Basic Modal Logic Research Articles

The expressive power of modal logic with inclusion atoms

Modal inclusion logic is the extension of basic modal logic with inclusion atoms, and its semantics is defined on Kripke models with teams. A team of a Kripke model is just a subset of its domain. In this paper we give a complete characterisation for the expressive power of modal inclusion logic: a class of Kripke models with teams is definable in modal inclusion logic if and only if it is closed under k-bisimulation for some integer k, it is closed under unions, and it has the empty team property. We also prove that the same expressive power can be obtained by adding a single unary nonemptiness operator to modal logic. Furthermore, we establish an exponential lower bound for the size of the translation from modal inclusion logic to modal logic with the nonemptiness operator.

The complexity of one-agent refinement modal logic

We investigate the complexity of satisfiability for one-agent refinement modal logic (RML), a known extension of basic modal logic (ML) obtained by adding refinement quantifiers on structures. It is known that RML has the same expressiveness as ML, but the translation of RML into ML is of non-elementary complexity, and RML is at least doubly exponentially more succinct than ML. In this paper, we show that RML-satisfiability is ‘only’ singly exponentially harder than ML-satisfiability, the latter being a well-known PSPACE-complete problem. More precisely, we establish that RML-satisfiability is complete for the complexity class AEXPpol, i.e., the class of problems solvable by alternating Turing machines running in single exponential time but only with a polynomial number of alternations (note that NEXPTIME⊆AEXPpol⊆EXPSPACE).11This work is the revised and expanded version of [1].

A logic of plausible justifications

In this work, we combine features from Justification Logics and Logics of Plausibility-Based Beliefs to build a logic for Multi-Agent Systems where each agent can explicitly state his justification for believing in a given sentence. Our logic is a normal modal logic based on the standard Kripke semantics, where we provide a semantic definition for the evidence terms and define the notion of plausible evidence for an agent, based on plausibility relations in the model. As we deal with beliefs, justifications can be faulty and unreliable. In our logic, agents can disagree not only over whether a sentence is true or false, but also on whether some evidence is a valid justification for a sentence or not. After defining our logic and its semantics, we provide a strongly complete axiomatic system for it, show that it has the finite model property, analyze the complexity of its Model-Checking Problem and show that its Satisfiability Problem has the same complexity as the one from basic modal logics. Thus, this logic seems to be a good first step for the development of a dynamic logic that can model the processes of argumentation and debate in Multi-Agent Systems.

Symmetric blocking

We present three different techniques that use information about symmetries detected in the input formula to block the expansion of diamonds in a modal tableau. We show how these blocking techniques can be included in a standard tableaux calculus for the basic modal logic, and prove that they preserve soundness and completeness. We empirically evaluate these blocking mechanisms in different modal benchmarks.

CONTINGENCY AND KNOWING WHETHER

AbstractA proposition is noncontingent, if it is necessarily true or it is necessarily false. In an epistemic context, ‘a proposition is noncontingent’ means that you knowwhetherthe proposition is true. In this paper, we study contingency logic with the noncontingency operator Δ but without the necessity operator □. This logic is not a normal modal logic, because Δ(φ→ψ) → (Δφ→ Δψ) is not valid. Contingency logic cannot define many usual frame properties, and its expressive power is weaker than that of basic modal logic over classes of models without reflexivity. These features make axiomatizing contingency logics nontrivial, especially for the axiomatization oversymmetricframes. In this paper, we axiomatize contingency logics over various frame classes using a novel method other than the methods provided in the literature, based on the ‘almost-definability’ schema AD proposed in our previous work. We also present extensions of contingency logic with dynamic operators. Finally, we compare our work to the related work in the fields of contingency logic and ignorance logic, where the two research communities have similar results but are apparently unaware of each other’s work. One goal of our paper is to bridge this gap.

Propositional dynamic logics for communicating concurrent programs with CCS's parallel operator

This work presents three increasingly expressive Dynamic Logics in which the programs are described in a language based on CCS. Our goal is to build dynamic logics that are suitable for the description and verification of properties of communicating concurrent systems, in a similar way as PDL is used for the sequential case. In order to accomplish that, CCS’s operators and constructions are added to a basic modal logic. Doing this, the semantics of CCS’s parallel operator allows us to build dynamic logics that support communicating and concurrent programs. We build a simple Kripke semantics for these logics, provide complete axiomatizations for them and show that they have the finite model property. This contrasts with other dynamic logics with parallel operators presented in the literature, such as Peleg’s Concurrent PDL with Channels, where either the parallel programs cannot communicate, or at least one of the properties mentioned above (simple Kripke semantics, complete axiomatization and finite model property) is missing.

Blending margins: the modal logic K has nullary unification type

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 $\phi \to \Box\phi / \phi, \neg\phi$, 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, introduced by Ghilardi; in particular, projective formulas coincide for these logics with formulas that are admissibly saturated (i.e., derive all their multiple-conclusion admissible consequences) or exact (i.e., axiomatize a theory of a substitution). In contrast, we show that in K, the formula $p \to \Box p$ is admissibly saturated, but neither projective nor exact. All our results for K also apply to the basic description logic ALC.

A Goal-Directed Decision Procedure for Hybrid PDL

We present the first goal-directed decision procedure for hybrid PDL. The procedure is based on a modular approach that scales from basic modal logic with eventualities to hybrid PDL. The approach is designed so that nominals and eventualities are treated orthogonally. To deal with the complex programs of PDL, the approach employs a novel disjunctive program decomposition. In arguing the correctness of our approach, we employ the novel notion of support generalizing the standard notion of Hintikka sets.

Deciding the unguarded modal -calculus

The modal -calculus extends basic modal logic with second-order quantification in terms of arbitrarily nested fixpoint operators. Its satisfiability problem is EXPTIME-complete. Decision procedures for the modal -calculus are not easy to obtain though since the arbitrary nesting of fixpoint constructs requires some combinatorial arguments for showing the well-foundedness of least fixpoint unfoldings. The tableau-based decision procedures so far also make assumptions on the unfoldings of fixpoint formulas, e.g., explicitly require formulas to be in guarded normal form. In this paper we present a tableau calculus for deciding satisfiability of arbitrary, i.e., not necessarily guarded -calculus formulas. It is based on a new unfolding rule for greatest fixpoint formulas which allows unguarded formulas to be handled without an explicit transformation into guarded form, thus avoiding a (seemingly) exponential blow-up in formula size. We prove soundness and completeness of the calculus, and compare it empirically to using guarded transformation instead. The new unfolding rule can also be used to handle nested star operators in PDL formulas correctly.

The expressibility of fragments of Hybrid Graph Logic on finite digraphs

Hybrid Graph Logic is a logic designed for reasoning about graphs and is built from a basic modal logic, augmented with the use of nominals and a facility to verify the existence of paths in graphs. We study the finite model theory of Hybrid Graph Logic. In particular, we develop pebble games for Hybrid Graph Logic and use these games to exhibit strict infinite hierarchies involving fragments of Hybrid Graph Logic when the logic is used to define problems involving finite digraphs. These fragments are parameterized by the quantifier-rank of formulae along with the numbers of propositional symbols and nominals that are available. We ascertain exactly the relative definability of these parameterized fragments of the logic.

A General Lindström Theorem for Some Normal Modal Logics

There are several known Lindstrom-style characterization results for basic modal logic. This paper proves a generic Lindstrom theorem that covers any normal modal logic corresponding to a class of Kripke frames definable by a set of formulas called strict universal Horn formulas. The result is a generalization of a recent characterization of modal logic with the global modality. A negative result is also proved in an appendix showing that the result cannot be strengthened to cover every first-order elementary class of frames. This is shown by constructing an explicit counterexample.

IF Modal Logic and Classical Negation

The present paper provides novel results on the model theory of Independence friendly modal logic. We concentrate on its particularly well-behaved fragment that was introduced in Tulenheimo and Sevenster (Advances in Modal Logic, 2006). Here we refer to this fragment as `Simple IF modal logic' (IFML s ). A model-theoretic criterion is presented which serves to tell when a formula of IFML s is not equivalent to any formula of basic modal logic (ML). We generalize the notion of bisimulation familiar from ML; the resulting asymmetric simulation concept is used to prove that IFML s is not closed under complementation. In fact we obtain a much stronger result: the only IFML s formulas admitting their classical negation to be expressed in IFML s itself are those whose truth-condition is in fact expressible in ML.

THE EXPRESSIVE POWER OF MEMORY LOGICS

We investigate the expressive power of memory logics. These are modal logics extended with the possibility to store (or remove) the current node of evaluation in (or from) a memory, and to perform membership tests on the current memory. From this perspective, the hybrid logic ℋℒ (↓), for example, can be thought of as a particular case of a memory logic where the memory is an indexed list of elements of the domain.This work focuses in the case where the memory is a set, and we can test whether the current node belongs to the set or not. We prove that, in terms of expressive power, the memory logics we discuss here lie between the basic modal logic ${\cal K}$ and ℋℒ (↓). We show that the satisfiability problem of most of the logics we cover is undecidable. The only logic with a decidable satisfiability problem is obtained by imposing strong constraints on which elements can be memorized.

Does the deduction theorem fail for modal logic?

Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified in a way that restricts its use to cases in which the premiss does not depend on assumptions. This restriction is entirely analogous to the restriction of the rule of universal generalization of first-order logic. A necessitation rule with this restriction permits a proof of the deduction theorem in its usual formulation. Other suggestions presented in the literature to deal with the problem are reviewed, and the present solution is argued to be preferable to the other alternatives. A contraction- and cut-free sequent calculus equivalent to the Hilbert system for basic modal logic shows the standard failure argument untenable by proving the underivability of \({\square\,A}\) from A.

Spartacus: A Tableau Prover for Hybrid Logic

Spartacus is a tableau prover for hybrid multimodal logic with global modalities and reflexive and transitive relations. Spartacus is the first system to use pattern-based blocking for termination. To achieve a competitive performance, Spartacus implements a number of optimization techniques, including a new technique that we call lazy branching. We evaluate the practical impact of pattern-based blocking and lazy branching for the basic modal logic K and observe high effectiveness of both techniques.

Toward Model-Theoretic Modal Logics

Adding certain cardinality quantifiers into first-order language will give substantially more expressive languages. Thus, many mathematical concepts beyond first-order logic can be handled. Since basic modal logic can be seen as the bisimular invariant fragment of first-order logic on the level of models, it has no ability to handle modally these mathematical concepts beyond first-order logic. By adding modalities regarding the cardinalities of successor states, we can, in principle, investigate modal logics of all cardinalities. Thus ways of exploring model-theoretic logics can be transferred to modal logics.

Using modal logics to express and check global graph properties

Graphs are among the most frequently used structures in Computer Science. Some of the properties that must be checked in many applications are connectivity, acyclicity and the Eulerian and Hamiltonian properties. In this work, we analyze how we can express these four properties with modal logics. This involves two issues: whether each of the modal languages under consideration has enough expressive power to describe these properties and how complex (computationally) it is to use these logics to actually test whether a given graph has some desired property. First, we show that these properties are not definable in a basic modal logic or in any bisimulation-invariant extension of it, like the modal μ-calculus. We then show that it is possible to express some of the above properties in a basic hybrid logic. Unfortunately, the Hamiltonian and Eulerian properties still cannot be efficiently checked. In a second attempt, we propose an extension of CTL∗ with nominals and show that the Hamiltonian property can be more efficiently checked in this logic than in the previous one. In a third attempt, we extend the basic hybrid logic with the ↓ operator and show that we can check the Hamiltonian property with optimal (NP) complexity in this logic. Finally, we tackle the Eulerian property in two different ways. First, we develop a generic method to express edge-related properties in hybrid logics and use it to express the Eulerian property. Second, we express a necessary and sufficient condition for the Eulerian property to hold using a graded modal logic.

A Logical Approach to Hamiltonian Graphs

Graphs are among the most frequently used structures in computer science. A lot of problems can be modelled using a graph and can then be solved by checking whether the graph satisfies some property. In this work, we are interested in how to use logical frameworks as a generic tool to express and efficiently check graph properties. In order to reason about this, we choose to analyze the Hamiltonian property and choose the family of modal logics as our framework. Our analysis has to deal with two central issues: whether each of the modal languages under consideration has enough expressive power to describe this property and how complex (computationally) it is to use these logics to actually test whether a given graph has this property. First, we show that this property is not definable in a basic modal logic or in any bisimulation-invariant extension of it, like the modal μ -calculus. We then show that it is possible to express it in a basic hybrid logic. Unfortunately, the Hamiltonian property still cannot be efficiently checked in this logic. In a second attempt, we extend this basic hybrid logic with the ↓ operator and show that we can check the Hamiltonian property with optimal (NP-Complete) complexity in this logic.

Modal characterisation theorems over special classes of frames

We investigate model theoretic characterisations of the expressive power of modal logics in terms of bisimulation invariance. The paradigmatic result of this kind is van Benthem’s theorem, which says that a first-order formula is invariant under bisimulation if, and only if, it is equivalent to a formula of basic modal logic. The present investigation primarily concerns ramifications for specific classes of structures. We study in particular model classes defined through conditions on the underlying frames, with a focus on frame classes that play a major role in modal correspondence theory and often correspond to typical application domains of modal logics. Classical model theoretic arguments do not apply to many of the most interesting classes–for instance, rooted frames, finite rooted frames, finite transitive frames, well-founded transitive frames, finite equivalence frames–as these are not elementary. Instead we develop and extend the game-based analysis (first-order Ehrenfeucht–Fraïssé versus bisimulation games) over such classes and provide bisimulation preserving model constructions within these classes. Over most of the classes considered, we obtain finite model theory analogues of the classically expected characterisations, with new proofs also for the classical setting. The class of transitive frames is a notable exception, with a marked difference between the classical and the finite model theory of bisimulation invariant first-order properties. Over the class of all finite transitive frames in particular, we find that monadic second-order logic is no more expressive than first-order as far as bisimulation invariant properties are concerned — though both are more expressive here than basic modal logic. We obtain ramifications of the de Jongh–Sambin theorem and a new and specific analogue of the Janin–Walukiewicz characterisation of bisimulation invariant monadic second-order for finite transitive frames.

Model-theoretic and Computational Properties of Modal Dependence Logic

We study the basic modal language extended by an operator dep. If pi are propositional atoms, then dep(p1,…,pn−1;pn) expresses, intuitively, that pn only depends on p1,…,pn−1. The resulting language was baptized ‘modal dependence logic’ by Väänänen in his paper Modal Dependence Logic. The current article compares modal dependence logic with basic modal logic in terms of its model-theoretic and computational properties. We show that modal dependence logic is strictly more expressive than modal logic, but that under special conditions modal dependence logic can be translated into basic modal logic.We show that the complexity of modal dependence logic is NEXP-complete.