided the number of ground subjective literals is small. It is noted, however, that the number of candidate world views generated and checked is exponential in the number of distinct ground subjective literals using the algorithm presented. For many of the test programs we ran, the independence of various groups of subjective literals was clear, hinting at a possible approach for improving the algorithm for certain classes of programs. Nonetheless, the algorithm employed shows that computation of world views under the semantics given is certainly acceptable for a number of ELPS programs. The following table provides empirical results for the ELPS solver, along with comparative results for ESmodels [ 23 ] with equivalent input programs. Programs named eligibleN were encodings of the scholarship eligibility example with N different students. Programs named yaleN were encodings of various Yale Shooting Problems using the epistemic conformant planner and a horizon of N . Tests were run on an Intel i7 820QM @ 1.73 GHz platform with 4 GB RAM running 64-bit Windows 7 operating system. The run time is given in seconds (s), minutes (m), or hours (h) as appropriate, and represents the best time observed. 4 A dash (–) indicates no solution was output. Program ELPS ESmodels Program ELPS ESmodels eligible1 <1 s <1 s yale1 <1 s 6 s eligible2 <1 s <1 s yale2 <1 s 1 m 50 s eligible4 <1 s <1 s yale3 2 s 35 m eligible6 <1 s <1 s yale4 5 s – eligible8 20 h 27 m – yale5 4 h 23 m – Program ELPS ESmodels Program ELPS ESmodels eligible1 <1 s <1 s yale1 <1 s 6 s eligible2 <1 s <1 s yale2 <1 s 1 m 50 s eligible4 <1 s <1 s yale3 2 s 35 m eligible6 <1 s <1 s yale4 5 s – eligible8 20 h 27 m – yale5 4 h 23 m – View Large Program ELPS ESmodels Program ELPS ESmodels eligible1 <1 s <1 s yale1 <1 s 6 s eligible2 <1 s <1 s yale2 <1 s 1 m 50 s eligible4 <1 s <1 s yale3 2 s 35 m eligible6 <1 s <1 s yale4 5 s – eligible8 20 h 27 m – yale5 4 h 23 m – Program ELPS ESmodels Program ELPS ESmodels eligible1 <1 s <1 s yale1 <1 s 6 s eligible2 <1 s <1 s yale2 <1 s 1 m 50 s eligible4 <1 s <1 s yale3 2 s 35 m eligible6 <1 s <1 s yale4 5 s – eligible8 20 h 27 m – yale5 4 h 23 m – View Large 9 Applications (conformant planning) There are many potential applications of the language of Epistemic Specifications in domains with incomplete information. These include: In this section we briefly discuss one of these possibilities, that of conformant planning. Conformant planning [ 22 ] is the problem of finding a plan (a sequence of actions) that is guaranteed to achieve a goal in spite of incomplete knowledge about the initial state. Research on the use of ASP for developing conformant planners has been done in the past (see, for instance, [ 25 ] and [ 11 ]). In what follows, we outline an alternative approach that uses the language of Epistemic Specifications. A thorough study of this approach will be presented elsewhere. conformant planning; autonomous robot control; policy management; assisting legal professionals (e.g. logical analysis of criminal evidence); and any application for which ASP is currently used but where the additional power of Epistemic Specifications might be needed due to incomplete information. By a (conformant) planning problem we mean a triple P=〈D,Σ,g〉 which consists of a system description D of an action language AL , a collection of the possible initial states Σ , and a goal g where g is a set of fluent literals. A sequence α of actions is called a solution to P if α is executable in every state of Σ and the execution of α in every such state makes the goal g true. (For more details, see [ 25 ].) For simplicity we assume that our system description D is deterministic, i.e. uncertainty is caused only by incompleteness of information about the initial state. We will be interested in looking for the solutions of the planning problem that have exactly n steps with one action occurring at each step. Constant n is often referred to as the horizon . The proposed approach consists of the following steps: D′ is obtained from D by expanding its signature by a new inertial fluent is_impossible and replacing every impossibility condition impossible a if l1,…,ln by causal law a causes is_impossible if l1,…,ln. The new collection Σ′ of initial states and the new goal are defined as follows: Σ′=def{σ′:σ′=σ∪{¬is_impossible} for every σ∈Σ} g′=defg∪{¬is_impossible} The translation τn of P′ into an ELPS consists of several steps. We use f to denote a fluent atom, l a fluent literal, body a comma-delimited list of fluent literals, and h(body,S) the comma-delimited list of literals of the form holds(f,S) or ¬holds(f,S) for each l that is a member of body where l=f or l=¬f , respectively. S ranges over natural numbers (often called steps ) from 0 to the horizon n . This completes our construction of τn(P′) . (1) A planning problem P=〈D,Σ,g〉 is translated into planning problem P′=〈D′,Σ′,g′〉 such that D′ is a system description of AL that contains no impossibility condition and α is a solution of P iff it is a solution of P′ . (2) Planning problem P′ is then translated into an ELPS τn(P′) (where n is the horizon) such that the world views of τn(P′) correspond to the solutions of P′ having length n . (1) Sorts are given by first having the keyword sorts followed by the sort definitions: #s1={0..n} . for the time steps, #s2={a1,…,ak} . where a1,…,ak is the list of all actions of the domain, and #s3={f1,…,fm} . where f1,…,fm is the list of all fluents in the domain. (2) Predicate declarations are given by having the keyword predicates followed by: holds(#s3,#s1) . occurs((#s2,#s1) . goal(#s1) . success() . (3) The keyword rules followed by the translations of the rest of the domain as given in the steps that follow. (4) Causal laws of D′ are translated into logic programming rules. (a) Each dynamic causal law of the form a causes l if body for which l=f is translated into the rule and, where l=¬f , into the rule (b) Each static causal law (also referred to as a state constraint) of the form l if body for which l=f is translated into the rule and, where l=¬f , into the rule (5) Inertia axioms and awareness axiom are also included in D′ : where F ranges over fluents. (6) Problem dependent relation goal(S) is defined by the rule goal(S)←h(g1′,…,gk′,S). where g1′,…,gk′ is the list of all elements of g′ . (7) The initial situation is defined by the collection of disjunctions of the form where each h is a literal of the form holds(f,0) or ¬holds(f,0) . (8) A collection of rules, called the epistemic conformant planning module , is added: Here each A ranges over actions. Intuitively, the first two rules form the generate part of the program. Note the fact that the first rule can be used as a generator due to the new semantics for modal operator M . (Recall Example 3 .) Used together with the previously defined rules, these two rules allow for the generation of multiple world views. For every such world view W and every step s such that 0≤s<n , there is exactly one action a such that an atom of the form occurs(a,s) belongs to all belief sets of W . The next rule defines the success of the planning process. Since we are only considering solutions of length n (i.e. the last action in the solution sequence occurs at step n−1 ), the goal must be met at step n . The last two rules define constraints which must be satisfied by each world view of the program. The first says that it is impossible to achieve success in one belief set of a world view without achieving it in all its belief sets. The second guarantees existence of at least one belief set containing success . Together they ensure that success is achieved in all belief sets of a world view without otherwise eliminating individual belief sets. T heorem 3 [ Correctness of the Proposed Conformant Planning Algorithm ] 5 Given a conformant planning problem P=〈D,Σ,g〉 where D is both deterministic and consistent (see [ 25 ]), the following is true with respect to the approach given above. We conclude by illustrating this algorithm with a famous simple example from [ 18 ]. For every world view W of τn(P′) , belief sets of W coincide on atoms of the form occurs(a,s) . A sequence α=a0,…,an−1 of actions is a solution of P of length n iff there is a world view W of τn(P′) such that occurs(a,k) belongs to its belief sets iff a is the k th element ak of α . E xample 8 [Yale Shooting Problem] The agent is operating in a domain in which there is a turkey and a gun. The turkey can be alive or not. The gun may be loaded or not. If the gun is loaded and the trigger is pulled, then the turkey will be dead. Pulling the trigger will unload the gun. The agent can load the gun, but this action is impossible if the gun is already loaded. The domain can be represented by the following domain description D : For our example, we give the following instance of the problem. Initially the turkey is alive, but it is unknown if the gun is loaded or not. The set of initial states is Σ={{alive,loaded},{alive,¬loaded}} . The goal is to kill the turkey; i.e. g={¬alive} . pull_trigger causes ¬alive if loaded pull_trigger causes ¬loaded load causes loaded impossible load if loaded This planning problem P=〈D,Σ,g〉 is translated into P′ by replacing the impossibility condition by the causal law load causes is_impossible if loaded and adding ¬is_impossible to g and to each initial state in Σ . The planning problem P′ is then translated into the ELPS τ(P′) consisting of the following: If the horizon is set to n=3 , the program has one world view. The world view has two belief sets: one in which the gun is initially loaded and one in which it is not. Both, however, contain the same sequence of actions: occurs(pull_trigger,0) , occurs(load,1) , and occurs(pull_trigger,2) . Regardless of the initial situation, the goal is reached at the horizon since ¬holds(alive,3) is in both belief sets and ¬holds(is_impossible,3) is in neither; hence, the sequence of actions is a conformant plan. 10 Related work In [ 29 ], Yan Zhang investigated the computational properties of epistemic logic programs, leading him to the development of an algorithm for computing world views. This effort culminated in the development of a solver known as Wviews [ 27 ] implemented by Michael Kelly. Wviews is based on ES1994 and relies on the ASP solver DLV [ 10 ]. In [ 24 ], Truszczyński offered alternative semantics to ES1991 by using Kleene's three-valued logic [ 19 ]. It was not Truszczyński's goal to address the issue of unintended world views; nonetheless, his added requirement that a world view be non-empty eliminated one form of unintended world view, whereas his additional requirement that belief sets of a world view be consistent put the language more in sync with modern ASP semantics, eliminating the world view containing the set of all literals. Later in the paper, a two-valued logic was used to provide a semantic framework in which three different languages based on Epistemic Specifications were defined. Truszczyński went on to derive complexity results for Epistemic Specifications that suggest that the language is strictly more expressive than ASP. Finally, he elaborated on the use of the language for modeling various problems that require meta-reasoning. In [ 12 ], Faber and Woltran presented a framework for combining ASP programs in a manner that allows consequences of programs to be used as input to other programs. They also presented a technique for compiling such a framework into a single ASP program using the concept of manifold programs. A manifold program is an expanded encoding of an ASP program that allows for reasoning about the consequences of its answer sets from within the encoding, thus providing a limited capacity for introspective reasoning similar to that provided by modal operators in the language of Epistemic Specifications. Their approach relies on the ASP extension known as weak constraints [ 8 ]. For certain epistemic logic programs, the methods described by Faber and Woltran may provide an efficient method for finding world views. Gelfond's 2011 paper [ 16 ] was the latest publication proposing an update to the semantics of Epistemic Specifications, ES2011, with a stated goal of eliminating unintended world views. His new reduct solved the problem of unintended world views associated with K ; however, the problem of unintended world views associated with recursion through M remained. Our article can be considered a continuation of that work. Cui, Zhizheng Zhang and Zhao [ 9 ] investigated the problem of grounding an epistemic logic program. That work culminated in the development of a grounder known as ESParser . Their more recent efforts [ 30 , 31 ] include the development of an associated solver called ESmodels [ 23 ] that relies on ASP solver claspD [ 21 ]. Their work is based on ES2011. The algorithm given for ESmodels uses an approach that is, at its core, much like the one used in the algorithm presented in this article. They also include an optimization they refer to as the ‘EFFICIENT ESMODELS’ algorithm that may suggest an approach that is applicable to our algorithm as well. Balai, Gelfond, and Yuanlin Zhang [ 3 , 4 ] extended ASP by adding a sorted signature, expanding the earlier work of Balduccini [ 5 ]. Included was an algorithm for applying the sorted signature to rules. This work culminated in the development of a solver/inference engine called SPARC [ 2 ]. Our article extends their work to the language of Epistemic Specifications. 11 Conclusion The version of Epistemic Specifications defined herein provides a reasonable solution to the problem of unintended world views that had remained unresolved for the past two decades. The new semantics eliminates unintended world views stemming from recursion through modal operators, while the requirement that world views be non-empty and contain only consistent sets eliminates still other sources of unintended world views and puts Epistemic Specifications in sync with current ASP semantics. We believe these changes represent a significant improvement to the language, making it arguably closer to the intuitive meaning, and thus better suited for knowledge representation and reasoning, and the design of intelligent agents. We believe adding a sorted signature to epistemic logic programs is an important enhancement to the language of Epistemic Specifications. Benefits include the fact that a sorted signature can: eliminate rule safety concerns; help avoid common errors by allowing static type-checking of predicates in rules; help avoid belief sets with unintended/nonsensical term relations; promote good programming design practices; and allow for a straightforward algorithm for computing the world views of a program. The algorithm described herein shows that solving an ELPS can be achieved by a method that includes the use of an existing state-of-the-art ASP solver, thereby avoiding ‘reinventing the wheel’ with respect to some of the more difficult tasks. Our implementation and testing of that algorithm demonstrates that using ELPS to model certain problems can be a practical method of problem solving. The algorithm will hopefully lead to an efficient epistemic logic program solver that will allow the language to be of more practical use. Applications such as the conformant planner show the potential of Epistemic Specifications for modelling complex tasks. Such applications also show that programs with multiple world views can be useful. 12 Future work We are working to improve the solver algorithm in order to create a more efficient epistemic logic program solver. A proof-of-concept solver has been developed as a loosely coupled system using Potassco's clingo [ 21 ]. It is hoped that a more tightly-coupled system with improvements to the algorithm will be developed. Once a reliable and reasonably efficient implementation exists, our next goal will be to devise applications and methodologies for solving problems for which the language of Epistemic Specifications seems ideally suited. The proposed framework for conformant planning is one such application. Further investigation of the mathematical properties of epistemic logic programs is also important. For example, there seems to be a close relationship between the models of epistemic logic programs and S5 Kripke frames, where every world is accessible from every world. The exact nature of this relationship and determining which Kripke frames correspond to world views requires careful examination. Acknowledgements We would like to thank the members of the Texas Tech Knowledge Representation Lab who provided incite, support and technical/editorial comments, especially Greg Gelfond, Yulia Kahl and Edward Wertz. We would also like to thank the anonymous reviewers who provided invaluable suggestions for improvement. Funding Work by M. Gelfond and Y. Zhang was partially supported by NFS grants (IIS-1018031 and CNS-1359359). References [1] E. Balai. ELPS , Texas Tech University, software & documentation available for download at https://github.com/iensen/elps/wiki/ . [2] E. Balai. SPARC , Texas Tech University, software & documentation available for download at https://github.com/iensen/sparc/wiki/ . [3] E. Balai, M. Gelfond and Y. Zhang. S P A R C –Sorted ASP with consistency restoring rules. CoRR abs/1301.1386 (2013). [4] E. Balai, M. Gelfond and Y. Zhang. Towards answer set programming with sorts. In LPNMR 2013 . Vol. 8148, Lecture Notes in Artificial Intelligence , P. Cabalar and T. C. Son eds, Springer, 2013. [5] M. Balduccini. Modules and signature declarations for A-Prolog: progress report. In Software Engineering for Answer Set Programming Workshop (SEA07), 2007. [6] M. Balduccini and T. 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Appendices A Correctness of solver algorithm The following are presented in the hope that they may be useful in improving intuition concerning the new version of Epistemic Specifications as well as in the development of proofs of correctness for epistemic logic programs. O bservation 1 Given a program Π which contains subjective literals, the first step of the construction of the algorithm adds rules which, after grounding in step 2, are of the forms: As these are the only rules with k1_ℓ,k0_ℓ,m1_ℓ , and m0_ℓ in their heads and the program is head cycle free with respect to these rules, each pair behaves like an ‘or’ and therefore any answer set of the program contains either k1_ℓ or k0_ℓ but not both (as with the ‘m's’ as well). It can further be seen that the set of all such literals forms a splitting set for with these rules as the bottom and the rest of the program as the top. Therefore, there is a potential for any combination of choices for each ‘or’ to occur in an answer set. D efinition 6 [Base Extension w.r.t. W and Π ] Let Π be a ground epistemic logic program and W be an ES structure. The base extension w.r.t. W and Π , denoted BEx(W,Π) , is the set BEx(W,Π)={k1_ℓ|W satisfies K ℓ and there is a rule R in Π which contains either K ℓ or not K ℓ} ∪ {k0_ℓ|W does not satisfy K ℓ and there is a rule R in Π which contains either K ℓ or not K ℓ} ∪ {m1_ℓ|W satisfies M ℓ and there is a rule R in Π which contains either M ℓ or not M ℓ} ∪ {m0_ℓ|W does not satisfy M ℓ and there is a rule R in Π which contains either M ℓ or not M ℓ} . The following is an example of the base extension. Notice that W does not need to be a world view of Π . E xample 9 Suppose Π is the program: (A1) (A2) e←K c. (A3) (A4) g←M a,b. (A5) (A6) and W={{a,c,f},{c,d,g}} . Then BEx(W,Π)={k1_c,k0_b,m1_a,m0_e} . D efinition 7 [Extension of A1 w.r.t. A , W , and Π ] Let Π be a ground epistemic logic program, W be an ES structure, and A and A1 be consistent sets of literals. The extension of A1 w.r.t. A , W , and Π denoted ExA,W,Π(A1) , is the set ExA,W,Π(A1)=BEx(W,Π) ∪ {¬k_ℓ∣k0_ℓ∈BEx(W,Π) or (k1_ℓ∈BEx(W,Π) and ℓ∉A)} ∪ {m_ℓ|m1_ℓ∈BEx(W,Π) or (M ℓ∈Π and m0_ℓ∈BEx(W,Π) and ℓ∈A) or ( not M ℓ∈Π and m0_ℓ∈BEx(W,Π) and ℓ∈A1)} . E xample 10 Let Π and W be as in example 9, A be the set {a,b,d,e} and A1 be the set {a,b,d} . Then: ExA,W,Π(A)={k1_c,k0_b,m1_a,m0_e,¬k_c,¬k_b,m_a,m_e} and ExA,W,Π(A1)={k1_c,k0_b,m1_a,m0_e,¬k_c,¬k_b,m_a}. L emma 1 Let Π be a epistemic logic program, Add be the ground answer set program that consists of only the rules of with k-/m-literals in their heads, W be an ES structure, A and A1 be consistent sets of literals not containing any k-/m-literals, E′=ExA,W,Π(A) , and E1′=ExA,W,Π(A1) . Then: (1) A1∪E1′ satisfies the rules of AddA∪E′ , and (2) there does not exist a E1′′⊂E1′ such that A1∪E1′′ satisfies the rules of AddA∪E′ . P roof . Can easily be seen from the construction of ExA,W,Π(A1) .  ▀ L emma 2 Let Π be a epistemic logic program, W be an ES structure, and A be a consistent set of literals. If we expand the definition of the reduct w.r.t A to allow the rules to be rules of an epistemic logic program, but otherwise do not change the definition, then ((ΠA)W)A=(ΠW)A . P roof . Can easily be seen from the definitions of reduct and modal reduct.  ▀ L emma 3 Let Π be an epistemic logic program with only safe rules and an equivalent finite ground representation, be the ground version of Π , be the program created in step 2 of the algorithm, W be an ES structure, A be a consistent set of literals not containing k-/m-literals, and A′′=A∪ExA,W,Π(A) . If A1⊆A and A1′′=A1∪ExA,W,Π(A1) then A1 satisfies the rules of iff A1′′ satisfies the rules of . P roof . The proof will be based on relating the rules of the two programs to the rules of . First notice that by Lemma 2 we need only consider rules of (if a rule contains then either: ℓ∈A and the rule will not exist in either or ; or, ℓ∉A and, if the rules have not been otherwise removed, the not ℓ will have been removed from the body by the reduct). Hence in the following proof, except for the rules added by the algorithm, rules will not contain ‘ not ’ in the objective part of the body. For a rule R in we will denote by RW the rule's counterpart in and we will denote by R′′ the rule's counterpart in . For each rule R in : The only remaining rules are the added rules in . As shown in Lemma 1 these rules are satisfied by A1′′ .  ▀ (1) If the body of R only contains objective literals then R is in both and and hence is satisfied by A1 iff it is satisfied by A1′′ . (2) If the body contains both subjective and objective literals and at least one objective literal in R is not satisfied by A1 then the counterpart in either does not exist (because it was removed by the modal reduct) or has the the same objective part as R . The rule R′′ has the same objective part as R . If a counterpart does not exist it can be viewed as being satisfied. If the objective part of the body contains a literal that is not satisfied then the body is not satisfied, hence the rule is. Therefore the counterpart of R in both programs is satisfied. (3) If the objective part of the body of R is satisfied by A1 but there is at least one subjective literal in R that is not satisfied by W then: (a) if there is a literal of the form K ℓ (or not M ℓ ) that is not satisfied then in the rule is deleted by the modal reduct. In the literal in rule R′′ is replaced by not ¬k_ℓ (or not m_ℓ resp.) while, by construction of the extension, A′′ contains k_ℓ (or m_ℓ resp.) hence the rule is deleted by the reduct w.r.t. A′′ . Otherwise; (b) if there is a literal of the form not K ℓ that is not satisfied then in RW the literal is replaced by not l while in R′′ the literal is replaced by ¬k_ℓ . By the construction of ExA,W,Π(A) , since not K ℓ is not satisfied (i.e K ℓ is satisfied), k1_ℓ∈A1′′ and k0_ℓ is not. There are two possibilities: (i) ℓ∈A - then RW is removed from by the reduct (hence the rule can be viewed as being satisfied). By the construction of ExA,W,Π(A) , since ℓ∈A and k0_ℓ∉BEx(W,Π) , then ¬k_ℓ∉A′′ . Hence the body of R′′ cannot be satisfied by A′′ and therefore the rule is satisfied. (ii) ℓ∉A - In RW the not ℓ is removed by the reduct. For R′′ , since ℓ∉A then ℓ∉A1 . By the construction of ExA,W,Π(A1) , since ℓ∉A1 and k1_ℓ∈BEx(W,Π) then ¬k_ℓ∈A1′′ . Therefore the part of the rule related to not K ℓ is satisfied in both rules and hence satisfaction of the rules depends on the other literals in the rules. (c) if there is a literal of the form M ℓ that is not satisfied then in RW the literal is replaced by not not ℓ while in R′′ the literal is replaced by m_ℓ . By the construction of ExA,W,Π(A) , since M ℓ is not satisfied, then m0_ℓ∈BEx(W,Π) . There are two possibilities: (i) ℓ∈A - In RW the not not ℓ is removed by the reduct. For R′′ , by the construction of ExA,W,Π(A1) , since ℓ∈A and m0_ℓ∈BEx(W,Π) then m_ℓ∈A1′′ . Therefore the part of the rule related to M ℓ is satisfied in both rules and hence satisfaction of the rules depends on the other literals in the rule. (ii) ℓ∉A - then RW is removed from by the reduct (hence the rule can be viewed as being satisfied). By the construction of ExA,W,Π(A1) , since ℓ∉A and m1_ℓ∉BEx(W,Π) , then m_ℓ∉A1′′ . Hence the body of R′′ cannot be satisfied by A1′′ and therefore the rule is satisfied. (4) If the objective part of the body of R is satisfied by A , all subjective literals in the body are satisfied by W , and there is at least one subjective literal in the body then: (a) if there is a literal of the form K ℓ which is satisfied by W then in RW the literal K ℓ is replaced by ℓ while in R′′ it is replaced by not ¬k_ℓ,ℓ . There are two possibilities: (i) ℓ∈A - By construction of ExA,W,Π(A) , ¬k_ℓ∉A′′ , and therefore the not ¬k_ℓ is removed from R′′ by the reduct and the rules are identical. Therefore, one rule is satisfied if and only if the other is. (ii) ℓ∉A - then ℓ∉A1 . As RW contains ℓ , the body of the rule is not satisfied by A and therefore the rule is satisfied. For R′′ , by construction of ExA,W,Π(A) , ¬k_ℓ∈A′′ , and therefore the rule is removed by the reduct (and hence satisfied). (b) if there is a literal of the form not M ℓ which is satisfied by W then in RW the literal is replaced by not ℓ and in R′′ the literal is replaced by not m_ℓ . There are two possibilities: (i) ℓ∈A - By construction of ExA,W,Π(A) , m_ℓ∈A′′ and both RW and R′′ are removed by their respective reducts. (ii) ℓ∉A - By construction of ExA,W,Π(A) , m_ℓ∉A′′ so, by the respective reducts, the not ℓ is removed from RW and the not m_ℓ is removed from R′′ and hence the satisfaction of the rules depends solely on the other literals in the rules. (c) if there is a literal of the form not K ℓ that is satisfied by W then the literal is removed from RW and in R′′ the literal is replaced by ¬k_ℓ . By construction of ExW,Π(A1) , since not K ℓ is satisfied, ¬k_ℓ∈A1′′ and therefore satisfied. Hence, satisfaction of the rules RW and R′′ do not differ with respect to this literal and satisfaction of the rules depends solely on the other literals in the rules. (d) if there is a literal of the form M ℓ that is satisfied by W then the literal is removed from RW and in R′′ the literal is replaced by m_ℓ . By construction of ExW,Π(A1) , since M ℓ is satisfied, m_ℓ∈A′′ and therefore satisfied. Hence, satisfaction of the rules RW and R′′ do not differ with respect to this literal and satisfaction of the rules depends solely on the other literals in the rules. E xample 11 Consider Π , W , A , A1 , ExW,Π(A) , and ExW,Π(A1) from example 10. Then, with respect to the rules of Π , the programs ΠWA and are: Recall that A1={a,b,d} and A1∪ExA,W,Π(A1)={a,b,d,k1_c,k0_b,m1_a , m0_e,¬k_c,¬k_b,m_a} . All the rules except those from (5) are satisfied by their respective set. The rules from (5) are not satisfied in either set. T heorem 1 [ Algorithm 1 Is Sound ] Given as input an epistemic logic program Π with only safe rules and an equivalent finite ground representation, if W is output by Algorithm 1 then W is a world view of Π . P roof . Let W be a set output by Algorithm 1. By the algorithm's construction, W met steps 3 and 4 of the algorithm. In order to be in the same grouping in step 3, W must have come from a candidate world view W′′={A∪ExW,Π(A)|A∈W} . For each A∈W , by Lemma 3, A satisfies all the rules of hence either A is an answer set of or there exists and A′⊂A which satisfies the rules of . However, by Lemma 3, then A′∪ExW,Π(A′) would satisfy all the rules of . Hence A∪ExW,Π(A) could not be an answer set of . This is a contradiction, therefore A is an answer set of ΠW . Next, suppose there was an answer set A′ of which was not in the W returned by the algorithm. By Lemma 3, A′∪ExW,Π(A′) would satisfy all the rules of (and by the construction of ExW,Π(A′) it would be in the same grouping in step 3 of the algorithm) and therefore either the expanded grouping would violate step 4 of the algorithm and W would not be returned, or A′∪ExW,Π(A′) (or a subset of it) would be in W′′ thus either A′∈W (or a subset of it) would be. Hence another contradiction. Therefore W is the set of all answer sets of and is then, by definition, a world view of Π .  ▀ T heorem 2 [ Algorithm 1 Is Complete ] Given as input an epistemic logic program Π with only safe rules and an equivalent finite ground representation, if W is a world view of Π then W is output by Algorithm 1. P roof . Let W be a world view of Π . By Lemma 3, for each A∈W , A∪ExW,Π(A) satisfies the rules of therefore either A∪ExW,Π(A) is an answer set of or there exists a proper subset of A∪ExW,Π(A) which is an answer set of . From Lemma 1, it can be seen that such a subset would have to be of the form A′∪ExW,Π(A′) where A′⊂A . But, by Lemma 3, A′ would satisfy the rules of and therefore A could not be an answer set of . This is a contradiction; therefore, A∪ExW,Π(A) is an answer set of . By the construction of BEx(W,Π) it can be seen that, for a given W , if A∈W each A∪ExW,Π(A) will be part of the same grouping in step 3 of the algorithm. Step 4 of the algorithm eliminates candidate world views if the k-/m-literals disagree with W . But, by the construction of ExW,Π(A) , since W is a world view of Π , the set of A∪ExW,Π(A) must pass the conditions of this step. Next, suppose there was an A′∪ExW,Π(A′) that was an answer set of where A′∉W . By Lemma 3, A′ would satisfy the rules of and therefore either A′∈W or a subset of it is. Hence another contradiction. Therefore if W is a world view of Π then W is output by Algorithm 1.  ▀ B Correctness of epistemic conformant planning module L emma 4 [Constraints Remove Answer Sets] Let P be an ASP program and c an ASP constraint (i.e. a headless rule). A set A of literals is an answer set of P∪c , iff A is an answer set of P and A satisfies c . P roof . The above lemma is essentially a rewording of Proposition 2.4.3 in [ 17 ].  ▀ P roposition 1 [ Constraints that Eliminate World Views that Don't Satisfy K ℓ ] Let Π be an epistemic logic program with at least one predicate symbol in its signature. Let Π′ be the epistemic logic program resulting from adding to Π the following two constraints: where ℓ is a ground literal in the language of Π . For any ES structure W , W is a world view of Π′ iff W is a world views of Π and W⊨K ℓ . . . P roof . Let W be an ES structure. There are only the following three cases. For any W , if W is a world view of Π′ , only case 1 above holds. Therefore, W is a world view of Π and W⊨K ℓ . If W is a world view of Π and W⊨K ℓ , then case 1 above holds. Therefore, W is a world view of Π′ . Hence, the proposition holds.  ▀ Case 1: W⊨K ℓ and W⊨M ℓ (i.e. W⊭   not   K ℓ and W⊭   not   M ℓ ) By Definition 3 (modal reduct), (Π′)W=ΠW∪{←ℓ , not   ℓ} . ←ℓ ,  not   ℓ is satisfied by any set of objective literals because its body is never satisfied. Hence, by Lemma 4, ΠW and (Π′)W have the same collection of answer sets. Therefore, W is a world view of Π′ iff W is a world view of Π . Case 2: W⊭K ℓ and W⊨M ℓ (i.e. W⊨   not   K ℓ and W⊭   not   M ℓ ) By Definition 3 (modal reduct), (Π′)W=ΠW∪{←ℓ} . Since W⊨M ℓ , there exists S∈W such that ℓ∈S by Definition 1. Therefore, S does not satisfy ←ℓ , and thus S is not an answer set of ΠW∪{←ℓ} . By Lemma 4, S is not an answer set of (Π′)W . Hence, W is not a world view of Π′ . Case 3: W⊭K ℓ and W⊭M ℓ (i.e. W⊨   not   K ℓ and W⊨   not   M ℓ ) By Definition 3 (modal reduct), (Π′)W=ΠW∪{←ℓ,←   not   ℓ} . Since no set of objective literals can satisfy the two rules ←ℓ and ←   not   ℓ , ΠW∪{←ℓ,←   not ℓ} has no answer set. Therefore, W is not a world view of Π′ . P roposition 2 [ Constraint that Eliminates World Views that Don't Satisfynot   K ℓ ] Let Π be an epistemic logic program with at least one predicate symbol in its signature. Let Π′ be the epistemic logic program resulting from adding to Π the following constraint: where ℓ is a ground literal in the language of Π . For any ES structure W , W is a world view of Π′ iff W is a world view of Π that satisfies not K ℓ . ←K ℓ , P roof . Let W be an ES structure. There are only two cases. For any W , if W is a world view of Π′ , only case 1 above holds. Therefore, W is a world view of Π and W⊨   not   K ℓ . If W is a world view of Π and W⊨   not   K ℓ , then case 1 above holds. Therefore, W is a world view of Π′ . Hence, the proposition holds.  ▀ Case 1: W⊨   not   K ℓ (i.e. W⊭K ℓ ) By Definition 3 (modal reduct), (Π′)W=ΠW . Therefore, by Definition 4 (world view), W is a world view of Π′ iff W is a world view of Π . Case 2: W⊭   not   K ℓ (i.e. W⊨K ℓ ) By Definition 3 (modal reduct), (Π′)W=ΠW∪{←ℓ} . Since W⊨K ℓ , every belief set in W contains ℓ by Definition 1. Hence, no belief set of W satisfies ←ℓ , and thus no belief set of W is an answer set of ΠW∪{←ℓ} , i.e. (Π′)W . Therefore, W is not a world view of Π′ . P roposition 3 [ Constraint that Eliminates World Views that Don't Satisfy Mℓ ] Let Π be an epistemic logic program with at least one predicate symbol in its signature. Let Π′ be the epistemic logic program resulting from adding to Π the following constraint: where ℓ is a ground literal in the language of Π . For any ES structure W , W is a world view of Π′ iff W is a world view of Π that satisfies Mℓ . ←   not   M ℓ , P roof . Let W be an ES structure. For any W , if W is a world view of Π′ , only case 1 above holds. Therefore, W is a world view of Π and W⊨M ℓ . If W is a world view of Π and W⊨M ℓ , then case 1 above holds. Therefore, W is a world view of Π′ . Hence, the proposition holds.  ▀ Case 1: W⊨M ℓ (i.e. W⊭   not   M ℓ ) By Definition 3 (modal reduct), (Π′)W=ΠW . Therefore, by Definition 4 (world view), W is a world view of Π′ iff W is a world view of Π . Case 2: W⊭M ℓ (i.e. W⊨   not   M ℓ ) By Definition 1 (modal reduct), (Π′)W=ΠW∪   not   ←ℓ} . Since W⊨   not   M ℓ , every belief set in W is missing ℓ by Definition 1. Therefore, no belief set of W satisfies ←   not   ℓ , and thus no belief set of W is an answer set of ΠW∪   not   ←ℓ , i.e. (Π′)W . Therefore, W is not a world view of Π′ . P roposition 4 [ Constraints that Eliminate World Views that Don't Satisfynot   Mℓ ] Let Π be an epistemic logic program with at least one predicate symbol in its signature. Let Π′ be the epistemic logic program resulting from adding to Π the following two constraints: where ℓ is a ground literal in the language of Π . For any ES structure W , W is a world view of Π′ iff W is a world view of Π that satisfies not   Mℓ . ←   not   ℓ,M ℓ . ←K ℓ . P roof . Let W be an ES structure. For any W , if W is a world view of Π′ , only case 1 above holds. Therefore, W is a world view of Π and W⊨   not   Mℓ . If W is a world view of Π and W⊨   not   Mℓ , then case 1 above holds. Therefore, W is a world view of Π′ . Hence, the proposition holds.  ▀ Case 1: W⊨   not   M ℓ and W⊨   not   K ℓ (i.e. W⊭M ℓ and W⊭K ℓ ) By Definition 3 (modal reduct), (Π′)W=ΠW∪{←   not   ℓ ,  not   not ℓ} . The constraint ←   not   ℓ , notnot   ℓ is always satisfied. Therefore, by Lemma 4, W is a world view of Π′ iff W is a world view of Π . Case 2: W⊭   not   M ℓ and W⊨   not   K ℓ (i.e. W⊨M ℓ and W⊭K ℓ ) By Definition 3 (modal reduct), (Π′)W=ΠW∪{←   not ℓ} . However, W⊨M ℓ , by Definition 1, means that there is one belief set S in W contains ℓ . By the definition of answer sets, S is not an answer set of ΠW∪{←   not ℓ} , i.e. (Π′)W . Therefore, W is not a world view of Π′ . Case 3: W⊭   not   M ℓ and W⊭   not   K ℓ (i.e. W⊨M ℓ and W⊨K ℓ ) By Definition 3 (modal reduct), (Π′)W=ΠW∪{←   not   ℓ.←ℓ} . {←   not   ℓ.←ℓ} is not satisfied by any consistent set of objective literals. Therefore, there is no answer set for ΠW∪{←   not   ℓ.←ℓ} , i.e. (Π′)W . Hence, W is not a world view of Π′ . The following terminology of AL [ 7 , 26 ] is needed in our proof. Action language AL is parameterized by a sorted signature containing three special sorts: statics , fluents and actions . The fluents are partitioned into two sorts: inertial and defined . Statics and fluents are called domain properties . A domain literal is a domain property or its classical negation. Given a set σ of domain literals, σnd denotes the collection of domain literals of σ formed by inertial fluents and statics. A system description is a set of AL statements. Given a system description SD , Π(SD) denotes the standard encoding of SD statements into an ASP program as in [ 17 ]. In the encoding, a relation holds(f,i) says that fluent f is true at step i . If f is a fluent then by h(l,i) , we denote holds(f,i) if l=f , ¬holds(f,i) if l=¬f , and l if l is a static literal. A relation occurs(a,i) where a is an action says that the action a occurs at step i . A complete and consistent set σ of domain literals is a state of the transition diagram defined by a system description if σ is the unique answer set of program Πc(SD)∪σnd where Πc denotes only the encoding of the state constraint statements. Given a system description SD , let a be a collection of actions and σ0 and σ1 be states of the transition diagram defined by SD . A triple 〈σ0,a,σ1〉 is a transition of T(SD) if Π(SD,σ0,a) has an answer set A such that σ1={l:h(l,1)∈A} . We use Π(SD,σ,a) , where σ is a state and a an action, to denote the ASP program Π(SD)∪{h(l,0):l∈σ}∪{occurs(a,0)} . An action a is said to be prohibited in a state σ of a transition diagram T(D) defined by a system description D if D contains an impossibility condition for action a0⊆a whose body is satisfied by σ ; otherwise, a is said to be executable in σ . A system description D is consistent if for any state σ1 and action a executable in σ1 , there exists at least one state σ2 such that 〈σ1,a,σ2〉∈T(D) . A system description is deterministic if for any state σ1 and action a , there is at most one state σ2 such that 〈σ1,a,σ2〉 is a transition defined by T(D) . We denote the algorithm for conformant planning by E . We first present a lemma about the relationship between a conformant planning problem and the one produced by the first step of E . L emma 5 [Equivalence of P and P′ ] Let the planning problem P′=〈D′,Σ′,g′〉 be constructed from P=〈D,Σ,g〉 by the first step of E . α is a solution of P iff it is a solution of P′ . P roof . Let E be the executability statements of D . Define f(E) as Π({a causes is_impossible if l1,…,ln: (impossible a if l1,…,ln)∈E})∪{inertial laws for is_impossible} . Let impF={h(¬is_impossible,0),h(¬is_impossible,1)} . For any states σ1 , σ2 and any action a of D , let σ1′=σ1∪{¬is_impossible} and σ2′=σ2∪{¬is_impossible} . According to the construction of P′ from P by E , we have (1) Π(D′)=(Π(D)−Π(E))∪Π(f(E)) , and (2) Π(D′,σ1′,a)=Π(D′)∪{h(l,0):l∈σ1}∪{h(¬is_impossible,0)}∪{occurs(a,0)} . We now show an intermediate result: 〈σ1,a,σ2〉 is a transition of T(D) iff 〈σ1′,a,σ2′〉 is a transition of T(D′) . ⟹ : Let (3) A be an answer set of Π(D,σ1,a) such that σ2={l:h(l,1)∈A} . Define (4) nonOccur(A)={¬occurs(a,i):occurs(a,i)∈A} . Let (5) B=(A−nonOccur(A))∪impF . We show B is an answer set of Π(D′,σ1′,a) such that σ2′={l:h(l,1)∈B} (and thus 〈σ1',a,σ2'〉 is a transition of T(D′) ). Since Π(E) are the only rules of Π(D) that define occurs , we take Π(E) as the top program of Π(D,σ1,a) . By splitting lemma of ASP, (3) implies that (6) A−nonOccur(A) is an answer set of (Π(D)−Π(E))∪{h(l,0):l∈σ1}∪{occurs(a)} . Since is_impossible is not a fluent of D , by (2), Π(f(E))∪{h(¬is_impossible,0)} defining is_impossible can be taken as the top program of Π(D′,σ1′,a) . Let the executability condition on a be (7) impossible aif l1,…,ln . It is encoded in Π(D) as rule r : (8) ¬occurs(a,i) :- h(l1,i),…,h(ln,i) . Since A is an answer set of Π(D,σ1,a) , r is satisfied by A . Since occurs(a,0)∈A , (9) A does not satisfy body(r) . By (7) and the construction of D′ , the following rule r′ belongs to Π(D′) (10) h(is_impossible,i) :- h(l1,i),…,h(ln,i),occurs(a,i) . By (9), (11) A does not satisfy body(r′) because body(r)⊂body(r′) and body(r′) contains no not . Since is_impossible is new to D and B−impF⊆A (by (5)), (11) implies (12) B does not satisfy body(r′) . Let (Π(f(E))∪{h(¬is_impossible,0)})B be the top program of Π(D′,σ1′,a)B . By (6), A−nonOccur(A) is an answer set of the bottom program of Π(D′,σ1′,a)B . By splitting lemma and (12), we can verify that (A−nonOccur(A))∪impF , i.e. B , is an answer set of Π(D′,σ1′,a)B . Therefore, B is an answer set of Π(D′,σ1′,a) . We can also verify that σ2′={l:h(l,1)∈B} . Hence, 〈σ1′,a,σ2′〉 is a transition of T(D′) . ⟸ : the proof of this direction is similar to the sufficient condition. Assume 〈σ1′,a,σ2′〉 is a transition of T(D′) . Let (13) B be an answer set of Π(D′,σ1′,a) such that (14) σ2′={l:h(l,1)∈B} . By definition of σ2′ , (15) h(¬is_impossible,1)∈B . Let all rules of Π(D′,σ1′,a) that define is_impossible be the top program of Π(D′,σ1′,a) , denoted by Ptop′ . Without loss of generality, assume there is only one executability condition for a be of the form (7). Let rule r and r′ of the form (8) and (10) respectively be rules, corresponding to the executability condition, in Π(D,σ1,a) and Π(D′,σ1′,a) respectively. (15) implies that (16) body(r′) is not satisfied by B . Now let A1=(B−impF) . (13) implies that (17) A1 is the answer set of the bottom program of Π(D′,σ1′,a) which is denoted by Pbot′ . Let all rules of Π(D,σ1,a) that define occurs be the top program of Π(D,σ1,a) , denoted by Ptop . Let the bottom program of Π(D,σ1,a) be denoted by Pbot . We can verify Pbot=Pbot′ . So, A1 is an answer set of Pbot by (17). (16) implies that body(r′) is not satisfied by A1 too. Therefore, body(r) (8) is not satisfied by A1 . Hence, by splitting lemma and the rules of Ptop , there is a set A such that A1⊆A and A is an answer set of Π(D,σ1,a) . Note that A−A1 contains only occurs(a,0) and some ¬occurs . Therefore, {l:h(l,1)∈A}={l:h(l,1)∈A1}={l:h(l,1)∈(B−impF)}=σ2′−{¬is_impossible}=σ2 . Hence, 〈σ1,a,σ2〉 is a transition of T(D) . Now we are in a position to prove the proposition. α=a0,…,an−1 is a solution of P iff for every initial state σ0 of P , σ0a0…an−1σn is a path of T(D) and g⊆σn iff for every initial state σ0∪{¬is_impossible} of P′ , (σ0∪{¬is_impossible})a0…an−1(σn∪{¬is_impossible}) is a path of T(D′) and (g∪{¬is_impossible})⊆(σn∪{¬is_impossible}) iff α is a solution of P′ .  ▀ The next proposition addresses the correctness of our algorithm for conformant planning. T heorem 3 [Correctness of the Proposed Conformant Planning Algorithm] Let P=〈D,Σ,g〉 be a conformant planning problem where D is consistent and deterministic and let τn(P′) be the epistemic program obtained from P by E . Let gP(n) denote the first two rules of part 5 of E where step S is ground to n . Then (a) For every world view W of τn(P′) , belief sets of W coincide on atoms of the form occurs(a,i) . (b) A sequence α=a0,…,an−1 of actions is a solution of P of length n iff there is a world view W of τn(P′)−gP(n) such that occurs(a,k) belongs to its belief sets iff a is the k th element ak of α . P roof . In this proof, we divide the program τn(P′) into three parts: Π1 consisting of the first two rules in part 8 of the description τn . Π2 consisting of rules described in parts 4–7 and rule 3 of part 8. Π3 consisting of the last two rules of part 8. (a) For any belief set S of W , assume occurs(a,i)∈S for some action a and step i . The modal reduct of (Π1∪Π2)W contains occurs(a,i). Since W is the collection of all answer sets of (Π1∪Π2)W , for any S1∈W , the fact occurs(a,i) belongs to S1 . Therefore, (18) belief sets of W coincide on atoms of the form occurs(a,i) . (b) (19) ⟹ : we prove first if a sequence α=a0,…,an−1 of actions is a solution of P′ of length n , then there is a world view W of τn(P′)−gP(n) such that occurs(a,k) belongs to its belief sets iff a is the k th element ak of α . Π2 is further partitioned as follows: Π21 consisting of rules in parts 4, 5 and 7 of E , and Π22 consisting of rules in part 6 (i.e. goal(i) :- h(g′,i) and rule 3 (i.e. success :- goal(n) ) of part 8. Clearly τn(P′)=Π1∪Π21∪Π22∪Π3 . Let Πi where i∈0..n be the rules of Π where the maximum step number involved is i . Let Σ′={σ10,…,σm0} . Since α=a0,…,an−1 is a solution of P′ , there are m paths of T(D′) and let the ith path be σi0,a0,…,an−1,σin . Let complement(a0,i) be defined as {¬occurs(a1,i):a1≠a0} . For a state σ , h(σ,i)={h(l,i):l∈σ} . For a set of states Σ , h(Σ,i)={h(σ,i):σ∈Σ} . For j∈0..n and i∈1..m , let Sij=h(σi0,0)∪…∪h(σij,j)∪{occurs(a0,0)}∪…∪{occurs(aj−1,j−1)}∪complement(a0,0)∪…∪complement(aj−1,j−1) (Note {occurs(a−1,−1)} and complement(a−1,−1) denote {} ), and Wj={S1j,…,Smj} . As the last state of every path achieves goal g′ , for any S of Wn , h(¬is_impossible,n)∈S . Since is_impossible is inertial and no action can cause ¬is_impossible , we have (20) for any j and any belief set S of Wj and any step i(≤j) , h(¬is_impossible,i)∈S . Let P=Π1∪Π21 . We next prove by induction on j that for all j∈0..n , Wj is the world view of Pj−gP(j) . Base case: j=0 . (P0−gP(0))W0=(Π21)0 . By part 4 of E , h(Σ′,0) , i.e. W0 , is all the answer sets of (Π21)0 . So, W0 is the answer sets of (P0−gP(0))W0 and thus the world view of P0−gP(0) . Inductive assumption: let the statement hold for k<n . We will prove that the statement holds for k+1 . We first prove for any Si(k+1)∈Wk+1 , it is an answer set of (Pk+1−gP(k+1))Wk+1 (22). We can verify that (21) (Pk+1−gP(k+1))Wk+1={occurs(ak,k)}∪{¬occurs(a1,k)←occurs(a2,k) , a1≠a2}∪((Π21)k+1−(Π21)k)∪(Pk−gP(k))Wk . Let ((Π21)k+1−(Π21)k) be the top program of (Pk+1−gP(k+1))Wk+1 . By inductive assumption, Wk is a world view of Pk−gP(k) , and we can show Sik is an answer set of (Pk−gP(k))Wk . Since there is a transition 〈σk,ak,σk+1〉 in T(D′) , Π(D′,σk,ak) has an answer set A . After renaming step 0 to k and 1 to k+1 in Π(D′,σk,ak) , Π(D′,σk,ak)=((Π21)k+1−(Π21)k)∪kP∪h(σk)∪{occurs(ak,k)} where kP represents the rules for state constraint at step k . h(σk)∪{occurs(ak,k)}∪h(σk+1) is an answer set of Π(D′,σk,ak) . Since kP⊆Pk , h(σk)⊆Sik and no rule of (Π21)k+1−(Π21)k produces any element of h(σk) , by splitting lemma, Sik∪{occurs(ak,k)}∪complement(ak,k)∪h(σk+1) , i.e. Si(k+1) , is an answer set of ((Π21)k+1−(Π21)k)∪{occurs(ak,k)}∪{¬occurs(a1,k)←occurs(a2,k),a1≠a2}∪Sik , i.e. (Pk+1−gP(k+1))Wk+1 , and thus (22) Si(k+1) is an answer set of (Pk+1−gP(k+1))Wk+1 . We now show that for any answer set A of (Pk+1−gP(k+1))Wk+1 (see (21)), A∈Wk+1 . Let {occurs(ak,k)}∪complement(ak,k) be the answer set of {occurs(ak,k)}∪{¬occurs(a1,k)←occurs(a2,k),a1≠a2} . Let ((Π21)k+1−(Π21)k) be the top program of ((Π21)k+1−(Π21)k)∪(Pk−gP(k))Wk∪({occurs(ak,k)}∪{¬occurs(a1,k) ←occurs(a2,k),a1≠a2}) . By splitting lemma and the inductive assumption (i.e. Wk is the world view of Pk ), there exists Sik∈Wk such that (23) A is an answer set of Sik∪((Π21)k+1−(Π21)k)∪{occurs(ak,k)}∪complement(ak,k) . By definition of Sik , (24) Sik=Si(k−1)∪h(σik,k)∪{occurs(ak−1,k−1)}∪complement(ak−1,k−1) . (Note Si(−1)={} .) Since 〈σk,ak,σk+1〉 is a transition of T(D′) , Π(D′,σk,a) , i.e. (after renaming steps 0 and 1 by k and k+1 respectively) ((Π21)k+1−(Π21)k)∪h(σik,k)∪{occurs(ak,k)} has an answer set h(σik,k)∪{occurs(ak,k)}∪h(σi(k+1),k+1) . By (20), ak is executable in σk . Since D is consistent and deterministic, h(σik,k)∪{occurs(ak,k)}∪h(σi(k+1),k+1) is the unique answer set of Π(D′,σk,a) , which, together with (23) and (24), implies Sik∪((Π21)k+1−(Π21)k)∪({occurs(ak,k)}∪complement(ak,k)) , i.e. (Pk+1−gP(k+1))Wk+1 , has a unique answer set. Hence, A=Sik∪h(σi(k+1),k+1)∪({occurs(ak,k)}∪complement(ak,k))=Si(k+1) . Therefore, (25) A∈Wk+1 , which, together with (22), implies (26) Wk+1 is the world view of Pk+1−gP(k+1) . Hence (27) For any j∈0..n , Wj is the world view of Pj−gP(j) . Now let Π22 be the top program of (Pn−gP(n))∪Π22 . We construct W′ from Wn as follows: for any Sin∈Wn , for any j , if h(g′,j)⊆h(σij,j) , add to Sin goal(j) . If goal(n)∈Sin , add success to Sin . Add Sin to W′ . Since g′ is achieved in σin for all i∈1..m , every belief set of W′ contains success . It can be shown that W′ is the collection of the answer sets of ((Pn−gP(n))∪Π22)W′ by splitting lemma, i.e. W′ is a world view of (Pn−gP(n))∪Π22 . Since W′⊨Ksuccess , by Proposition 1, W′ is the world view of (Pn−gP(n))∪Π22∪Π3 , i.e. τn(P′)−gP(n) . By the construction of W′ , we can verify that occurs(a,k) belongs to its belief sets iff a is the k th element ak of α . (28) ⟸ : we prove now the following. Assume there is a world view W of τn(P′)−gP(n) . By (a) of the proposition, the seqence of actions α=a0,…,an−1 where occurs(ai,i)∈S for some S∈W is unique. (For simplicity, if there is no action at step i in W , we assume there is a dummy action which doesn't have any effects or executbility condition.) We will prove α is a solution of P′ , i.e. for any initial state σ0∈Σ′ , we prove there is a path σ0,a0,…,an−1σn in T(D′) and g′⊆σn . Since W is a world view of τn(P′)−gP(n) , by Proposition 1, W is a world view of Π1∪Π2−gP(n) and (29) W⊨Ksuccess , which implies (30) for any belief set S∈W and any i∈0..n , h(¬is_impossible,i)∈S . In the following, we construct the path σ0,a0,…,an−1σn . Since σ0 is an initial state, (31) h(σ0,0) is an answer set of (Π21)0 . Let Pi be ((Π1∪Π2−gP(i))W)i . Let botP(i) be the rules of Pi that defining occurs and initial situation. Let botP(i) is the bottom program of Πi . h(σ0,0)∪{occurs(a0,0)}∪complement(a0,0) is an answer set of botP(i) . Let topP(i)=Pi−botP(i) and topP(i) be the top program. Let A0=h(σ0,0)∪{occurs(a0,0)}∪complement(a0,0) . We next show A0∪topP(1) has an answer set. Let the executibility condition of a0 be (32) impossible a0if l1,…,le , whose corresponding statement in P′ is (33) a0causes is_impossibleif l1,…,le . Consider case 1: l1,…,le is satisfied by σ0 , i.e. a0 is not executable at σ0 . By the ASP rule of P1 encoding the statement (33), h(is_impossible,1) has to be believed, which disables all rules obtained from causal laws that define a fluent in step 1 because noth(is_impossible,1) is in their body. We can verify that P1 has an answer set, in fact a unique answer set, which contains h(σ0),occurs(a0,0) and h(is_impossible,1) . Consider case 2: l1,…,le is not satisfied by σ0 , i.e. a0 is executable at σ0 . We now prove A0∪topP(1) has an answer set by contradiction. Assume (34) A0∪topP(1) has no answer set. Let P′ be obtained from A0∪topP(1) by removing literals involving is_impossible from the bodys of the rules that encode the dynamic and static causal laws. Since h(¬is_impossible,0)∈topP(1) and the body of the rule encoding (33) is not satisfied by h(σ0,0) (because a0 is not executable in σ0 ), (35) P′ has an answer iff A0∪topP(1) has an answer set. Let the rules of P′ defining is_impossible be the top program of P′ . We denote the top program by topP′ and the bottome btmP′ . By splitting lemma and the nature of topP′ , P′ has an answer set iff btmP′ has an answer set. Hence, by (34) and (35), (36) btmP′ has no answer set. (37) Π(D,σ0−{¬is_impossible},a0)=btmP′∪{¬occurs(a0,0)←l1,…,lk} . Since a0 is executable in σ0−{¬is_impossible} and D is consistent, (38) Π(D,σ0−{¬is_impossible},a0) has an answer set. However, as btmP′ has no answer set (36), one can verify that btmP′∪{¬occurs(a0,0) ←l1,…,lk} , i.e. Π(D,σ0−{¬is_impossible},a0) (37), has no answer set, contradicting (38). Hence, A0∪topP(1) has an answer set. Since D is deterministic, Π(D,σ0−{¬is_impossible},a0) has a unique answer set. Hence, by (37), btmP′ has a unique answer set. Given the nature of topP′ , by splitting lemma, A0∪topP(1) has a unique answer set. In summary, A0∪topP(1) has a unique answer set. Let it be A . Let σ1={l,h(l,1)∈A} . In a similar fashion, (39) we can construct σ2,…,σn . We next show for all i∈0..(n−1) ai is executable (41) and g′⊆σn . Let Sn be h(σ0)∪⋯∪h(σn)∪{occurs(a0,0)}∪⋯∪{occurs(an−1,n−1)∪ complement(a0,0)∪…∪complement(an−1,n−1) . Add goal(i)(i∈0..n) to Sn if g′⊆σi , and add success to Sn if goal(n)∈Sn . We can show, by induction on n , that (40) Sn is an answer set of ((Π1∪Π2−gP(n))W)n . The base case n=0 . S0=h(σ0) it is an answer set of Π20 . Assume (40) holds for k . We prove it holds for k+1 . ((Π1∪Π2−gP(k+1))W)k+1=((Π1∪Π2−gP(k))W)k∪{occurs(ak,k)}∪complement(ak,k)∪((Π2)k+1−(Π2)k) . By induction assumption, Sk is an answer set of ((Π1∪Π2−gP(k))W)k . Let ((Π1∪Π2−gP(k))W)k be the bottom program of ((Π1∪Π2−gP(k+1))W)k+1 . Similar to the proof that A0∪P1 has an answer set, we can show that Sik∪{occurs(ak,k)}∪complement(ak,k)∪((Π2)k+1−(Π2)k) , has an answer set. By the construction of σn , we have that Si(k+1) is the answer set of Sik∪{occurs(ak,k)}∪complement(ak,k)∪((Π2)k+1−(Π2)k) , Si(k+1) , and thus of ((Π1∪Π2−gP(k+1))W)k+1 . Therefore, Sn∈W . By (30), for any i∈0..n , h(¬is_impossible,i)∈Sn . Hence, (41) for all i∈0..n−1 , ai is executable at σi . Since for any i∈0..n−1 , ai is executable, Π(D′,σk,ak) has an answer set iff σk∪{occurs(ak,k)}∪complement(ak,k)∪((Π2)k+1−(Π2)k) has an answer set. As shown in the inductive proof above, the latter has a unique answer. Hence Π(D′,σk,ak) has the unique answer which coincides Si(k+1) with all literals of the form ¬occurs , goal or success removed. Therefore, (42) for all k∈0..n−1 , 〈σk,ak,σk+1〉 is a transition of T(D′) . By (29), g′⊆σn , which, together with (42) and (41), implies that a0,…,an−1 is a solution of P′ . Finally, by (19), (28) and Lemma 5 on the equivalence of P and P′ , the statement (b) of the proposition holds.  ▀ C Alternative modal reduct The following is due to an anonymous reviewer. It is provided in the hope that it offers further insight into the development of the semantics proposed in this article, as well as an alternative understanding of the new semantics that some may find preferable to that presented in Section 3.2 , in particular Definition 3. Maintain the same syntax as presented, but operator M is actually defined in terms of K as follows: 6 This just implies that K can be applied on some L of the form ℓ (an objective literal) or not   ℓ (its default negation). The reduct of KL with respect to a pointed ES structure W , written (KL)W , is defined as follows: 7 The modal reduct of an epistemic logic program Π with respect to W , written ΠW , is the result of replacing each occurrance of K L in Π with (K L)W . Finally, we apply the standard transformations for nested expressions in logic programs per [ 20 ] to remove triple negations notnotnot φ⇔   not   φ and truth constants not   ⊥⇔⊤ , etc. These transformations guarantee, for instance, that not M ℓ , whose definition is notnot Knot ℓ , becomes strongly equivalent to K   not ℓ . 1 See Appendix C for an alternative modal reduct that may provide further intuition. 2 By nested expressions we mean those of the form notnot ℓ in the body of a rule as defined in [ 20 ]. For ease of presentation, we may use ASP to mean ASP with nested expressions . 3 The proofs for soundness and completeness of the algorithm are given in Appendix A. 4 ELPS run times are not believed to have benefitted from having a sorted signature. 5 Proof of this theorem is presented in Appendix B. 6 Defining M in terms of K was first proposed in [ 16 ]. 7 Also add to Definition 1 the fact that 〈A,W〉⊨K   not   ℓ if ∀S∈W:ℓ∉S . © The Author, 2015. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com
TI - The language of epistemic specifications (refined) including a prototype solver
JF - Journal of Logic and Computation
DO - 10.1093/logcom/exv065
DA - 2015-09-09
UR - https://www.deepdyve.com/lp/oxford-university-press/the-language-of-epistemic-specifications-refined-including-a-prototype-g0o9fRHYzW
SP - 1
VL - Advance Article
IS - 
DP - DeepDyve
ER -