Heyting Algebras Research Articles

An intuitionistic set-theoretical model of fully dependent CC

AbstractWerner’s set-theoretical model is one of the simplest models of CIC. It combines a functional view of predicative universes with a collapsed view of the impredicative sort “ ${\tt Prop}$ ”. However, this model of ${\tt Prop}$ is so coarse that the principle of excluded middle $P \lor \neg P$ holds. Following our previous work, we interpret ${\tt Prop}$ into a topological space (a special case of Heyting algebra) to make the model more intuitionistic without sacrificing simplicity. We improve on that work by providing a full interpretation of dependent product types, using Alexandroff spaces. We also extend our approach to inductive types by adding support for ${\mathsf{list}}$ s.

Fuzzy relational Galois connections between fuzzy transitive digraphs

We present a fuzzy version of the notion of relational Galois connection between fuzzy transitive directed graphs (fuzzy T-digraphs) on the specific setting in which the underlying algebra of truth values is a complete Heyting algebra. The components of such fuzzy Galois connection are fuzzy relations satisfying certain reasonable properties expressed in terms of the so-called full powering. Moreover, we provide a necessary and sufficient condition under which it is possible to construct a right adjoint for a given fuzzy relation between a fuzzy T-digraph and an unstructured set.

T-rough symmetric Heyting algebras with tense operators

T-rough Heyting algebras were introduced by Eric SanJuan in 2008 as an algebraic formalism for reasoning on finite increasing sequences over Boolean algebras in general and on generalizations of rough set concepts in particular. In this paper, we introduce the variety of algebras, which we call T-rough symmetric Heyting algebras. These algebras constitute an extension of T-rough Heyting algebras and a generalization of symmetric Heyting algebras of order n. Our main interest is the representation theory of tense operators on T-rough symmetric Heyting algebras. In order to do this, a discrete-style duality for these algebras is developed.

L-fuzzy concept analysis using fuzzy categories

The purpose of this paper is to investigate topics in fuzzy concept analysis using the theory of arrow and fuzzy categories. Our approach deals with the original, fuzzy data using fuzzy subsets in order to obtain a notion of fuzzy concepts, fuzzy concept lattices, and attribute implications between fuzzy sets of attributes. Our approach is general in the sense that we only require a Heyting algebra together with a t-norm like operation as the lattice of truth values instead of the unit interval [0,1] or a substructure thereof. As a consequence, regular concept analysis, i.e., concept analysis based on regular sets and the Boolean truth values is just a special case of our approach. In addition, our approach differs from a lot of approaches in the literature, which usually first apply some kind of defuzzyfication, e.g., α-cuts, and then apply concept analysis in the classical (crisp) sense.

Hereditarily Structurally Complete Intermediate Logics: Citkin’s Theorem Via Duality

A deductive system is said to be structurally complete if its admissible rules are derivable. In addition, it is called hereditarily structurally complete if all its extensions are structurally complete. Citkin (1978) proved that an intermediate logic is hereditarily structurally complete if and only if the variety of Heyting algebras associated with it omits five finite algebras. Despite its importance in the theory of admissible rules, a direct proof of Citkin’s theorem is not widely accessible. In this paper we offer a self-contained proof of Citkin’s theorem, based on Esakia duality and the method of subframe formulas. As a corollary, we obtain a short proof of Citkin’s 2019 characterization of hereditarily structurally complete positive logics.

Brain and Its Universal Logical Model of Multi-Agent Biological Systems

We build a topological model, based on intuitionistic logic, for multi-agent biological systems (such as Physarum polycephalum, bacterial colonies or any other swarm), reacting to external nourishment stimuli. Our construction follows the topological description of brain activity, where particles (neurons) are activated by an external environment, represented by a topological space X with an open cover {U_i:iin I}. The brain builds the model of this external space via the nerve (trace) of a topological space X. Here the body of Physarum polycephalum or a swarm made of networks of tubular structures represents a nerve (trace) of X also which means that Physarum polycephalum or a swarm gains orientation in the space of external stimuli even in the absence of any neural system. The logic of living organisms is based on open subsets of X and thus can be represented by Heyting algebra (i.e. intuitionistically). We also consider the generalisation of the nerve construction to a categorical context, where the category is determined by the network structures of multi-agent biological system. This model can be generalised up to simulating the behaviour of any swarm by means of intuitionistic logic.

Fuzzy ideals and fuzzy congruences of distributive demi-p-algebras

In this article, we introduce *-fuzzy ideals of a distributive demi-p-algebra and its properties. We show that every *-fuzzy ideals of distributive demi-p-algebra L are precisely the kernels of the fuzzy congruences Θ on L if and only if its quotient is a Boolean algebra. Also we prove that the set of *-fuzzy ideals forms a complete Heyting algebra.

Computable Heyting Algebras with Distinguished Atoms and Coatoms

Computable Heyting Algebras with Distinguished Atoms and Coatoms

THE AUTOMORPHISM GROUP OF THE FRAÏSSÉ LIMIT OF FINITE HEYTING ALGEBRAS

AbstractRoelcke non-precompactness, simplicity, and non-amenability of the automorphism group of the Fraïssé limit of finite Heyting algebras are proved among others.

The variety of modal weak Gödel algebras

An algebra 〈A,∧,∨,→,□,0,1〉 of type (2,2,2,1,0,0) is said to be a modal weak Heyting algebra if 〈A,∧,∨,→,0,1〉 is a weak Heyting algebra and the following conditions are satisfied for every a,b∈A: M1) □(1)=1, M2) □(a∧b)=□(a)∧□(b) and M3) □(a→b)≤□(a)→□(b). If this algebra satisfies the inequality a∧(a→b)≤b then it is called modal RWH-algebra. In this paper we study the variety of modal RWH-algebras, which is denoted by KRWH, and some of its subvarieties. We focus our attention on the study of the lattice of congruences of any member of KRWH and some related properties. In particular, we give an equational basis for the subvariety of KRWH generated by the class of their totally ordered members.

On infinitary Gödel logics

Abstract We study propositional and first-order Gödel logics over infinitary languages, which are motivated semantically by corresponding interpretations into the unit interval $[0,1]$. We provide infinitary Hilbert-style calculi for the particular (propositional and first-order) cases with con-/disjunctions of countable length and prove corresponding completeness theorems by extending the usual Lindenbaum–Tarski construction to the infinitary case for a respective algebraic semantics via complete linear Heyting algebras. We provide infinitary hypersequent calculi and prove corresponding cut-elimination theorems in the Schütte–Tait style. Initial observations are made regarding truth-value sets other than $[0,1]$.

On intermediate justification logics

Abstract We study arbitrary intermediate propositional logics extended with a collection of axioms from (classical) justification logics. For these, we introduce various semantics by combining either Heyting algebras or Kripke frames with the usual semantic machinery used by Mkrtychev’s, Fitting’s or Lehmann and Studer’s models for classical justification logics. We prove unified completeness theorems for all intermediate justification logics and their corresponding semantics using a respective propositional completeness theorem of the underlying intermediate logic. Further, by a modification of a method of Fitting, we prove unified realization theorems for a large class of intermediate justification logics and accompanying intermediate modal logics.

Approximate positional analysis of fuzzy social networks

In order to facilitate the processing of vast amounts of data that emerges in the study of fuzzy social networks, scholars have developed various procedures for reducing these networks. Some procedures use regular and structural fuzzy relations to reduce such networks. In this article, we generalize the notions of regular and structural fuzzy relations to obtain even better reductions of fuzzy networks. More precisely, for a fuzzy social network given by the set of social entities and the family of fuzzy relations between them, we define μ-approximate regular fuzzy relations, where μ is the degree taken from the underlying set of truth values, which is a complete Heyting algebra. Using these specific fuzzy relations, we show that it is possible to reduce a fuzzy social network in some cases when previously developed algorithms fail to reduce it. We investigate the properties of μ-approximate regular fuzzy relations. We show that the blockmodel of a fuzzy social network, which is its reduced fuzzy social network built via μ-approximate regular fuzzy preorder, retains specific structure-preserving properties. We give a method for calculating the greatest μ-approximate regular fuzzy relation on a given fuzzy social network. For fuzzy social networks defined over the real-unit interval [0,1], we give a procedure that determines all subintervals of [0,1] that share the greatest μ-approximate regular fuzzy relation. Analogous results are provided for μ-approximate structural fuzzy relations.

Satisfiability in MultiValued Circuits

Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebras $\mathbf A$ there is a polynomial time algorithm that decides if an equation over $\mathbf A$ has a solution. We are also looking for polynomial time algorithms that decide if two circuits over a finite algebra compute the same function. Although we have not managed to solve these problems in the most general setting we have obtained such a characterization for a very broad class of algebras from congruence modular varieties. This class includes most known and well-studied algebras such as groups, rings, modules (and their generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie algebras), lattices (and their extensions like Boolean algebras, Heyting algebras or other algebras connected with multi-valued logics including MV-algebras). This paper seems to be the first systematic study of the computational complexity of satisfiability of non-Boolean circuits and solving equations over finite algebras. The characterization results provided by the paper is given in terms of nice structural properties of algebras for which the problems are solvable in polynomial time.

Residuated lattices derived from filters(ideals) in double Boolean algebras

Double Boolean algebras (dBas) are algebraic structures D = (D, v, ^, ‌, ', ⊥, T) of type (2, 2, 1, 1, 0, 0), introduced by Rudolf Wille to capture the equational theory of the algebra of protoconcepts. Our goal is an algebraic investigation of dBas, based on similar results on Boolean algebras. In this paper, first we characterize filters on dBas as deductive systems and we give many characterization of primary filters(ideals). Second, for a given dBa, we show that the set of its filters F(D) (resp.ideals I(D)) is endowed with the structure of distributive pseudo-complemented lattices, Heyting algebras and residuated lattices. We finish by introducing the notions of annihilators and co-annihilators on dBas and investigate some relalted properties of them. We show that pseudo-complement of an ideal I (filter F) is the annihilator I* of I ( co-annihilator F*) and the set of annihilators (co-annihilators) forms a Boolean algebra.

Computational Complexity of the Word Problem in Modal and Heyting Algebras with a Small Number of Generators

Computational Complexity of the Word Problem in Modal and Heyting Algebras with a Small Number of Generators

A characterization of Stone and linear Heyting algebras

A characterization of Stone and linear Heyting algebras

Nuclear ranges in implicative semilattices

A nucleus on a meet-semilattice A is a closure operation that preserves binary meets. The nuclei form a semilattice mathrm{N }A that is isomorphic to the system {mathcal {N}}A of all nuclear ranges, ordered by dual inclusion. The nuclear ranges are those closure ranges which are total subalgebras (l-ideals). Nuclei have been studied intensively in the case of complete Heyting algebras. We extend, as far as possible, results on nuclei and their ranges to the non-complete setting of implicative semilattices (whose unary meet translations have adjoints). A central tool are so-called r-morphisms, that is, residuated semilattice homomorphisms, and their adjoints, the l-morphisms. Such morphisms transport nuclear ranges and preserve implicativity. Certain completeness properties are necessary and sufficient for the existence of a least nucleus above a prenucleus or of a greatest nucleus below a weak nucleus. As in pointfree topology, of great importance for structural investigations are three specific kinds of l-ideals, called basic open, boolean and basic closed.

Contrapositionally complemented Heyting algebras and intuitionistic logic with minimal negation

AbstractTwo algebraic structures, the contrapositionally complemented Heyting algebra (ccHa) and the contrapositionally $\vee $ complemented Heyting algebra (c$\vee $cHa), are studied. The salient feature of these algebras is that there are two negations, one intuitionistic and another minimal in nature, along with a condition connecting the two operators. Properties of these algebras are discussed, examples are given and comparisons are made with relevant algebras. Intuitionistic Logic with Minimal Negation (ILM) corresponding to ccHas and its extension ${\textrm {ILM}}$-${\vee }$ for c$\vee $cHas are then investigated. Besides its relations with intuitionistic and minimal logics, ILM is observed to be related to Peirce’s logic and Vakarelov’s logic MIN. With a focus on properties of the two negations, relational semantics for ILM and ${\textrm {ILM}}$-${\vee }$ are obtained with respect to four classes of frames, and inter-translations between the classes preserving truth and validity are provided. ILM and ${\textrm {ILM}}$-${\vee }$ are shown to have the finite model property with respect to these classes of frames and proved to be decidable. Extracting features of the two negations in the algebras, a further investigation is made, following logical studies of negations that define the operators independently of the binary operator of implication. Using Dunn’s logical framework for the purpose, two logics $K_{im}$ and $K_{im-{\vee }}$ are discussed, where the language does not include implication. The $K_{im}$-algebras are reducts of ccHas and are different from relevant algebraic structures having two negations. The negations in the $K_{im}$-algebras and $K_{im-{\vee }}$-algebras are shown to occupy distinct positions in an enhanced form of Dunn’s kite of negations. Relational semantics for $K_{im}$ and $K_{im-{\vee }}$ is provided by a class of frames that are based on Dunn’s compatibility frames. It is observed that this class coincides with one of the four classes giving the relational semantics for ILM and ${\textrm {ILM}}$-${\vee }$.

L-algebras and three main non-classical logics

The semantics of three main branches of non-classical logic, intuitionistic, many-valued, and quantum logic, is unified by the concept of L-algebra. The corresponding three classes of algebras (Heyting algebras, MV-algebras, and orthomodular lattices) are associated to specializations of a bounded L-algebra, given by simple equations. Three basic specializations lead to three more classes of algebras, including quantized Heyting algebras which have not been considered before. All these algebras are obtained from a new class of L-algebras which simultaneously satisfy general versions of Glivenko's and Mundici's theorems.