The unification problem in a propositional logic is to determine, given a formula F, whether there exists a substitution s such that s(F) is in that logic. In that case, s is a unifier of F. When a unifiable formula has minimal complete sets of unifiers, the formula is either infinitary, finitary, or unitary, depending on the cardinality of its minimal complete sets of unifiers. In this paper, we study the unification types of some modal logics determined by classes of deterministic frames.
The unification problem in a normal modal logic is to determine, given a formula F, whether there exists a substitution s such that s(F) is in that logic. In that case, s is a unifier of F. We shall say that a set of unifiers of a unifiable formula F is complete if for all unifiers s of F, there exists a unifier t of F in that set such that t is more general than s. When a unifiable formula has no minimal complete set of unifiers, the formula is nullary. In this paper, we prove that KB, KDB and KTB possess nullary formulas.
Concolic testing is a popular dynamic validation technique that can be used for both model checking and automatic test case generation. We have recently introduced concolic testing in the context of logic programming. In contrast to previous approaches, the key ingredient in this setting is a technique to generate appropriate run-time goals by considering all possible ways an atom can unify with the heads of some program clauses. This is called "selective" unification. In this paper, we show that the existing algorithm is not complete and explore different alternatives in order to have a sound…
Matching Logic is a framework for specifying programming language semantics and reasoning about programs. Its formulas are called patterns and are built with variables, symbols, connectives and quantifiers. A pattern is a combination of structural components (term patterns), which must be matched, and constraints (predicate patterns), which must be satisfied. Dealing with more than one structural component in a pattern could be cumbersome because it involves multiple matching operations. A source for getting patterns with many structural components is the conjunction of patterns. Here, we prop…
We show that the unification problem `is there a substitution instance of a given formula that is provable in a given logic?' is undecidable for basic modal logics K and K4 extended with the universal modality. It follows that the admissibility problem for inference rules is undecidable for these logics as well. These are the first examples of standard decidable modal logics for which the unification and admissibility problems are undecidable. We also prove undecidability of the unification and admissibility problems for K and K4 with at least two modal operators and nominals (instead of the u…
The Description Logic EL has recently drawn considerable attention since, on the one hand, important inference problems such as the subsumption problem are polynomial. On the other hand, EL is used to define large biomedical ontologies. Unification in Description Logics has been proposed as a novel inference service that can, for example, be used to detect redundancies in ontologies. The main result of this paper is that unification in EL is decidable. More precisely, EL-unification is NP-complete, and thus has the same complexity as EL-matching. We also show that, w.r.t. the unification type,…
Analysis of (partial) groundness is an important application of abstract interpretation. There are several proposals for improving the precision of such an analysis by exploiting type information, icluding our own work with Hill and King, where we had shown how the information present in the type declarations of a program can be used to characterise the degree of instantiation of a term in a precise and yet inherently finite way. This approach worked for polymorphically typed programs as in Goedel or HAL. Here, we recast this approach following works by Codish, Lagoon and Stuckey. To formalise…
We investigate properties of the formula $p \to \Box p$ in the basic modal logic K. We show that K satisfies an infinitary weaker variant of the rule of margins $ϕ\to \Boxϕ/ ϕ, \negϕ$, and as a consequence, we obtain various negative results about admissibility and unification in K. We describe a complete set of unifiers (i.e., substitutions making the formula provable) of $p \to \Box p$, and use it to establish that K has the worst possible unification type: nullary. In well-behaved transitive modal logics, admissibility and unification can be analyzed in terms of projective formulas, introdu…
Manna and Waldinger's theory of substitutions and unification has been verified using the Cambridge LCF theorem prover. A proof of the monotonicity of substitution is presented in detail, as an example of interaction with LCF. Translating the theory into LCF's domain-theoretic logic is largely straightforward. Well-founded induction on a complex ordering is translated into nested structural inductions. Correctness of unification is expressed using predicates for such properties as idempotence and most-generality. The verification is presented as a series of lemmas. The LCF proofs are compared …
Nominal Logic is a version of first-order logic with equality, name-binding, renaming via name-swapping and freshness of names. Contrarily to higher-order logic, bindable names, called atoms, and instantiable variables are considered as distinct entities. Moreover, atoms are capturable by instantiations, breaking a fundamental principle of lambda-calculus. Despite these differences, nominal unification can be seen from a higher-order perspective. From this view, we show that nominal unification can be reduced to a particular fragment of higher-order unification problems: Higher-Order Pattern U…
We study admissibility of inference rules and unification with parameters in transitive modal logics (extensions of K4), in particular we generalize various results on parameter-free admissibility and unification to the setting with parameters. Specifically, we give a characterization of projective formulas generalizing Ghilardi's characterization in the parameter-free case, leading to new proofs of Rybakov's results that admissibility with parameters is decidable and unification is finitary for logics satisfying suitable frame extension properties (called cluster-extensible logics in this pap…
We propose a "modal linear logic" to reformulate intuitionistic modal logic S4 (IS4) in terms of linear logic, establishing an S4-version of Girard translation from IS4 to it. While the Girard translation from intuitionistic logic to linear logic is well-known, its extension to modal logic is non-trivial since a naive combination of the S4 modality and the exponential modality causes an undesirable interaction between the two modalities. To solve the problem, we introduce an extension of intuitionistic multiplicative exponential linear logic with a modality combining the S4 modality and the ex…
Propositional term modal logic is interpreted over Kripke structures with unboundedly many accessibility relations and hence the syntax admits variables indexing modalities and quantification over them. This logic is undecidable, and we consider a variable-free propositional bi-modal logic with implicit quantification. Thus $[\forall] α$ asserts necessity over all accessibility relations and $[\exists] α$ is classical necessity over some accessibility relation. The logic is associated with a natural bisimulation relation over models and we show that the logic is exactly the bisimulation invari…
Building on our previous work on hybrid polyadic modal logic we identify modal logic equivalents for Matching Logic, a logic for program specification and verification. This provides a rigorous way to transfer results between the two approaches, which should benefit both systems.
We show undecidability of the satisfiability problem of what is arguably the simplest non-sub-Boolean modal logic with an implicit notion of binding. This work enriches the series of existing results of undecidability of modal logics with binders, which started with Hybrid Logics and continued with Memory Logics.
Extending and generalizing the approach of 2-sequents (Masini, 1992), we present sequent calculi for the classical modal logics in the K, D, T, S4 spectrum. The systems are presented in a uniform way-different logics are obtained by tuning a single parameter, namely a constraint on the applicability of a rule. Cut-elimination is proved only once, since the proof goes through independently from the constraints giving rise to the different systems. A sequent calculus for the discrete linear temporal logic ltl is also given and proved complete. Leitmotiv of the paper is the formal analogy between…
Term modal logics (TML) are modal logics with unboundedly many modalities, with quantification over modal indices, so that we can have formulas of the form $\exists y. \forall x. (\Box_x P(x,y) \supset\Diamond_y P(y,x))$. Like First order modal logic, TML is also "notoriously" undecidable, in the sense that even very simple fragments are undecidable. In this paper, we show the decidability of one interesting fragment, that of two variable TML. This is in contrast to two-variable First order modal logic, which is undecidable.
We present a syntactic abstraction method to reason about first-order modal logics by using theorem provers for standard first-order logic and for propositional modal logic.
We investigate the expressivity and computational complexity of two modal logics on finite forests equipped with operators to reason on submodels. The logic ML(|) extends the basic modal logic ML with the composition operator | from static ambient logic, whereas ML(*) contains the separating conjunction * from separation logic. Though both operators are second-order in nature, we show that ML(|) is as expressive as the graded modal logic GML (on finite trees) whereas ML(*) lies strictly between ML and GML. Moreover, we establish that the satisfiability problem for ML(*) is Tower-complete, wher…
The unification algorithm is at the core of the logic programming paradigm, the first unification algorithm being developed by Robinson [5]. More efficient algorithms were developed later [3] and I introduce here yet another efficient unification algorithm centered on a specific data structure, called the Unification Table.
Higher dimensional automata (HDA) are a model of concurrency that can express most of the traditional partial order models like Mazurkiewicz traces, pomsets, event structures, or Petri nets. Modal logics, interpreted over Kripke structures, are the logics for reasoning about sequential behavior and interleaved concurrency. Modal logic is a well behaved subset of first-order logic; many variants of modal logic are decidable. However, there are no modal-like logics for the more expressive HDA models. In this paper we introduce and investigate a modal logic over HDAs which incorporates two modali…
We prove a generic completeness result for a class of modal fixpoint logics corresponding to flat fragments of the two-way mu-calculus, extending earlier work by Santocanale and Venema. We observe that Santocanale and Venema's proof that least fixpoints in the Lindenbaum-Tarski algebra of certain flat fixpoint logics are constructive, using finitary adjoints, no longer works when the converse modality is introduced. Instead, our completeness proof directly constructs a model for a consistent formula, using the induction rule in a way that is similar to the standard completeness proof for propo…
In this paper we consider Modal Team Logic, a generalization of Classical Modal Logic in which it is possible to describe dependence phenomena between data. We prove that most known fragment of Full Modal Team Logic allow the elimination of the so called "existential bisimulation quantifiers", where the existence of a certain set is made modulo bisimulation. As a consequence, we prove that these fragments enjoy the Uniform Interpolation Property.
We study extensions of standard description logics to the framework of polyadic modal logic. We promote a natural approach to such logics via general relation algebras that can be used to define operations on relations of all arities. As a concrete system to illustrate our approach, we investigate the polyadic version of ALC extended with relational permutation operators and tuple counting. The focus of the paper is conceptual rather than technical.
The syntax of modal graphs is defined in terms of the continuous cut and broken cut following Charles Peirce's notation in the gamma part of his graphical logic of existential graphs. Graphical calculi for normal modal logics are developed based on a reformulation of the graphical calculus for classical propositional logic. These graphical calculi are of the nature of deep inference. The relationship between graphical calculi and sequent calculi for modal logics is shown by translations between graphs and modal formulas.
Cathoristic logic is a multi-modal logic where negation is replaced by a novel operator allowing the expression of incompatible sentences. We present the syntax and semantics of the logic including complete proof rules, and establish a number of results such as compactness, a semantic characterisation of elementary equivalence, the existence of a quadratic-time decision procedure, and Brandom's incompatibility semantics property. We demonstrate the usefulness of the logic as a language for knowledge representation.
Canonical models are of central importance in modal logic, in particular as they witness strong completeness and hence compactness. While the canonical model construction is well understood for Kripke semantics, non-normal modal logics often present subtle difficulties - up to the point that canonical models may fail to exist, as is the case e.g. in most probabilistic logics. Here, we present a generic canonical model construction in the semantic framework of coalgebraic modal logic, which pinpoints coherence conditions between syntax and semantics of modal logics that guarantee strong complet…
A logic has uniform interpolation if its formulas can be projected down to given subsignatures, preserving all logical consequences that do not mention the removed symbols; the weaker property of (Craig) interpolation allows the projected formula -- the interpolant -- to be different for each logical consequence of the original formula. These properties are of importance, e.g., in the modularization of logical theories. We study interpolation in the context of coalgebraic modal logics, i.e. modal logics axiomatized in rank 1, restricting for clarity to the case with finitely many modalities. E…
We present a fuzzy (or quantitative) version of the van Benthem theorem, which characterizes propositional modal logic as the bisimulation-invariant fragment of first-order logic. Specifically, we consider a first-order fuzzy predicate logic along with its modal fragment, and show that the fuzzy first-order formulas that are non-expansive w.r.t. the natural notion of bisimulation distance are exactly those that can be approximated by fuzzy modal formulas.
In probabilistic transition systems, behavioural metrics provide a more fine-grained and stable measure of system equivalence than crisp notions of bisimilarity. They correlate strongly to quantitative probabilistic logics, and in fact the distance induced by a probabilistic modal logic taking values in the real unit interval has been shown to coincide with behavioural distance. For probabilistic systems, probabilistic modal logic thus plays an analogous role to that of Hennessy-Milner logic on classical labelled transition systems. In the quantitative setting, invariance of modal logic under …
This paper studies the complexity of classical modal logics and of their extension with fixed-point operators, using translations to transfer results across logics. In particular, we show several complexity results for multi-agent logics via translations to and from the mu-calculus and modal logic, which allow us to transfer known upper and lower bounds. We also use these translations to introduce a terminating tableau system for the logics we study, based on Kozen's tableau for the mu-calculus, and the one of Fitting and Massacci for modal logic.
The classical propositional logic is known to be sound and complete with respect to the set semantics that interprets connectives as set operations. The paper extends propositional language by a new binary modality that corresponds to partial recursive function type constructor under the above interpretation. The cases of deterministic and non-deterministic functions are considered and for both of them semantically complete modal logics are described and decidability of these logics is established.
This paper is focused on the study of modal logics defined from valued Kripke frames, and particularly, on computability and expressibility questions of modal logics of transitive Kripke frames evaluated over certain residuated lattices. It is shown that a large family of those logics -- including the ones arising from the standard MV and Product algebras -- yields an undecidable consequence relation. Later on, the behaviour of transitive modal Lukasiewicz logic is compared with that of its non transitive counterpart, exhibiting some particulars concerning computability and equivalence with ot…
We present deductive systems for various modal logics that can be obtained from the constructive variant of the normal modal logic CK by adding combinations of the axioms d, t, b, 4, and 5. This includes the constructive variants of the standard modal logics K4, S4, and S5. We use for our presentation the formalism of nested sequents and give a syntactic proof of cut elimination.
Propositional and modal inclusion logic are formalisms that belong to the family of logics based on team semantics. This article investigates the model checking and validity problems of these logics. We identify complexity bounds for both problems, covering both lax and strict team semantics. By doing so, we come close to finalising the programme that ultimately aims to classify the complexities of the central reasoning problems for modal and propositional dependence, independence, and inclusion logics.
Anti-unification refers to the process of generalizing two (or more) goals into a single, more general, goal that captures some of the structure that is common to all initial goals. In general one is typically interested in computing what is often called a most specific generalization, that is a generalization that captures a maximal amount of shared structure. In this work we address the problem of anti-unification in CLP, where goals can be seen as unordered sets of atoms and/or constraints. We show that while the concept of a most specific generalization can easily be defined in this contex…
One way of proving theorems in modal logics is translating them into the predicate calculus and then using conventional resolution-style theorem provers. This approach has been regarded as inappropriate in practice, because the resulting formulas are too lengthy and it is impossible to show the non-theoremhood of modal formulas. In this paper, we demonstrate the practical feasibility of the (relational) translation method. Using a state-of-the-art theorem prover for first-order predicate logic, we proved many benchmark theorems available from the modal logic literature. We show the invalidity …
The Poison Game is a two-player game played on a graph in which one player can influence which edges the other player is able to traverse. It operationalizes the notion of existence of credulously admissible sets in an argumentation framework or, in graph-theoretic terminology, the existence of non-trivial semi-kernels. We develop a modal logic (poison modal logic, PML) tailored to represent winning positions in such a game, thereby identifying the precise modal reasoning that underlies the notion of credulous admissibility in argumentation. We study model-theoretic and decidability properties…
Inquisitive modal logic, InqML, is a generalisation of standard Kripke-style modal logic. In its epistemic incarnation, it extends standard epistemic logic to capture not just the information that agents have, but also the questions that they are interested in. Technically, InqML fits within the family of logics based on team semantics. From a model-theoretic perspective, it takes us a step in the direction of monadic second-order logic, as inquisitive modal operators involve quantification over sets of worlds. We introduce and investigate the natural notion of bisimulation equivalence in the …
A modal logic is \emph{non-iterative} if it can be defined by axioms that do not nest modal operators, and \emph{rank-1} if additionally all propositional variables in axioms are in scope of a modal operator. It is known that every syntactically defined rank-1 modal logic can be equipped with a canonical coalgebraic semantics, ensuring soundness and strong completeness. In the present work, we extend this result to non-iterative modal logics, showing that every non-iterative modal logic can be equipped with a canonical coalgebraic semantics defined in terms of a copointed functor, again ensuri…
This paper presents a many-sorted polyadic modal logic that generalizes some of the existing approaches. The algebraic semantics has led us to a many-sorted generalization of boolean algebras with operators, for which we prove the analogue of the Jónsson-Tarski theorem. While the transition from the mono-sorted logic to many-sorted one is a smooth process, we see our system as a step towards deepening the connection between modal logic and program verification, since our system can be seen as the propositional fragment of Matching logic, a first-order logic for specifying and reasoning about p…
We formulate and prove logical characterizations of crisp simulations and crisp directed simulations between fuzzy labeled transition systems with respect to fuzzy modal logics that use a general t-norm-based semantics. The considered logics are fragments of the fuzzy propositional dynamic logic with the Baaz projection operator. The logical characterizations concern preservation of positive existential (respectively, positive) modal formulas under crisp simulations (respectively, crisp directed simulations), as well as the Hennessy-Milner property of such simulations.
In this article, we start with a two-player game that models communication under adverse circumstances in everyday life and study it from the perspective of a modal logic of graphs, where links can be deleted locally according to definitions available to the adversarial player. We first introduce a new language, semantics, and some typical validities. We then formulate a new type of first-order translation for this modal logic and prove its correctness. Then, a novel notion of bisimulation is proposed which leads to a characterization theorem for the logic as a fragment of first-order logic, a…
We study the computational complexity of model checking and satisfiability problems of polyadic modal logics extended with permutations and Boolean operators on accessibility relations. First, we show that the combined complexity of the model checking problem for the resulting logic is PTime-complete. Secondly, we show that the satisfiability problem of polyadic modal logic extended with negation on accessibility relations is ExpTime-complete. Finally, we show that the satisfiability problem of polyadic modal logic with permutations and Boolean operators on accessibility relations is ExpTime-c…
In this paper, we investigate the many-valued version of coalgebraic modal logic through predicate lifting approach. Coalgebras, understood as generic transition systems, can serve as semantic structures for various kinds of modal logics. A well-known result in coalgebraic modal logic is that its completeness can be determined at the one-step level. We generalize the result to the finitely many-valued case by using the canonical model construction method. We prove the result for coalgebraic modal logics based on three different many-valued algebraic structures, including the finitely-valued Łu…
Non-normal modal logics, interpreted on neighbourhood models which generalise the usual relational semantics, have found application in several areas, such as epistemic, deontic, and coalitional reasoning. We present here preliminary results on reasoning in a family of modal description logics obtained by combining ALC with non-normal modal operators. First, we provide a framework of terminating, correct, and complete tableau algorithms to check satisfiability of formulas in such logics with the semantics based on varying domains. We then investigate the satisfiability problems in fragments of…
We present an algebraic characterization of the complexity classes Logspace and NLogspace, using an algebra with a composition law based on unification. This new bridge between unification and complexity classes is inspired from proof theory and more specifically linear logic and Geometry of Interaction. We show how unification can be used to build a model of computation by means of specific subalgebras associated to finite permutations groups. We then prove that whether an observation (the algebraic counterpart of a program) accepts a word can be decided within logarithmic space. We also show…
