Product Logic

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Lluis Godo - One of the best experts on this subject based on the ideXlab platform.

Sara Ugolini - One of the best experts on this subject based on the ideXlab platform.

Franco Montagna - One of the best experts on this subject based on the ideXlab platform.

  • Algebraic and Proof-theoretic Aspects of Non-classical Logics - Rényi-Ulam game semantics for Product Logic and for the Logic of cancellative hoops
    Lecture Notes in Computer Science, 2020
    Co-Authors: Sándor Jenei, Franco Montagna
    Abstract:

    Connections between games and Logic are quite common in the literature: for example, to every analytic proof system with the subformula property (hence admitting cut-elimination) one can associate a game in which a player tries to find a cut-free proof and his opponent can attack parts of the proof constructed since then. Along these lines, formulas correspond to games and proofs correspond to winning strategies. A first connection between many-valued Logic and games was discovered by Giles in [9]. A variant of such semantics was used in [4] in order to obtain a uniform proof system with a game-theoretical interpretation for Łukasiewicz, Product and Godel Logics. The above mentioned papers are extremely interesting, but we would say that the interest of this game semantics is more proof-theoretical than game-theoretical.

  • Proof search and Co-NP completeness for many-valued Logics
    Fuzzy Sets and Systems, 2016
    Co-Authors: Mattia Bongini, Agata Ciabattoni, Franco Montagna
    Abstract:

    We provide a methodology to introduce proof search oriented calculi for a large class of many-valued Logics, and a sufficient condition for their Co-NP completeness. Our results apply to many well known Logics including Godel, Łukasiewicz and Product Logic, as well as Hajek's Basic Fuzzy Logic.

  • A Categorical Equivalence for Product Algebras
    Studia Logica, 2015
    Co-Authors: Franco Montagna, Sara Ugolini
    Abstract:

    In this paper we provide a categorical equivalence for the category $${\mathcal{P}}$$ P of Product algebras, with morphisms the homomorphisms. The equivalence is shown with respect to a category whose objects are triplets consisting of a Boolean algebra B , a cancellative hoop C and a map $${\vee_e}$$ ∨ e from B  × C into C satisfying suitable properties. To every Product algebra P , the equivalence associates the triplet consisting of the maximum boolean subalgebra B ( P ), the maximum cancellative subhoop C ( P ), of P , and the restriction of the join operation to B  × C . Although several equivalences are known for special subcategories of $${\mathcal{P}}$$ P , up to our knowledge, this is the first equivalence theorem which involves the whole category of Product algebras. The syntactic counterpart of this equivalence is a syntactic reduction of classical Logic CL and of cancellative hoop Logic CHL to Product Logic, and viceversa.

  • Product Logic and probabilistic Ulam games
    Fuzzy Sets and Systems, 2007
    Co-Authors: Franco Montagna, Claudio Marini, Giulia Simi
    Abstract:

    There is a well-known game semantics for Lukasiewicz Logic, introduced by Daniele Mundici, namely the Renyi-Ulam game. Records in a Reny-Ulam game are coded by functions, which constitute an MV-algebra, and it is possible to prove a completeness theorem with respect to this semantics. In this paper we investigate some probabilistic variants of the Renyi-Ulam game, and we prove that some of them constitute a complete game semantics for Product Logic, whilst some other constitute a game semantics for a Logic between @PMTL and Product Logic.

  • Algebraic and Proof-theoretic Aspects of Non-classical Logics - Notes on strong completeness in łukasiewicz, Product and BL Logics and in their first-order extensions
    Lecture Notes in Computer Science, 2007
    Co-Authors: Franco Montagna
    Abstract:

    In this paper we investigate the problem of characterizing infinite consequence relation in standard BL-algebras by the adding of new rules. First of all, we note that finitary rules do not help, therefore we need at least one infinitary rule. In fact we show that one infinitary rule is sufficient to obtain strong standard completeness, also in the first-order case. Similar results are obtained for Product Logic and for Łukasiewicz Logic. Finally, we show some applications of our results to probabilistic Logic over many-valued events and to first-order many-valued Logic. In particular, we show a tight bound to the complexity of BL first-order formulas which are valid in the standard semantics.

Francesc Esteva - One of the best experts on this subject based on the ideXlab platform.

  • about strong standard completeness of Product Logic
    2014
    Co-Authors: Amanda Vidal, Francesc Esteva, Lluis Godo
    Abstract:

    The authors acknowledge support of the Spanish projects EdeTRI (TIN2012-39348-C02-01) and AT (CONSOLIDER CSD 2007- 0022). Amanda Vidal is supported by a CSIC JAE Predoc grant

  • on the implementation of a fuzzy dl solver over infinite valued Product Logic with smt solvers
    Scalable Uncertainty Management, 2013
    Co-Authors: Teresa Alsinet, Marco Cerami, David Barroso, Ramon Bejar, Francesc Esteva
    Abstract:

    In this paper we explain the design and preliminary implementation of a solver for the positive satisfiability problem of concepts in a fuzzy description Logic over the infinite-valued Product Logic. This very solver also answers 1-satisfiability in quasi-witnessed models. The solver works by first performing a direct reduction of the problem to a satisfiability problem of a quantifier free boolean formula with non-linear real arithmetic properties, and secondly solves the resulting formula with an SMT solver. We show that the satisfiability problem for such formulas is still a very challenging problem for even the most advanced SMT solvers, and so it represents an interesting problem for the community working on the theory and practice of SMT solvers.

  • SUM - On the Implementation of a Fuzzy DL Solver over Infinite-Valued Product Logic with SMT Solvers
    Lecture Notes in Computer Science, 2013
    Co-Authors: Teresa Alsinet, Marco Cerami, David Barroso, Ramon Bejar, Francesc Esteva
    Abstract:

    In this paper we explain the design and preliminary implementation of a solver for the positive satisfiability problem of concepts in a fuzzy description Logic over the infinite-valued Product Logic. This very solver also answers 1-satisfiability in quasi-witnessed models. The solver works by first performing a direct reduction of the problem to a satisfiability problem of a quantifier free boolean formula with non-linear real arithmetic properties, and secondly solves the resulting formula with an SMT solver. We show that the satisfiability problem for such formulas is still a very challenging problem for even the most advanced SMT solvers, and so it represents an interesting problem for the community working on the theory and practice of SMT solvers.

  • On Completeness Results for Predicate Lukasiewicz, Product, Godel and Nilpotent Minimum Logics Expanded with Truth-constants
    2011
    Co-Authors: Francesc Esteva, Lluis Godo, Carles Noguera
    Abstract:

    In this paper we deal with generic expansions of rst-order predicate log- ics of some left-continuous t-norms with a countable set of truth-constants. Besides already known results for the case of Lukasiewicz Logic, we obtain new conservativeness and completenesss results for some other expansions. Namely, we prove that the expansions of predicate Product, Godel and Nilpotent Minimum Logics with truth-constants are conservative, which already implies the failure of standard completeness for the case of Product Logic. In contrast, the expansions of predicate Godel and Nilpotent Minimum Logics are proved to be strong standard complete but, when the semantics is restricted to the canonical algebra, they are proved to be complete only for tautologies. Moreover, when the language is restricted to evaluated formulae we prove canonical completeness for deductions from nite sets of premises.

  • decidability of a description Logic over infinite valued Product Logic
    Principles of Knowledge Representation and Reasoning, 2010
    Co-Authors: Marco Cerami, Francesc Esteva
    Abstract:

    This paper proves that validity and satisfiability of assertions in the Fuzzy Description Logic based on infinite-valued Product Logic with universal and existential quantifiers (which are non-interdefinable) is decidable when we only consider quasi-witnessed interpretations. We prove that this restriction is neither necessary for the validity problem (i.e., the validity of assertions in the Fuzzy Description Logic based on infinite-valued Product Logic is decidable) nor for the positive satisfiability problem, because quasi-witnessed interpretations are particularly adequate for the infinite-valued Product Logic. We give an algorithm that reduces the problem of validity (and satisfiability) of assertions in our Fuzzy Description Logic (restricted to quasi-witnessed interpretations) to a semantic consequence problem, with finite number of hypothesis, on infinite-valued propositional Product Logic.

Tommaso Flaminio - One of the best experts on this subject based on the ideXlab platform.

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