Some Remarks on Relations between Proofs and Games

International audienceThis paper aims at studying relations between proof systems and games in a given logic and at analyzing what can be the interest and limits of a game formulation as an alternative semantic framework for modelling proof search and also for understanding relations between logics. In this perspective, we firstly study proofs and games at an abstract level which is neither related to a particular logic nor adopts a specific focus on their relations. Then, in order to instantiate such an analysis, we describe a dialogue game for intu-itionistic logic and emphasize the adequateness between proofs and winning strategies in this game. Finally, we consider how games can be seen to provide an alternative formulation for proof search and we stress on the possible mix of logical rules and search strategies inside games rules. We conclude on the merits and limits of the game semantics as a tool for studying logics, validity in these logics and some relations between them. 2 Proofs and Games In this section, we present a common terminology to present both proof systems and games at a relatively abstract level. Our aim consists in obtaining tools on which bridges can be built between the proof-theoretical approach and the game semantics approach in establishing the (universal) validity of logical formulae. We explain how proofs and games can be viewed as complementary notions. We illustrate how proof trees in calculi correspond to winning strategies in games and vice-versa

Proof Search in Multi-Agent Dialogues for Modal Logic

In computer science, and also in philosophy, modal logics play an important role in various areas. They can be used to model knowledge structures among software-agents, behaviour of computer systems, or ontologies. They also provide mathematical tools to perform reasoning in these models, e.g., to extract common knowledge of agents, check whether security-relevant problems might occur when running a program, or to detect contradictions in a set of terminological definitions. Intuitionistic or constructive propositional logic can be considered as a special kind of modal logic. Constructive modal logics, as a combination of intuitionistic propositional logic and classical modal logics, describe a family of modal systems which are, compared to the classical variant, more restrictive concerning the validity of formulas. To prove validity of a statement formalized in such a logic, various reasoning procedures (also called calculi) have been investigated. There are especially many variants of sequent and tableau systems which can be used easily to find proofs by applying given syntactical rules one after another. Sometimes there are different possibilities to find a proof for the same formula within the same calculus. It also happens that a bad choice of non-invertible rule applications at the wrong time makes it impossible to finish the proof successfully, although the formula is provable. For this reason, a normalization of deductions in a calculus is desired. This restricts the possibilities to apply rules arbitrarily and emphasizes the situations in which significant, non-invertible rule applications are necessary. Such a normalization is enforced in so-called focused sequent systems. Another attempt to find a normalized calculus leads to dialogical logic, a game-theoretic reasoning technique. Usually, two players, one proponent and one opponent, argue about an assertion, expressed as a formula and stated by the proponent at the beginning of the play. The kinds of arguments, namely attacks and defences, are bound to special game rules. These are designed in such a way that the proponent has a winning strategy in the game if and only if his initial statement is a valid formula. The dialogical approach is very flexible as the game rules can be adjusted easily. Sets of rules exist to perform reasoning in many different kinds of logic, however proving soundness and completeness of dialogical calculi is complex and, if at all, often only considered very roughly in the literature. The standard two-player dialogues do not have much potential to enforce normalization like focus sequent systems. However, it turns out that introducing further proponent-players who fight against one opponent in a round-based setting leads to a normalization as described above. The flexibility of two-player games is largely preserved in multi-proponent dialogues. Other ordinary sequent systems can easily be transferred into the dialectic setting to achieve a normalization. Further, the round-based scheduling induces a method to parallelize the reasoning process. Modifying the game rules makes it possible to construct new intermediate or even more restrictive logics. In this work, dialogical systems with multiple proponents are presented for intuitionistic propositional logic and modal logics S4 and CS4. Starting with the former one, it is shown that the normalization can be transferred easily to both the latter systems. Informal game rules are introduced and, to make them concrete and unambiguous, translated into the dialogical sequent-style calculi DiaSeqI, DiaSeqS4, and DiaSeqCS4. An extra system for intuitionistic logic, which guarantees termination in proof searches, even if the target formula is not valid, is also provided. Soundness and completeness of all these presented dialogical sequent calculi is proven formally, by showing that it is always possible to translate derivations in the game-oriented approach into another sound and complete sequent system and vice versa. Thereby, a new (ordinary) multi-conclusion sequent calculus for CS4 is introduced for which adequateness is shown, too. The multi-proponent dialogical systems of this work are compared to different sequent calculi and other dialogical attempts found in literature. A comprehensive survey of such approaches is also part of this thesis

Neutrality and Many-Valued Logics

In this book, we consider various many-valued logics: standard, linear, hyperbolic, parabolic, non-Archimedean, p-adic, interval, neutrosophic, etc. We survey also results which show the tree different proof-theoretic frameworks for many-valued logics, e.g. frameworks of the following deductive calculi: Hilbert's style, sequent, and hypersequent. We present a general way that allows to construct systematically analytic calculi for a large family of non-Archimedean many-valued logics: hyperrational-valued, hyperreal-valued, and p-adic valued logics characterized by a special format of semantics with an appropriate rejection of Archimedes' axiom. These logics are built as different extensions of standard many-valued logics (namely, Lukasiewicz's, Goedel's, Product, and Post's logics). The informal sense of Archimedes' axiom is that anything can be measured by a ruler. Also logical multiple-validity without Archimedes' axiom consists in that the set of truth values is infinite and it is not well-founded and well-ordered. On the base of non-Archimedean valued logics, we construct non-Archimedean valued interval neutrosophic logic INL by which we can describe neutrality phenomena.Comment: 119 page

From Semantic Games to Provability: The Case of Gödel Logic

We present a semantic game for Gödel logic and its extensions, where the players’ interaction stepwise reduces arbitrary claims about the relative order of truth degrees of complex formulas to atomic ones. The paper builds on a previously developed game for Gödel logic with projection operator in Fermüller et al. (in: M.-J. Lesot, S. Vieira, M.Z. Reformat, J.P. Carvalho, A. Wilbik, B. Bouchon-Meunier, and R.R. Yager, (eds.), Information processing and management of uncertainty in knowledge-based systems, Springer, Cham, 2020, pp. 257–270). This game is extended to cover Gödel logic with involutive negations and constants, and then lifted to a provability game using the concept of disjunctive strategies. Winning strategies in the provability game, with and without constants and involutive negations, turn out to correspond to analytic proofs in a version of SeqGZL (A. Ciabattoni, and T. Vetterlein, Fuzzy Sets and Systems 161(14):1941–1958, 2010) and in a sequent-of-relations calculus (M. Baaz, and Ch.G. Fermüller, in: N.V. Murray, (ed.), Automated reasoning with analytic tableaux and related methods, Springer, Berlin, 1999, pp. 36–51) respectively

Hyper Natural Deduction

Paper introduces a Hyper Natural Deduction system as an extension of Gentzen's Natural Deduction system, by adding additional rules providing means for communication between derivations. It is shown that the Hyper Natural Deduction system is sound and complete for infinite-valued propositional Gödel Logic, by giving translations to and from Avron's Hyper sequent Calculus. The paper also provides conversions for normalisation and prove the existence of normal forms for the Hyper Natural Deduction system

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established

Hyper Natural Deduction for Gödel Logic—A natural deduction system for parallel reasoning

We introduce a system of Hyper Natural Deduction for Gödel Logic as an extension of Gentzen’s system of Natural Deduction. A deduction in this system consists of a finite set of derivations which uses the typical rules of Natural Deduction, plus additional rules providing means for communication between derivations. We show that our system is sound and complete for infinite-valued propositional Gödel Logic, by giving translations to and from Avron’s Hypersequent Calculus. We provide conversions for normalization extending usual conversions for Natural Deduction and prove the existence of normal forms for Hyper Natural Deduction for Gödel Logic. We show that normal deductions satisfy the subformula property