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In this note we provide a straightforward translation ${{\mathcal{C}^{{\Gamma}}_{p}}}(T)$ for sets of formulas T and $H_\Gamma(\exists x A(x))$ for existential formulas $\exists x A(x)$ s.t. ${{\mathcal{C}^{{\Gamma}}_{p}}}(T) \vdash H_\Gamma(\exists x A(x))$ expresses "$\exists x A(x)$ is derivable constructively from T iff it is derivable at all". [ABSTRACT FROM AUTHOR]

It is shown that a conservative expansion of infinite valued Łukasiewicz logic by new connectives univocally determined by their axioms does not necessarily have a complete semantics in the real interval [0,1]. However, such extensions are always complete with respect to valuations in a family of MV-chains. Rational Łukasiewicz logic being the largest one that has a complete semantics in [0,1]. In addition, this logic does not admit expansions by axiomatic implicit connectives that are not already explicit. Similar results are obtained for n-valued Łukasiewicz logic and for the logic of abelian lattice ordered groups. These and related results are obtained by the study of compatible operations implicitly defined by identities in the varieties of MV-algebras and abelian ℓ-groups; the pertaining algebraic results having independent interest. [ABSTRACT FROM AUTHOR]

An outline of the history of the algebras corresponding to Łukasiewicz many-valued logic from the pioneering work by G. Moisil in 1940 until D. Mundici's work in 1986. [ABSTRACT FROM AUTHOR]

We introduce the class of Symmetric MV-algebras. Such algebras have a suitable behavior with respect to a family of MV-polynomials. It turns out that the class of Symmetric MV-algebras can be characterized as the class of MV-algebras having homomorphic image in the variety generated by a single MV-chain with p + 1 elements, where p = 1 or p is a prime number. Also, using symmetric MV-algebras, we provide a new characterization of the above mentioned varieties. [ABSTRACT FROM AUTHOR]

Basic constructions of two different theories are presented. The first one is based on the Łukasiewicz connectives, the second on the max - min connectives. In both cases the joint observable is constructed. As an application the central limit theorem is proved. [ABSTRACT FROM AUTHOR]

A few things are known, and many are unknown, on the automorphism group of the free MV-algebra over n − 1 generators. In this paper we show that this group appears as the stabilizer of ${\rm 1\mskip-4mu l}$ in the larger group of all automorphisms of the free cancellative hoop over n generators. Both groups have a dual action on the same space, namely the (n − 1)-dimensional cube. The larger group has a richer dynamics, at the expense of loosing the two key features of the McNaughton homeomorphisms: preservation of denominators of rational points, and preservation of the Lebesgue measure. We present here some basic results, some examples, and some problems. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 Gödel 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. [ABSTRACT FROM AUTHOR]

In this paper we define the concept of a profile, which is a characteristic clause set, corresponding to an LK-proof in first-order logic, which is invariant under rule permutations. It is shown (via cut-elimination) that the profile is even invariant under a large class of proof transformations (called "simple transformations"), which includes transformations to negation normal form. As proofs having the same profile show the same behavior w.r.t. cut-elimination (which can be formally defined via the method CERES), proofs obtained by simple transformations can be considered as equal in this sense. A comparison with related results based on proof nets is given: in particular it is shown that proofs having the same profile define a larger equivalence class than those having the same proof net. [ABSTRACT FROM AUTHOR]

If T is a consistent theory over a fuzzy predicate logic with Gödel negation (e.g. Gödel logic, product logic) then T remains consistent after adding the schema tertium non datur (ϕ ∨ ¬ϕ) for sentences (closed formulas). We prove and discuss this. [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for the first-order theory of a finitely presented abelian lattice-ordered group to be decidable. We also show that if the number of generators is at most 3, then elementary equivalence implies isomorphism. We deduce from our methods that the theory of the free MV-algebra on at least 2 generators is undecidable. [ABSTRACT FROM AUTHOR]

Loops and other unbound control structures constitute a major bottleneck in formal software verification, because correctness proofs over such control structures generally require user interaction: typically, induction hypotheses or invariants must be found or modified by hand. Such interaction involves expert knowledge of the underlying calculus and proof engine. We show that one can replace interactive proof techniques, such as induction, with automated first-order reasoning in order to deal with parallelizable loops. A loop can be parallelized, whenever the execution of a generic iteration of its body depends only on the step parameter and not on other iterations. We use a symbolic dependence analysis that ensures parallelizability. This guarantees soundness of a proof rule that transforms a loop into a universally quantified update of the state change information effected by the loop body. This rule makes it possible to employ automatic first-order reasoning techniques to deal with loops. The method has been implemented in the KeY verification tool. We evaluated its applicability with representative case studies from the Java Card domain. [ABSTRACT FROM AUTHOR]

Mundici's Γ functor establishes a categorical equivalence between MV-algebras and lattice-ordered Abelian groups with a strong unit. In this short note we present a first step towards the generalization of such a relationship when we replace MV-algebras by weaker structures obtained by dropping the divisibility condition. These structures are the so-called involutive monoidal t-norm based algebras, IMTL-algebras for short. In this paper we restrict ourselves to linearly ordered IMTL-algebras, for which we show a one-to-one correspondence with a kind of ordered grupoid-like structures with a strong unit. A key feature is that the associativity property in such a new structure related to a IMTL-chain is lost as soon the IMTL-chain is no longer a MV-chain and the strong unit used in Mundici's Γ functor is required here to have stronger properties. Moreover we define a functor between the category of such structures and the category of IMTL algebras that is a generalization of Mundici's functor Γ and, restricted to their linearly ordered objects, a categorical equivalence. [ABSTRACT FROM AUTHOR]

We try to make a distinction between the idea of representing and that of interpreting a mathematical structure. We present a slight generalization of Di Nola's Representation Theorem as to incorporate this point of view. Furthermore, we examine some preservation and functorial aspects of the Boolean power construction. [ABSTRACT FROM AUTHOR]

Generalizations of the Cantor-Bernstein theorem have been proved for different types of algebras, starting from σ-complete orthomodular lattices and σ-complete MV-algebras and continuing with more general structures, including (pseudo) effect algebras and (pseudo) BCK-algebras. E.g., for σ-complete MV-algebras a version of the Cantor-Bernstein theorem has been proved which assumes that the bounds of isomorphic intervals are boolean. There is another direction of research which has been paid less attention. We ask which algebras satisfy the Cantor-Bernstein theorem in the same form as for σ-complete boolean algebras, without any additional assumption. In the case of orthomodular lattices, it has been proved that this class is rather large. E.g., every orthomodular lattice can be embedded as a subalgebra or expressed as an epimorphic image of a member of this class. On the other hand, also the complement of this class is large in the same sense. We study the analogous question for MV-algebras and we find out interesting examples of MV-algebras which possess or do not possess this property. This contributes to the investigations of the scope of validity of the Cantor-Bernstein theorem in its original form. [ABSTRACT FROM AUTHOR]

A characteristic feature of quantum computation is the use of reversible logical operations. These correspond to quantum logical gates that are mathematically represented by unitary operators defined on convenient Hilbert spaces. Two questions arise: 1) to what extent is quantum computation bound to the use of reversible logical operations? 2) How to identify the logical operations that admit a quantum computational simulation by means of appropriate gates? We introduce the notion of quantum computational simulation of a binary function defined on the real interval [0,1], and we prove that for any binary Boolean function there exists a unique fuzzy extension admitting a quantum computational simulation. As a consequence, the Łukasiewicz conjunction and disjunction do not admit a quantum computational simulation. [ABSTRACT FROM AUTHOR]

A paraconsistent logic is a logic which allows non-trivial inconsistent theories. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa's approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely differently when contradictions are involved. da Costa's approach has led to the family of Logics of Formal (In)consistency (LFIs). In this paper we provide non-deterministic semantics for a very large family of first-order LFIs (which includes da Costa's original system $C^*_1$, as well as thousands of other logics). We show that our semantics is effective and modular, and we use this effectiveness to derive some important properties of logics in this family. [ABSTRACT FROM AUTHOR]

In this work we present a MATLAB implementation of a fuzzy controller with Conditionally Firing Rules (CFR). The performance of Mamdani-Assilian, Takagi-Sugeno-Kang and CFR inferences are compared and analyzed on two test examples. [ABSTRACT FROM AUTHOR]

MTL is the logic of all left-continuous t-norms and their residua. Its algebraic semantics is constituted by the variety (MTL) of MTL-algebras. Among schematic extensions of MTL there are infinite-valued logics such that the finitely generated free algebras in the corresponding subvariety of (MTL) are finite. In this paper we focus on Gödel and Nilpotent Minimum logics. We give concrete representations of their associated free algebras in terms of finite algebras of sections over finite posets. [ABSTRACT FROM AUTHOR]