Intuitionistic Modal Logic Research Articles

Universal proof theory: Feasible admissibility in intuitionistic modal logics

We introduce a general and syntactically defined family of sequent-style calculi over the propositional language with the modalities {□,◇} and its fragments as a formalization for constructively acceptable systems. Calling these calculi constructive, we show that any strong enough constructive sequent calculus, satisfying a mild technical condition, feasibly admits all Visser's rules. This means that there exists a polynomial-time algorithm that, given a proof of the premise of a Visser's rule, provides a proof for its conclusion. As a positive application, we establish the feasible admissibility of Visser's rules in sequent calculi for several intuitionistic modal logics, including CK, IK, their extensions by the modal axioms T, B, 4, 5, and the axioms for bounded width and depth and their fragments CK□, propositional lax logic and IPC. On the negative side, we show that if a strong enough intuitionistic modal logic (satisfying a mild technical condition) does not admit at least one of Visser's rules, it cannot have a constructive sequent calculus. Consequently, no intermediate logic other than IPC has a constructive sequent calculus.

Intuitionistic S4 as a logic of topological spaces

Abstract We design and study various topological semantics for the diamond-free intuitionistic modal logic $\textsf{iS4}$, an intuitionistic analogue of $\textsf{S4}$. Ultimately we prove that ordinary topological spaces can be used as semantics, using the specialization order to interpret intuitionistic implication and the interior for the modality. Some of our soundness and completeness results are mechanised in Coq.

Natural deduction calculi for classical and intuitionistic S5

We propose an indexed natural deduction system for the modal logic , ideally following Wansing's previous work in the context of tableaux sequents. The system, given both in the classical and intuitionistic versions (called and respectively), is designed to match as much as possible the structure and properties of the standard system of natural deduction for first-order logic, exploiting the formal analogy between modalities and quantifiers. We study a (syntactical) normalization theorem for both and and its main consequences, the sub-formula principle and the consistency theorem. In particular, we propose an intuitionistic encoding of classical (via a suitable extension of the Gödel translation for first-order classical logic). Moreover, via the BHK interpretation of intuitionistic proofs, we propose a suitable Curry–Howard isomorphism for . By translation into the natural deduction system given by Galmiche and Salhi in [(2010b). Label-free proof systems for intuitionistic modal logic is5. In E. M. Clarke & A. Voronkov (Eds.), Logic for programming, artificial intelligence, and reasoning (pp. 255–271). Springer Berlin Heidelberg.], we prove the equivalence of w.r.t. an Hilbert-style axiomatization of . However, when considering the sheer provability of labelled formulas, our system is comparable to the one presented by Simpson in [(1993). The proof theory and semantics of intuitionistic modal logic [PhD thesis], University of Edinburgh, UK.]. Nevertheless, it remains uncertain whether it is feasible to establish a translation between the corresponding derivations.

When privacy fails, a formula describes an attack: A complete and compositional verification method for the applied π-calculus

We prove that three semantics for the applied π-calculus coincide – a testing semantics, a labelled equivalence, and a modal logic – and explain how design decisions in each of these semantics are objectively supported by this convergence. Each of these semantics has a distinct role when verifying security protocols. Testing semantics are suited to security and privacy problems expressed in terms of a process equivalence problem, where tests define the space of attack strategies. A labelled equivalence is suited to proving the given equivalence problem when such security and privacy properties hold; while, dually, the modal logic describes attacks when such security or privacy properties are violated. For this particular investigation into this good-practice design pattern, we select what is perhaps the most powerful (interleaving) testing semantics: open barbed bisimilarity. Selecting open barbed bisimilarity comes with the bonus that it is a congruence hence is suited to compositional reasoning. This reference testing semantics leads to a notion of quasi-open bisimilarity and an intuitionistic modal logic for the applied π-calculus. We argue that, since we characterise the finest (interleaving) testing semantics and tests are attack strategies, our intuitionistic modal logic can describe all attacks whenever a privacy property expressed as an equivalence problem fails.

Admissible rules for six intuitionistic modal logics

This paper characterizes the admissible rules for six interesting intuitionistic modal logics: iCK4, iCS4≡IPC, strong Löb logic iSL, modalized Heyting calculus mHC, Kuznetsov-Muravitsky logic KM, and propositional lax logic PLL. Admissible rules are rules that can be added to a logic without changing the set of theorems of the logic. We provide a Gentzen-style proof theory for admissibility that combines methods known for intuitionistic propositional logic and classical modal logic. From this proof theory, we extract bases for the admissible rules, i.e., sets of admissible rules that derive all other admissible rules. In addition, we show that admissibility is decidable for these logics.

The G4i Analogue of a G3i Sequent Calculus

This paper provides a method to obtain terminating analytic calculi for a large class of intuitionistic modal logics. For a given logic L with a cut-free calculus G that is an extension of G3ip the method produces a terminating analytic calculus that is an extension of G4ip and equivalent to G. G4ip was introduced by Roy Dyckhoff in 1992 as a terminating analogue of the calculus G3ip for intuitionistic propositional logic. Thus this paper can be viewed as an extension of Dyckhoff’s work to intuitionistic modal logic.

A van Benthem Theorem for Atomic and Molecular Logics

After recalling the definitions of atomic and molecular logics, we show how notions of bisimulation can be automatically defined from the truth conditions of the connectives of any of these logics. Then, we prove a generalization of van Benthem modal characterization theorem for molecular logics. Our molecular connectives should be uniform and contain all conjunctions and disjunctions. We use modal logic, the Lambek calculus and modal intuitionistic logic as case study and compare in particular our work with Olkhovikov's work.

The predicate version of the joint logic of problems and propositions

We consider the joint logic of problems and propositions $\mathrm{QHC}$ introduced by Melikhov. We construct Kripke models with audit worlds for this logic and prove the soundness and completeness of $\mathrm{QHC}$ with respect to this type of model. The conservativity of the logic $\mathrm{QHC}$ over the intuitionistic modal logic $\mathrm{QH4}$, which coincides with the ‘lax logic’ $\mathrm{QLL}^+$, is established. We construct Kripke models with audit worlds for the logic $\mathrm{QH4}$ and prove the corresponding soundness and completeness theorems. We also prove that the logics $\mathrm{QHC}$ and $\mathrm{QH4}$ have the disjunction and existence properties. Bibliography: 33 titles.

TEMPORAL INTERPRETATION OF MONADIC INTUITIONISTIC QUANTIFIERS

AbstractWe show that monadic intuitionistic quantifiers admit the following temporal interpretation: “always in the future” (for $\forall $ ) and “sometime in the past” (for $\exists $ ). It is well known that Prior’s intuitionistic modal logic ${\sf MIPC}$ axiomatizes the monadic fragment of the intuitionistic predicate logic, and that ${\sf MIPC}$ is translated fully and faithfully into the monadic fragment ${\sf MS4}$ of the predicate ${\sf S4}$ via the Gödel translation. To realize the temporal interpretation mentioned above, we introduce a new tense extension ${\sf TS4}$ of ${\sf S4}$ and provide a full and faithful translation of ${\sf MIPC}$ into ${\sf TS4}$ . We compare this new translation of ${\sf MIPC}$ with the Gödel translation by showing that both ${\sf TS4}$ and ${\sf MS4}$ can be translated fully and faithfully into a tense extension of ${\sf MS4}$ , which we denote by ${\sf MS4.t}$ . This is done by utilizing the relational semantics for these logics. As a result, we arrive at the diagram of full and faithful translations shown in Figure 1 which is commutative up to logical equivalence. We prove the finite model property (fmp) for ${\sf MS4.t}$ using algebraic semantics, and show that the fmp for the other logics involved can be derived as a consequence of the fullness and faithfulness of the translations considered.

One-Step Modal Logics, Intuitionistic and Classical, Part 2

Hodes (2021) “looked under the hood” of the familiar versions of the classical propositional modal logic K and its intuitionistic counterpart (see Plotkin & Sterling 1986). This paper continues that project, addressing some familiar classical strengthenings of K (D, T, K4, KB, K5, Dio (the Diodorian strengthening of K) and GL), and their intuitionistic counterparts (see Plotkin & Sterling 1986 for some of these counterparts). Section 9 associates two intuitionistic one-step proof-theoretic systems to each of the just mentioned intuitionistic logics, this by adding for each a new rule to those which generated IK in Hodes (2021). For the systems associated with the intuitionistic counterparts of D and T, these rules are “pure one-step”: their schematic formulations does not use □ or ♢. For the systems associated with the intuitionistic counterparts of K4, etc., these rules meet these conditions: neither □ nor ♢ is iterated; none use both □ and ♢. The join of the two systems associated with each of these familiar logics is the full one-step system for that intuitionistic logic. And further “blended” intuitionistic systems arise from joining these systems in various ways. Adding the 0-version of Excluded Middle to their intuitionistic counterparts yields the one-step systems corresponding to the familiar classical logics. Each proof-theoretic system defines a consequence relation in the obvious way. Section 10 examines inclusions between these consequence relations. Section 11 associates each of the above consequence relations with an appropriate class of models, and proves them sound with respect to their appropriate class. This allows proofs of some failures of inclusion between consequence relations. (Sections 10 and 11 provide an exhaustive study of a variety of intuitionistic modal logics.) Section 12 proves that the each consequence relation is complete or (for those corresponding to GL) weakly complete, that relative to its appropriate class of models. The Appendix presents three further results about some of the intuitionistic consequence relations discussed in the body of the paper.

A Characterisation of Open Bisimilarity using an Intuitionistic Modal Logic

Open bisimilarity is defined for open process terms in which free variables may appear. The insight is, in order to characterise open bisimilarity, we move to the setting of intuitionistic modal logics. The intuitionistic modal logic introduced, called $\mathcal{OM}$, is such that modalities are closed under substitutions, which induces a property known as intuitionistic hereditary. Intuitionistic hereditary reflects in logic the lazy instantiation of free variables performed when checking open bisimilarity. The soundness proof for open bisimilarity with respect to our intuitionistic modal logic is mechanised in Abella. The constructive content of the completeness proof provides an algorithm for generating distinguishing formulae, which we have implemented. We draw attention to the fact that there is a spectrum of bisimilarity congruences that can be characterised by intuitionistic modal logics.

A fully labelled proof system for intuitionistic modal logics

Abstract Labelled proof theory has been famously successful for modal logics by mimicking their relational semantics within deductive systems. Simpson in particular designed a framework to study a variety of intuitionistic modal logics integrating a binary relation symbol in the syntax. In this paper, we present a labelled sequent system for intuitionistic modal logics such that there is not only one but two relation symbols appearing in sequents: one for the accessibility relation associated with the Kripke semantics for normal modal logics and one for the pre-order relation associated with the Kripke semantics for intuitionistic logic. This puts our system in close correspondence with the standard birelational Kripke semantics for intuitionistic modal logics. As a consequence, it can be extended with arbitrary intuitionistic Scott–Lemmon axioms. We show soundness and completeness, together with an internal cut elimination proof, encompassing a wider array of intuitionistic modal logics than any existing labelled system.

Categorical and algebraic aspects of the intuitionistic modal logic IEL― and its predicate extensions

Abstract The system of intuitionistic modal logic $\textbf{IEL}^{-}$ was proposed by S. Artemov and T. Protopopescu as the intuitionistic version of belief logic (S. Artemov and T. Protopopescu. Intuitionistic epistemic logic. The Review of Symbolic Logic, 9, 266–298, 2016). We construct the modal lambda calculus, which is Curry–Howard isomorphic to $\textbf{IEL}^{-}$ as the type-theoretical representation of applicative computation widely known in functional programming.We also provide a categorical interpretation of this modal lambda calculus considering coalgebras associated with a monoidal functor on a Cartesian closed category. Finally, we study Heyting algebras and locales with corresponding operators. Such operators are used in point-free topology as well. We study complete Kripke–Joyal-style semantics for predicate extensions of $\textbf{IEL}^{-}$ and related logics using Dedekind–MacNeille completions and modal cover systems introduced by Goldblatt (R. Goldblatt. Cover semantics for quantified lax logic. Journal of Logic and Computation, 21, 1035–1063, 2011). The paper extends the conference paper published in the LFCS’20 volume (D. Rogozin. Modal type theory based on the intuitionistic modal logic IEL. In International Symposium on Logical Foundations of Computer Science, pp. 236–248. Springer, 2020).

Algebraic semantics of the $$\left\{ \rightarrow ,\square \right\} $$-fragment of Propositional Lax Logic

In this paper, we will study a particular subvariety of Hilbert algebras with a modal operator $$\square $$, called Lax Hilbert algebras. These algebras are the algebraic semantic of the $$\left\{ \square ,\rightarrow \right\} $$-fragment of a particular intuitionistic modal logic, called Propositional Lax Logic ($$\mathcal {PLL}$$), which has applications to the formal verification of computer hardware. These algebras turn to be a generalization of the variety of Heyting algebras with a modal operator studied, under different names, by Macnab (Algebra Univ 12:5–29, 1981), Goldblatt (Math Logic Q 27(31–35):495–529, 1981; J Logic Comput 21(6):1035–1063, 2010) and by Bezhanishvili and Ghilardi (Ann Pure Appl Logic 147:84–100, 2007). We shall prove that the set of fixpoints of a Lax Hilbert algebra $$\left\langle A,\square \right\rangle $$ is a Hilbert algebra such that its dual space is homeomorphic to the subspace of reflexive elements of the dual space of A. We will define the notion of subframe of a Hilbert space $$\left\langle X,{\mathcal {K}}\right\rangle $$, and we will prove that there is a 1–1 correspondence between subframes of $$\left\langle X,{\mathcal {K}}\right\rangle $$ and binary relations $$Q\subseteq X\times X$$ such that $$\left\langle X,{\mathcal {K}},Q\right\rangle $$ is a Lax Hilbert space. In addition, we will define the notion of subframe variety and we will prove that any variety of Hilbert algebras is a subframe variety.

Topological and Multi-Topological Frames in the Context of Intuitionistic Modal Logic

We present three examples of topological semantics for intuitionistic modal logic with one modal operator □. We show that it is possible to treat neighborhood models, introduced earlier, as topological or multi-topological. From the neighborhood point of view, our method is based on differences between properties of minimal and maximal neighborhoods. Also we propose transformation of multitopological spaces into the neighborhood structures.

Uniform interpolation and the existence of sequent calculi

This paper presents a uniform and modular method to prove uniform interpolation for several intermediate and intuitionistic modal logics. The proof-theoretic method uses sequent calculi that are extensions of the terminating sequent calculus G4ip for intuitionistic propositional logic. It is shown that whenever the rules in a calculus satisfy certain structural properties, the corresponding logic has uniform interpolation. It follows that the intuitionistic versions of K and KD (without the diamond operator) have uniform interpolation. It also follows that no intermediate or intuitionistic modal logic without uniform interpolation has a sequent calculus satisfying those structural properties, thereby establishing that except for the seven intermediate logics that have uniform interpolation, no intermediate logic has such a sequent calculus.

Gentzen sequent calculi for some intuitionistic modal logics

Abstract Intuitionistic modal logics are extensions of intuitionistic propositional logic with modal axioms. We treat with two modal languages ${\mathscr{L}}_\Diamond $ and $\mathscr{L}_{\Diamond ,\Box }$ which extend the intuitionistic propositional language with $\Diamond $ and $\Diamond ,\Box $, respectively. Gentzen sequent calculi are established for several intuitionistic modal logics. In particular, we introduce a Gentzen sequent calculus for the well-known intuitionistic modal logic $\textsf{MIPC}$. These sequent calculi admit cut elimination and subformula property. They are decidable.

Algorithmic correspondence and canonicity for non-distributive logics

We extend the theory of unified correspondence to a broad class of logics with algebraic semantics given by varieties of normal lattice expansions (LEs), also known as ‘lattices with operators’. Specifically, we introduce a syntactic definition of the class of Sahlqvist formulas and inequalities which applies uniformly to each LE-signature and is given purely in terms of the order-theoretic properties of the algebraic interpretations of the logical connectives. We also introduce the algorithm ALBA, parametric in each LE-setting, which effectively computes first-order correspondents of LE-inequalities, and is guaranteed to succeed on a wide class of inequalities (the so-called inductive inequalities) which significantly extend the Sahlqvist class. Further, we show that every inequality on which ALBA succeeds is canonical. Projecting these results on specific signatures yields state-of-the-art correspondence and canonicity theory for many well known modal expansions of classical and intuitionistic logic and for substructural logics, from classical poly-modal logics to (bi-)intuitionistic modal logics to the Lambek calculus and its extensions, the Lambek-Grishin calculus, orthologic, the logic of (not necessarily distributive) De Morgan lattices, and the multiplicative-additive fragment of linear logic.

UNIFICATION IN SUPERINTUITIONISTIC PREDICATE LOGICS AND ITS APPLICATIONS

AbstractWe introduce unification in first-order logic. In propositional logic, unification was introduced by S. Ghilardi, see Ghilardi (1997, 1999, 2000). He successfully applied it in solving systematically the problem of admissibility of inference rules in intuitionistic and transitive modal propositional logics. Here we focus on superintuitionistic predicate logics and apply unification to some old and new problems: definability of disjunction and existential quantifier, disjunction and existential quantifier under implication, admissible rules, a basis for the passive rules, (almost) structural completeness, etc. For this aim we apply modified specific notions, introduced in propositional logic by Ghilardi, such as projective formulas, projective unifiers, etc.Unification in predicate logic seems to be harder than in the propositional case. Any definition of the key concept of substitution for predicate variables must take care of individual variables. We allow adding new free individual variables by substitutions (contrary to Pogorzelski & Prucnal (1975)). Moreover, since predicate logic is not as close to algebra as propositional logic, direct application of useful algebraic notions of finitely presented algebras, projective algebras, etc., is not possible.

Terminating sequent calculi for two intuitionistic modal logics

This paper presents sequent calculi in which proof search is terminating for two intuitionistic modal logics, the intuitionistic versions of the classical modal logics K and KD without a diamond operator. The calculi are extensions of the terminating sequent calculus G4ip for intuitionistic propositional logic that was discovered independently by Dyckhoff and Hudelmaier around 1990. It is shown by proof–theoretic means that these terminating calculi are equivalent to the cutfree extensions of G3ip that form some of the standard calculi for intuitionistic modal logics.