Countermodel Construction Research Articles

PROOF SYSTEMS FOR TWO-WAY MODAL MU-CALCULUS

Abstract We present sound and complete sequent calculi for the modal mu-calculus with converse modalities, aka two-way modal mu-calculus. Notably, we introduce a cyclic proof system wherein proofs can be represented as finite trees with back-edges, i.e., finite graphs. The sequent calculi incorporate ordinal annotations and structural rules for managing them. Soundness is proved with relative ease as is the case for the modal mu-calculus with explicit ordinals. The main ingredients in the proof of completeness are isolating a class of non-wellfounded proofs with sequents of bounded size, called slim proofs, and a counter-model construction that shows slimness suffices to capture all validities. Slim proofs are further transformed into cyclic proofs by means of re-assigning ordinal annotations.

Proof theory for the logics of bringing-it-about: Ability, coalitions and means-end relationship

Abstract The logic of bringing-it-about (BIAT) aims to capture a notion of agency in which actions are analysed in terms of their results: ‘An agent does something’ means that the agent brings it about that something takes place. Our starting point is the basic BIAT logic as introduced by Elgesem in the ‘90s: this logic contains only a modal operator to express BIAT statements by single agents. Several extensions have been proposed by Elgesem himself and others, notably with the capability operator, coalitions of agents and means-end BIAT statements (i.e. of the form ‘the agent does B by doing A’). We first propose a variant of the neighbourhood semantics, called bi-neighbourhood semantics, for the basic BIAT logic and the mentioned extensions, in which a world is equipped by a set of pairs or neighbourhoods. Differently from the semantics defined in the literature, this reformulation is well suited for countermodel construction. We then introduce modular hypersequent calculi for all logics considered in this work. Our calculi enjoy the fundamental property of cut admissibility, from which it follows their completeness with respect to the axiomatization. Moreover, our calculi provide at the same time a decision procedure, as well as the first practical countermodel extraction procedure: from a single failed proof it is possible to build directly a finite countermodel of the formula under verification in the bi-neighbourhood semantics. By this last result, we obtain constructive proofs of the semantic completeness of the calculi and consequently of the finite model property for all logics.

Interpolation for intermediate logics via injective nested sequents

AbstractWe introduce a novel, semantically inspired method of constructing nested sequent calculi for propositional intermediate logics. Applying recently developed methods for proving Craig interpolation to these nested sequent calculi, we obtain constructive proofs of the interpolation property for most non-trivial interpolable intermediate logics, as well as Lyndon interpolation for Gödel logic. Finally, we provide a prototype implementation combining proof search and countermodel construction.

Maehara-style modal nested calculi

We develop multi-conclusion nested sequent calculi for the fifteen logics of the intuitionistic modal cube between IK and IS5. The proof of cut-free completeness for all logics is provided both syntactically via a Maehara-style translation and semantically by constructing an infinite birelational countermodel from a failed proof search. Interestingly, the Maehara-style translation for proving soundness syntactically fails due to the hierarchical structure of nested sequents. Consequently, we only provide the semantic proof of soundness. The countermodel construction used to prove completeness required a completely novel approach to deal with two independent sources of non-termination in the proof search present in the case of transitive and Euclidean logics.

Counter-Example Construction with Euler Diagrams

One of the traditional applications of Euler diagrams is as a representation or counterpart of the usual set-theoretical models of given sentences. However, Euler diagrams have recently been investigated as the counterparts of logical formulas, which constitute formal proofs. Euler diagrams are rigorously defined as syntactic objects, and their inference systems, which are equivalent to some symbolic logical systems, are formalized. Based on this observation, we investigate both counter-model construction and proof-construction in the framework of Euler diagrams. We introduce the notion of "counter-diagrammatic proof", which shows the invalidity of a given inference, and which is defined as a syntactic manipulation of diagrams of the same sort as inference rules to construct proofs. Thus, in our Euler diagrammatic framework, the completeness theorem can be formalized in terms of the existence of a diagrammatic proof or a counter-diagrammatic proof.

Proofs and Countermodels in Non-Classical Logics

Proofs and countermodels are the two sides of completeness proofs, but, in general, failure to find one does not automatically give the other. The limitation is encountered also for decidable non-classical logics in traditional completeness proofs based on Henkin’s method of maximal consistent sets of formulas. A method is presented that makes it possible to establish completeness in a direct way: For any given sequent either a proof in the given logical system or a countermodel in the corresponding frame class is found. The method is a synthesis of a generation of calculi with internalized relational semantics, a Tait–Schutte–Takeuti style completeness proof, and procedures to finitize the countermodel construction. Finitizations for intuitionistic propositional logic are obtained through the search for a minimal derivation, through pruning of infinite branches in search trees by means of a suitable syntactic counterpart of semantic filtration, or through a proof-theoretic embedding into an appropriate provability logic. A number of examples illustrates the method, its subtleties, challenges, and present scope.

A cut-free sequent system for Grzegorczyk logic, with an application to the Gödel–McKinsey–Tarski embedding

It is well-known that intuitionistic propositional logic Int may be faithfully embedded not just into the modal logic S4 but also into the provability logics GL and Grz of Godel-Lob and Grzegorczyk, and also that there is a similar embedding of Grz into GL. Known proofs of these faithfulness results are short but model-theoretic and thus non-constructive. Here a labelled sequent system Grz for Grzegorczyk logic is presented and shown to be complete and therefore closed with respect to Cut. The completeness proof, being constructive, yields a constructive decision procedure, i.e. both a proof procedure for derivable sequents and a countermodel construction for underivable sequents. As an application, a constructive proof of the faithfulness of the embedding of Int into Grz and hence a constructive decision procedure for Int are obtained.

A Connection-based Characterization of Bi-intuitionistic Validity

We give a connection-based characterization of validity in propositional bi-intuitionistic logic in terms of specific directed graphs called R-graphs. Such a characterization is well-suited for deriving labelled proof-systems with counter-model construction facilities. We first define the notion of bi-intuitionistic R-graphs from which we then obtain a connection-based characterization of propositional bi-intuitionistic validity and derive a sound and complete cut-free free-variable labelled sequent calculus that enjoys variable splitting.

A phase semantics for polarized linear logic and second order conservativity

AbstractThis paper presents a polarized phase semantics, with respect to which the linear fragment of second order polarized linear logic of Laurent [15] is complete. This is done by adding a topological structure to Girard's phase semantics [9], The topological structure results naturally from the categorical construction developed by Hamano–Scott [12]. The polarity shifting operator ↓ (resp. ↑) is interpreted as an interior (resp. closure) operator in such a manner that positive (resp. negative) formulas correspond to open (resp. closed) facts. By accommodating the exponentials of linear logic, our model is extended to the polarized fragment of the second order linear logic. Strong forms of completeness theorems are given to yield cut-eliminations for the both second order systems. As an application of our semantics, the first order conservativity of linear logic is studied over its polarized fragment of Laurent [16]. Using a counter model construction, the extension of this conservativity is shown to fail into the second order, whose solution is posed as an open problem in [16]. After this negative result, a second order conservativity theorem is proved for an eta expanded fragment of the second order linear logic, which fragment retains a focalized sequent property of [3].

Reductions for non-clausal theorem proving

This paper presents the TAS methodology 1 1 TAS stands for Transformaciones de Árboles Sintácticos, Spanish translation of Syntactic Trees Transformations. as a new framework for generating non-clausal Automated Theorem Provers. We present a complete description of the ATP for Classical Propositional Logic, named TAS-D, but the ideas, which make use of implicants and implicates can be extended in a natural manner to first-order logic, and non-classical logics. The method is based on the application of a number of reduction strategies on subformulas, in a rewrite-system style, in order to reduce the complexity of the formula as much as possible before branching. Specifically, we introduce the concept of complete reduction, and extensions of the pure literal rule and of the collapsibility theorems; these strategies allow to limit the size of the search space. In addition, TAS-D is a syntactical countermodel construction. As an example of the power of TAS-D we study a class of formulas which has linear proofs (in the number of branchings) when either resolution or dissolution with factoring is applied. When applying our method to these formulas we get proofs without branching. In addition, some experimental results are reported.