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

Achieving while maintaining:A logic of knowing how with intermediate constraints

In this paper, we propose a ternary knowing how operator to express that the agent knows how to achieve $\phi$ given $\psi$ while maintaining $\chi$ in-between. It generalizes the logic of goal-directed knowing how proposed by Yanjing Wang 2015 'A logic of knowing how'. We give a sound and complete axiomatization of this logic.Comment: appear in Proceedings of ICLA 201

On some axiomatic extensions of the monoidal T-norm based logic MTL : an analysis in the propositional and in the first-order case

The scientific area this book belongs to are many-valued logics: in particular, the logic MTL and some of its extensions, in the propositional and in the first-order case. The book is divided in two parts: in the first one the necessary background about these logics, with some minor new results, are presented. The second part is devoted to more specific topics: there are five chapters, each one about a different problem. In chapter 6 a temporal semantics for Basic Logic BL is presented. In chapter 7 we move to first-order logics, by studying the supersoundness property: we have improved some previous works about this theme, by expanding the analysis to many extensions of the first-order version of MTL. Chapter 8 is dedicated to four different families of n-contractive axiomatic extensions of BL, analyzed in the propositional and in the first-order case: completeness, computational and arithmetical complexity, amalgamation and interpolation properties are studied. Finally, chapters 9 and 10 are about Nilpotent Minimum logic: in chapter 9 the sets of tautologies of some NM-chains (subalgebras of [0,1]_NM) are studied, compared and the problems of axiomatization and undecidability are tackled. Chapter 10, instead, concerns some logical and algebraic properties of (propositional) Nilpotent Minimum logic. The results (or an extended version of them) of these last chapters have been also presented in papers

ON SOME AXIOMATIC EXTENSIONS OF THE MONOIDAL T-NORM BASED LOGIC MTL: AN ANALYSIS IN THE PROPOSITIONAL AND IN THE FIRST-ORDER CASE

The scientific area this thesis belongs to are many-valued logics: in particular, the logic MTL and some of its extensions, in the propositional and in the first-order case (see [8],[9],[6],[7]). The thesis is divided in two parts: in the first one the necessary background about these logics, with some minor new results, are presented. The second part is devoted to more specific topics: there are five chapters, each one about a different problem. In chapter 6 a temporal semantics for Basic Logic BL is presented. In chapter 7 we move to first-order logics, by studying the supersoundness property: we have improved some previous works about this theme, by expanding the analysis to many extensions of the first-order version of MTL. Chapter 8 is dedicated to four different families of n-contractive axiomatic extensions of BL, analyzed in the propositional and in the first-order case: completeness, computational and arithmetical complexity, amalgamation and interpolation properties are studied. Finally, chapters 9 and 10 are about Nilpotent Minimum logic (NM, see [8]): in chapter 9 the sets of tautologies of some NM-chains (subalgebras of [0,1]_NM) are studied, compared and the problems of axiomatization and undecidability are tackled. Chapter 10, instead, concerns some logical and algebraic properties of (propositional) Nilpotent Minimum logic. The results (or an extended version of them) of these last chapters have been also presented in papers [1, 4, 5, 2, 3]. ---------------------------------References--------------------------------------------- [1] S. Aguzzoli, M. Bianchi, and V. Marra. A temporal semantics for Basic Logic. Studia Logica, 92(2), 147-162, 2009. doi:10.1007/s11225-009-9192-3. [2] M. Bianchi. First-order Nilpotent Minimum Logics: first steps. Submitted for publication,2010. [3] M. Bianchi. On some logical and algebraic properties of Nilpotent Minimum logic and its relation with G\uf6del logic. Submitted for publication, 2010. [4] M. Bianchi and F. Montagna. Supersound many-valued logics and Dedekind-MacNeille completions. Arch. Math. Log., 48(8), 719-736, 2009. doi:10.1007/s00153-009-0145-3. [5] M. Bianchi and F. Montagna. n-contractive BL-logics. Arch. Math. Log., 2010. doi:10.1007/s00153-010-0213-8. [6] P. Cintula, F. Esteva, J. Gispert, L. Godo, F. Montagna, and C. Noguera. Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalencies. Ann. Pure Appl. Log., 160(1), 53-81, 2009. doi:10.1016/j.apal.2009.01.012. [7] P. Cintula and P. H\ue1jek. Triangular norm predicate fuzzy logics. Fuzzy Sets Syst., 161(3), 311-346, 2010. doi:10.1016/j.fss.2009.09.006. [8] F. Esteva and L. Godo. Monoidal t-norm based logic: Towards a logic for left-continuous t-norms. Fuzzy sets Syst., 124(3), 271-288, 2001. doi:10.1016/S0165-0114(01)00098-7. [9] P. H\ue1jek. Metamathematics of Fuzzy Logic, volume 4 of Trends in Logic. Kluwer Academic Publishers, paperback edition, 1998. ISBN:9781402003707

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