Tarski Algebra Research Articles

Transitivity of implicative aBE algebras

We prove that every implicative aBE algebra satisfies the transitivity property. This means that every implicative aBE algebra is a Tarski algebra, and thus is also a commutative BCK algebra.

FIBRED ALGEBRAIC SEMANTICS FOR A VARIETY OF NON-CLASSICAL FIRST-ORDER LOGICS AND TOPOLOGICAL LOGICAL TRANSLATION

AbstractLawvere hyperdoctrines give categorical algebraic semantics for intuitionistic predicate logic. Here we extend the hyperdoctrinal semantics to a broad variety of substructural predicate logics over the Typed Full Lambek Calculus, verifying their completeness with respect to the extended hyperdoctrinal semantics. This yields uniform hyperdoctrinal completeness results for numerous logics such as different types of relevant predicate logics and beyond, which are new results on their own; i.e., we give uniform categorical semantics for a broad variety of non-classical predicate logics. And we introduce an analogue of Lawvere–Tierney topology and cotopology in the hyperdoctrinal setting, which gives a unifying perspective on different logical translations, in particular allowing for a uniform treatment of Girard’s exponential translation between linear and intuitionistic logics and of Kolmogorov’s double negation translation between intuitionistic and classical logics. In the hyerdoctrinal conception, type theories are categories, logics over type theories are functors, and logical translations between them, then, are natural transformations, in particular Lawvere–Tierney topologies and cotopologies on hyperdoctrines. The view of logical translations as hyperdoctrinal Lawvere–Tierney topologies and cotopologies has not been elucidated before, and may be seen as a novel contribution of the present work. From a broader perspective, this work may be regarded as taking first steps towards interplay between algebraic and categorical logics; it is, technically, a combination of substructural (or Lambekian) algebraic logic and hyperdoctrinal (or Lawverian) categorical logic, as the hyperdoctrinal completeness theorem is shown via the integration of the Lindenbaum–Tarski algebra construction with the syntactic category construction. As such this work lays a foundation for further interactions between algebraic and categorical logics.

Inquisitive Heyting Algebras

In this paper we introduce a class of inquisitive Heyting algebras as algebraic structures that are isomorphic to algebras of finite antichains of bounded implicative meet semilattices. It is argued that these structures are suitable for algebraic semantics of inquisitive superintuitionistic logics, i.e. logics of questions based on intuitionistic logic and its extensions. We explain how questions are represented in these structures (prime elements represent declarative propositions, non-prime elements represent questions, join is a question-forming operation) and provide several alternative characterizations of these algebras. For instance, it is shown that a Heyting algebra is inquisitive if and only if its prime filters and filters generated by sets of prime elements coincide and prime elements are closed under relative pseudocomplement. We prove that the weakest inquisitive superintuitionistic logic is sound with respect to a Heyting algebra iff the algebra is what we call a homomorphic p-image of some inquisitive Heyting algebra. It is also shown that a logic is inquisitive iff its Lindenbaum–Tarski algebra is an inquisitive Heyting algebra.

Subordination Tarski algebras

ABSTRACTIn this work we will study Tarski algebras endowed with a subordination, called subordination Tarski algebras. We will define the notion of round filters, and we will study the class of irreducible round filters and the maximal round filters, called ends. We will prove that the poset of all round filters is a lattice isomorphic to the lattice of the congruences that are compatible with the subordination. We will prove that every end is an irreducible round filter, and that in a topological subordination Tarski algebra A, every irreducible round filter is an end iff A is a monadic subordination Tarski algebra. As corollary of this result we have that the variety of monadic Tarski algebra can be characterised as the topological Tarski algebras where every irreducible open filter is a maximal open filter.

Reflection calculus and conservativity spectra

Strictly positive logics have recently attracted attention both in the description logic and in the provability logic communities for their combination of efficiency and sufficient expressivity. The language of Reflection Calculus, , consists of implications between formulas built up from propositional variables and the constant `true' using only conjunction and diamond modalities which are interpreted in Peano arithmetic as restricted uniform reflection principles. The language of is extended by another series of modalities representing the operators associating with a given arithmetical theory its fragment axiomatized by all theorems of of arithmetical complexity for all . It is noted that such operators, in a strong sense, cannot be represented in the full language of modal logic. A formal system is formulated that extends and is sound and (it is conjectured) complete under this interpretation. It is shown that in this system one is able to express the iterations of reflection principles up to any ordinal . Second, normal forms are provided for its variable-free fragment. This fragment is thereby shown to be algorithmically decidable and complete with respect to its natural arithmetical semantics. In the last part of the paper the Lindenbaum–Tarski algebra of the variable-free fragment of and its dual Kripke structure are characterized in several natural ways. Most importantly, elements of this algebra correspond to the sequences of proof-theoretic -ordinals of bounded fragments of Peano arithmetic called conservativity spectra, as well as to points of Ignatiev's well-known Kripke model. Bibliography: 46 titles.

A transfer method from bounded existential Diophantine equations to Tarski algebra formulas

We identify a transfer method from bounded existentially quantified Diophantine equations to formulas of Tarski algebra, the first order theory of the real field. The method is applied to show that NP is contained in ⋃n=1∞Dtime(2a⋅logO(1)n), where a depends only on the given Diophantine equation.

Weak Implication on Monadic Heyting Algebras

In this paper, we introduce an implication operation, called weak implication, which will be quite useful in order to characterize subdirectly irreducible monadic Heyting algebras. Furthermore, it is shown that deductively semisimple algebras are the non trivial ones such that the subalgebra of constants is a Tarski algebra with first element, i.e. a Boolean algebra, as it is mentioned by A. Monteiro and O. Varsavsky in 1957 (Algebras de Heyting monadicas, Actas de las X Jornadas de la Union Matematica Argentina, Bahia Blanca, (1957), (52–62). Finally, it is stated that some of the results established for monadic Heyting algebras are also valid for monadic generalized Heyting algebras.

Algebraization of Jaśkowski’s Paraconsistent Logic D2

Abstract The aim of this paper is to present an algebraic approach to Jaśkowski’s paraconsistent logic D2. We present: a D2-discursive algebra, Lindenbaum- Tarski algebra for D2 and D2-matrices. The analysis is mainly based on the results obtained by Jerzy Kotas in the 70s.

Beyond Rasiowan Systems: Unital Deductive Systems

We deal with monotone structural deductive systems in an unspecified propositional language $${\mathcal{L}}$$ . These systems fall into several overlapping classes, forming a hierarchy. Along with well-known classes of deductive systems such as those of implicative, Fregean and equivalential systems, we consider new classes of unital and weakly implicative systems. The latter class is auxiliary, while the former is central in our discussion. Our analysis of unital systems leads to the concept of Lindenbaum–Tarski algebra which, under some natural conditions, is a free algebra in a variety closely related to the deductive system in focus.

Stochastic game logic

Stochastic game logic (SGL) is a new temporal logic for multi-agent systems modeled by turn-based multi-player games with discrete transition probabilities. It combines features of alternating-time temporal logic (ATL), probabilistic computation tree logic and extended temporal logic. SGL contains an ATL-like modality to specify the individual cooperation and reaction facilities of agents in the multi-player game to enforce a certain winning objective. While the standard ATL modality states the existence of a strategy for a certain coalition of agents without restricting the range of strategies for the semantics of inner SGL formulae, we deal with a more general modality. It also requires the existence of a strategy for some coalition, but imposes some kind of strategy binding to inner SGL formulae. This paper presents the syntax and semantics of SGL and discusses its model checking problem for different types of strategies. The model checking problem of SGL turns out to be undecidable when dealing with the full class of history-dependent strategies. We show that the SGL model checking problem for memoryless deterministic strategies as well as the model checking problem of the qualitative fragment of SGL for memoryless randomized strategies is PSPACE-complete, and we establish a close link between natural syntactic fragments of SGL and the polynomial hierarchy. Further, we give a reduction from the SGL model checking problem under memoryless randomized strategies into the Tarski algebra which proves the problem to be in EXPSPACE.

Fuzzifications of generalized Tarski filters in Tarski algebras

The fuzzification of generalized Tarski filters of generalized Tarski algebras is considered, and related properties are investigated. Characterizations of a fuzzy generalized Tarski filter are established. Based on the theory of a falling shadow, the notion of a falling fuzzy generalized Tarski filter of a generalized Tarski algebra is introduced. Relations between fuzzy generalized Tarski filters and falling fuzzy generalized Tarski filters are provided. A characterization of a falling fuzzy generalized Tarski filter is established.

Physical Propositions and Quantum Languages

The word proposition is used in physics with different meanings, which must be distinguished to avoid interpretational problems. We construct two languages ℒ * (x) and ℒ(x) with classical set-theoretical semantics which allow us to illustrate those meanings and to show that the non-Boolean lattice of propositions of quantum logic (QL) can be obtained by selecting a subset of p-testable propositions within the Boolean lattice of all propositions associated with sentences of ℒ(x). Yet, the aforesaid semantics is incompatible with the standard interpretation of quantum mechanics (QM) because of known no-go theorems. But if one accepts our criticism of these theorems and the ensuing SR (semantic realism) interpretation of QM, the incompatibility disappears, and the classical and quantum notions of truth can coexist, since they refer to different metalinguistic concepts (truth and verifiability according to QM, respectively). Moreover one can construct a quantum language ℒ TQ (x) whose Lindenbaum–Tarski algebra is isomorphic to QL, the sentences of which state (testable) properties of individual samples of physical systems, while standard QL does not bear this interpretation.

Decomposability of free Tarski algebras

In this paper we study the direct decomposability of free Tarski algebras. We show that infinite freely generated Tarski algebras are directly indecomposable, whereas finite freely generated Tarski algebras can only be decomposed into a direct product of two factors, one of which is the two-element Tarski algebra.

Free monadic Tarski algebras

The free monadic Tarski algebra FMT(n) with a finite set G of n free generators, was determined by A. Figallo in [7]. In this paper we indicate a formula, as a function of n, which is an easier way of calculating the number of elements of FMT(n).

Heuristics for constructing logic programs

The Tarski algebra of relations offers a bridge between logic programming and algorithmic programming because it captures the semantics of both brands of programming. In this article we discuss the application to logic programming of heuristics designed to develop algorithmic programs. Not only did these heuristics prove applicable to logic programs, they proved in fact to be more easily applicable to logic programs than to algorithmic programs in the following sense: some steps of the program construction process, which are carried out by hand in the construction of an algorithmic program, are automated in logic programming. This automation is made possible by the ability of logic programming to represent relations and operations on relations.

Complexity of deciding Tarski algebra

Let a formula of Tarski algebra contain k atomic subformulas of the kind ( f i ⩾ 0), 1 ⩽ t ⩽ k , where the polynomials f i ∈ ℤ [ X 1 ,..., X n ] have degrees deg ( f i ) < d , let 2 M be an upper bound for the absolute value of every coeffieient of the polynomials f i , 1 ⩽ i ⩽ k , let a ⩽ n be the number of quantifier alternations in the prenex form of the formula. A decision method for Tarski algebra is described with the running time polynomial in M ( k d ) ( O ( n ) ) 4 n − 2 . Previously known decision procedures have a time complexity polynomial in ( M k d ) 2 O ( n ) .