BCS © 2017
Formal Aspects of Computing (2017) 29: 805-832
Birkhoff style calculi for hybrid logics
Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan
Abstract. We develop an abstract proof calculus for hybrid logics whose sentences are (hybrid) Horn clauses,and
we prove a Birkhoff completeness theorem for hybrid logics in the general setting provided by the institution theory.
This result is then applied to particular cases of hybrid logics with user-deﬁned sharing, where the ﬁrst-order
variables in quantiﬁed sentences are interpreted uniformly across worlds.
Keywords: Institution, Horn clause, Birkhoff calculus, Hybrid logic, Reconﬁgurable system
In 1935, Birkhoff [Bir35] ﬁrst proved a completeness theorem for equational logic, in the unsorted case. Goguen
and Meseguer [GM85], giving a sound and complete system of proof rules for many-sorted equational deduction,
generalised the completeness theorem of Birkhoff to the completeness of many-sorted equational logic and
provided simultaneously a full algebraisation of many-sorted equational deduction. Codescu and G
cast the result in the category-based setting of the institution theory [GB92], separating clearly the details of
concrete logics from the logical-independent aspects of the completeness property. Institution theory is a category-
based model theory that arose about three decades ago within formal methods as a response to the explosion in
the population of logics in use there; its original aim is to develop as much computing science as possible in a
general uniform way independently of particular logical systems. In this paper, we deﬁne an abstract notion of
Horn clause and we prove a Birkhoff completeness result for hybrid logics in the general setting provided by the
Hybrid logics [Bla00] are extensions of standard modal logics, involving symbols that name individual states
in models. Their history can be traced back to work of Arthur Prior in the ﬁfties [Pri67]. The subject was further
developed in contributions such as [PT91, AB01a, Bra11]. Recently, hybrid logics were developed at an abstract
institution theoretic level in works such as [MMDB11, Dia16b, DM16, G
15b]. The ability to refer to
speciﬁc states has several advantages from the point of views of logic and formal speciﬁcation. For example, it
has been argued [Bra11] that hybrid logics allow a more uniform proof theory than non-hybrid modal logics.
From a software engineering perspective, hybrid logics offer a generic framework to approach the speciﬁcation
of reconﬁgurable systems, i.e. systems with reconﬁgurable features managing the dynamic evolution of their
conﬁgurations in response to external stimuli or internal performance indicators. See [SC11] for an overview of
the software reconﬁguration paradigm.
Correspondence and offprint requests to:D.G
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