Infinite-valued Logic Research Articles

On White's Expansion of Lukasiewicz Logic

Avariant of Łukasiewicz logic defined in a remark in White's 1979 paper on the consistency of the axiom of comprehension in the infinite-valued predicate logic of Łukasiewicz is studied; it is shown that the logic in question is an interesting expansion of Łukasiewicz logic which deserves further investigation together with its further expansions. Several formulas are proved in it and some model theory is offered. In the Appendix A, it is shown that a set theory with full comprehension over this logic (claimed consistent by White) is contradictory.

A problem of annihilator primes in [formula omitted]-algebras

Using Mundici’s equivalence [D. Mundici, Interpretation of AF C ∗ -algebras in Łukasiewicz sentential calculus, J. Funct. Anal. 65 (1) (1986) 15–63] this note generalizes the result of [R. Cignoli, A. Torrens, The poset of prime l -ideals of an abelian l -group with strong unit, J. Algebra 184 (2) (1996) 604–612] for prime ideals and polars. Moreover, the generalization characterizes M V -algebras for which every prime ideal is the annihilator of a linear ordered ideal, and thus gives a negative answer to a problem of [L.P., Belluce, Semisimple algebras of infinite valued logic and bold fuzzy set theory, Canad. J. Math. 38 (6) (1986) 1356–1379]

Strong equivalence of logic programs under the infinite-valued semantics

We consider the notion of strong equivalence [V. Lifschitz, D. Pearce, A. Valverde, Strongly equivalent logic programs, ACM Transactions on Computational Logic 2 (4) (2001) 526–541] of normal propositional logic programs under the infinite-valued semantics [P. Rondogiannis, W.W. Wadge, Minimum model semantics for logic programs with negation-as-failure, ACM Transactions on Computational Logic 6 (2) (2005) 441–467] (which is a purely model-theoretic semantics that is compatible with the well-founded one). We demonstrate that two such programs are strongly equivalent under the infinite-valued semantics if and only if they are logically equivalent in the corresponding infinite-valued logic. In particular, we show that strong equivalence of normal propositional logic programs is decidable, and more specifically coNP-complete. Our results have a direct implication for the well-founded semantics since, as we demonstrate, if two programs are strongly equivalent under the infinite-valued semantics, then they are also strongly equivalent under the well-founded semantics.

Invariant Measures in Free MV-Algebras

MV-algebras can be viewed either as the Lindenbaum algebras of Łukasiewicz infinite-valued logic, or as unit intervals of lattice-ordered abelian groups in which a strong order unit has been fixed. The free n-generated MV-algebra Free n is representable as an algebra of continuous piecewise-linear functions with integer coefficients over the unit cube [0, 1] n . The maximal spectrum of Free n is canonically homeomorphic to [0, 1] n , and the automorphisms of the algebra are in 1–1 correspondence with the pwl homeomorphisms with integer coefficients of the unit cube. In this article, we prove that the only probability measure on [0, 1] n which is null on underdimensioned 0-sets and is invariant under the group of all such homeomorphisms is the Lebesgue measure. From the viewpoint of lattice-ordered abelian groups, this fact means that, in relevant cases, fixing an automorphism-invariant strong unit implies fixing a distinguished probability measure on the maximal spectrum. From the viewpoint of algebraic logic, it means that the only automorphism-invariant truth averaging process that detects pseudotrue propositions is the integral with respect to Lebesgue measure.

Monadic GMV-algebras

Monadic MV-algebras are an algebraic model of the predicate calculus of the Łukasiewicz infinite valued logic in which only a single individual variable occurs. GMV-algebras are a non-commutative generalization of MV-algebras and are an algebraic counterpart of the non-commutative Łukasiewicz infinite valued logic. We introduce monadic GMV-algebras and describe their connections to certain couples of GMV-algebras and to left adjoint mappings of canonical embeddings of GMV-algebras. Furthermore, functional MGMV-algebras are studied and polyadic GMV-algebras are introduced and discussed.

Monadic Bounded Commutative Residuated ℓ-monoids

Bounded commutative residuated l-monoids are a generalization of algebras of propositional logics such as BL-algebras, i.e. algebraic counterparts of the basic fuzzy logic (and hence consequently MV-algebras, i.e. algebras of the Łukasiewicz infinite valued logic) and Heyting algebras, i.e. algebras of the intuitionistic logic. Monadic MV-algebras are an algebraic model of the predicate calculus of the Łukasiewicz infinite valued logic in which only a single individual variable occurs. We introduce and study monadic residuated l-monoids as a generalization of monadic MV-algebras.

Convergence with a fixed regulator in perfect MV-algebras

AbstractMV-algebras were introduced by Chang as an algebraic counterpart of the Łukasiewicz infinite-valued logic. D. Mundici proved that the category of MV-algebras is equivalent to the category of abelian

Formal systems of fuzzy logic and their fragments

Formal systems of fuzzy logic (including the well-known Łukasiewicz and Gödel–Dummett infinite-valued logics) are well-established logical systems and respected members of the broad family of the so-called substructural logics closely related to the famous logic BCK. The study of fragments of logical systems is an important issue of research in any class of non-classical logics. Here we study the fragments of nine prominent fuzzy logics to all sublanguages containing implication. However, the results achieved in the paper for those nine logics are usually corollaries of theorems with much wider scope of applicability. In particular, we show how many of these fragments are really distinct and we find axiomatic systems for most of them. In fact, we construct strongly separable axiomatic systems for eight of our nine logics. We also fully answer the question for which of the studied fragments the corresponding class of algebras forms a variety. Finally, we solve the problem how to axiomatize predicate versions of logics without the lattice disjunction (an essential connective in the usual axiomatic system of fuzzy predicate logics).

First-order Gödel logics

First-order Gödel logics are a family of finite- or infinite-valued logics where the sets of truth values V are closed subsets of [ 0 , 1 ] containing both 0 and 1. Different such sets V in general determine different Gödel logics G V (sets of those formulas which evaluate to 1 in every interpretation into V ). It is shown that G V is axiomatizable iff V is finite, V is uncountable with 0 isolated in V , or every neighborhood of 0 in V is uncountable. Complete axiomatizations for each of these cases are given. The r.e. prenex, negation-free, and existential fragments of all first-order Gödel logics are also characterized.

On the infinite-valued Łukasiewicz logic that preserves degrees of truth

Łukasiewicz’s infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the Łukasiewicz algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Łukasiewicz algebra by using a “truth-preserving” scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there exists no Gentzen calculus fully adequate for it. Another presentation of the same deductive system can be obtained from a substructural Gentzen calculus. In this paper we use the framework of abstract algebraic logic to study a different deductive system which uses the aforementioned algebra under a scheme of “preservation of degrees of truth”. We characterize the resulting deductive system in a natural way by using the lattice filters of Wajsberg algebras, and also by using a structural Gentzen calculus, which is shown to be fully adequate for it. This logic is an interesting example for the general theory: it is selfextensional, non-protoalgebraic, and satisfies a “graded” deduction theorem. Moreover, the Gentzen system is algebraizable. The first deductive system mentioned turns out to be the extension of the second by the rule of Modus Ponens.

Canonical extensions of double quasioperator algebras: An algebraic perspective on duality for certain algebras with binary operations

The context for this paper is a class of distributive lattice expansions, called double quasioperator algebras (DQAs). The distinctive feature of these algebras is that their operations preserve or reverse both join and meet in each coordinate. Algebras of this type provide algebraic semantics for certain non-classical propositional logics. In particular, MV-algebras, which model the Łukasiewicz infinite-valued logic, are DQAs. Varieties of DQAs are here studied through their canonical extensions. A variety of this type having additional operations of arity at least 2 may fail to be canonical; it is already known, for example, that the variety of MV-algebras is not. Non-canonicity occurs when basic operations have two distinct canonical extensions and both are necessary to capture the structure of the original algebra. This obstruction to canonicity is different in nature from that customarily found in other settings. A generalized notion of canonicity is introduced which is shown to circumvent the problem. In addition, generalized canonicity allows one to capture on the canonical extensions of DQAs the algebraic operations in such a way that the laws that these obey may be translated into first-order conditions on suitable frames. This correspondence may be seen as the algebraic component of duality, in a way which is made precise. In many cases of interest, binary residuated operations are present. An operation h which, coordinatewise, preserves ∨ and 0 lifts to an operation which is residuated, even when h is not. If h also preserves binary meet then the upper adjoints behave in a functional way on the frames.

Bernoulli automorphisms of finitely generated free MV-algebras

MV-algebras can be viewed either as the Lindenbaum algebras of Łukasiewicz infinite-valued logic, or as unit intervals [ 0 , u ] of lattice-ordered abelian groups in which a strong order unit u > 0 has been fixed. They form an equational class, and the free n -generated free MV-algebra is representable as an algebra of piecewise-linear continuous functions with integer coefficients over the unit n -dimensional cube. In this paper we show that the automorphism group of such a free algebra contains elements having strongly chaotic behaviour, in the sense that their duals are measure-theoretically isomorphic to a Bernoulli shift. This fact is noteworthy from the viewpoint of algebraic logic, since it gives a distinguished status to Lebesgue measure as an averaging measure on the space of valuations. As an ergodic theory fact, it provides explicit examples of volume-preserving homeomorphisms of the unit cube which are piecewise-linear with integer coefficients, preserve the denominators of rational points, and enjoy the Bernoulli property.

Distribution of Computer Jobs in Automatic Control Systems on the Basis of Infinite-valued Logic

The problem of optimal distribution of job groups among the computers in the networks of various automatic control systems was formulated and solved. The optimization algorithm is based on the infinite-valued logic.

H�jek basic fuzzy logic and ?ukasiewicz infinite-valued logic

Using the theory of BL-algebras, it is shown that a propositional formula ϕ is derivable in Łukasiewicz infinite valued Logic if and only if its double negation ˜˜ϕ is derivable in Hajek Basic Fuzzy logic. If SBL is the extension of Basic Logic by the axiom (φ & (φ→˜φ)) → ψ, then ϕ is derivable in in classical logic if and only if ˜˜ ϕ is derivable in SBL. Axiomatic extensions of Basic Logic are in correspondence with subvarieties of the variety of BL-algebras. It is shown that the MV-algebra of regular elements of a free algebra in a subvariety of BL-algebras is free in the corresponding subvariety of MV-algebras, with the same number of free generators. Similar results are obtained for the generalized BL-algebras of dense elements of free BL-algebras.

Tableaux for Łukasiewicz Infinite-valued Logic

In this work we propose a labelled tableau method for Łukasiewicz infinite-valued logic Lω. The method is based on the Kripke semantics of this logic developed by Urquhart [25] and Scott [24]. On the one hand, our method falls under the general paradigm of labelled deduction [8] and it is rather close to the tableau systems for sub-structural logics proposed in [4]. On the other hand, it provides a CoNP decision procedure for Lω validity by reducing the check of branch closure to linear programming

Finite-valued reductions of infinite-valued logics

In this paper we present a method to reduce the decision problem of several infinite-valued propositional logics to their finite-valued counterparts. We apply our method to Łukasiewicz, Godel and Product logics and to some of their combinations. As a byproduct we define sequent calculi for all these infinite-valued logics and we give an alternative proof that their tautology problems are in co-NP.

On the Beth Property in Extension of Lukasiewicz Logics

It is proved that a necessary condition for conservative axiomatic extensions of the infinite-valued (or n-valued) Lukasiewicz logic to possess the Beth definability property consists in the presence in the language of these extensions of a countable set (resp. a set of power n) of constant terms nonequivalent with respect to the given extensions.

Adding structure to MV-algebras

We study the algebras corresponding to various extensions of Łukasiewicz infinite-valued logics. In particular, we study the structures resulting from adding the characteristic function for truth, adding an arithmetical product, and the corresponding residuation operator. We characterize the free algebras in the relative equational classes, and we discuss their spectral spaces.

Decidable and undecidable prime theories in infinite-valued logic

In classical propositional logic, a theory T is prime (i.e., for every pair of formulas F,G, either T⊢F→G or T⊢G→F) iff it is complete. In Łukasiewicz infinite-valued logic the two notions split, completeness being stronger than primeness. Using toric desingularization algorithms and the fine structure of prime ideal spaces of free ℓ-groups, in this paper we shall characterize prime theories in infinite-valued logic. We will show that recursively enumerable (r.e.) prime theories over a finite number of variables are decidable, and we will exhibit an example of an undecidable r.e. prime theory over countably many variables.

Natural Dualities for Varieties of MV-Algebras, I

MV-algebras are the Lindenbaum algebras for Łukasiewicz's infinite-valued logic, just as Boolean algebras correspond to the classical propositional calculus. The finitely generated subvarieties of the variety M of all MV-algebras are generated by finite chains. We develop a natural duality, in the sense of Davey and Werner, for each subvariety generated by a finite chain, and use it to describe the free and the injective members of these classes. Finally, we point out the relations between the dualities and some categorical equivalences discovered by A. Di Nola and A. Lettieri.