A connection between concurrency and language theory

We show that three fixed point structures equipped with (sequential) composition, a sum operation, and a fixed point operation share the same valid equations. These are the theories of (context-free) languages, (regular) tree languages, and simulation equivalence classes of (regular) synchronization trees (or processes). The results reveal a close relationship between classical language theory and process algebra

Completeness for Flat Modal Fixpoint Logics

This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint logics. Given a set \Gamma of modal formulas of the form \gamma(x, p1, . . ., pn), where x occurs only positively in \gamma, the language L\sharp (\Gamma) is obtained by adding to the language of polymodal logic a connective \sharp\_\gamma for each \gamma \epsilon. The term \sharp\_\gamma (\varphi1, . . ., \varphin) is meant to be interpreted as the least fixed point of the functional interpretation of the term \gamma(x, \varphi 1, . . ., \varphi n). We consider the following problem: given \Gamma, construct an axiom system which is sound and complete with respect to the concrete interpretation of the language L\sharp (\Gamma) on Kripke frames. We prove two results that solve this problem. First, let K\sharp (\Gamma) be the logic obtained from the basic polymodal K by adding a Kozen-Park style fixpoint axiom and a least fixpoint rule, for each fixpoint connective \sharp\_\gamma. Provided that each indexing formula \gamma satisfies the syntactic criterion of being untied in x, we prove this axiom system to be complete. Second, addressing the general case, we prove the soundness and completeness of an extension K+ (\Gamma) of K\_\sharp (\Gamma). This extension is obtained via an effective procedure that, given an indexing formula \gamma as input, returns a finite set of axioms and derivation rules for \sharp\_\gamma, of size bounded by the length of \gamma. Thus the axiom system K+ (\Gamma) is finite whenever \Gamma is finite

Fuzzy Sets, Fuzzy Logic and Their Applications

The present book contains 20 articles collected from amongst the 53 total submitted manuscripts for the Special Issue “Fuzzy Sets, Fuzzy Loigic and Their Applications” of the MDPI journal Mathematics. The articles, which appear in the book in the series in which they were accepted, published in Volumes 7 (2019) and 8 (2020) of the journal, cover a wide range of topics connected to the theory and applications of fuzzy systems and their extensions and generalizations. This range includes, among others, management of the uncertainty in a fuzzy environment; fuzzy assessment methods of human-machine performance; fuzzy graphs; fuzzy topological and convergence spaces; bipolar fuzzy relations; type-2 fuzzy; and intuitionistic, interval-valued, complex, picture, and Pythagorean fuzzy sets, soft sets and algebras, etc. The applications presented are oriented to finance, fuzzy analytic hierarchy, green supply chain industries, smart health practice, and hotel selection. This wide range of topics makes the book interesting for all those working in the wider area of Fuzzy sets and systems and of fuzzy logic and for those who have the proper mathematical background who wish to become familiar with recent advances in fuzzy mathematics, which has entered to almost all sectors of human life and activity

Automaták , fixpontok, és logika = Automata, fixed points, and logic

Megmutattuk, hogy a véges automaták (faautomaták, súlyozott automaták, stb.) viselkedése végesen leírható a fixpont művelet általános tulajdonságainak felhasználásával. Teljes axiomatizálást adtunk a véges automaták viselkedését leíró racionális hatványsorokra és fasorokra, ill. a véges automaták biszimuláció alapú viselkedésére. Megmutattuk, hogy az automaták elméletének alapvető Kleene tétele és általánosításai a fixpont művelet azonosságainak következménye. Algebrai eszközökkel vizsgáltuk az elágazó idejű temporális logikák és a monadikus másodrendű logika frágmenseinek kifejező erejét fákon. Fő eredményünk egy olyan kölcsönösen egyértelmű kapcsolat kimutatása, amely ezen logikák kifejező erejének vizsgálatát visszavezeti véges algebrák és preklónok bizonyos pszeudovarietásainak vizsgálatára. Jellemeztük a reguláris és környezetfüggetlen nyelvek lexikografikus rendezéseit, végtelen szavakra általánosítottuk a környezetfüggetlen nyelv fogalmát, és tisztáztuk ezek számos algoritmikus tulajdonságát. | We have proved that the the bahavior of finite automata (tree automata, weighted automata, etc.) has a finite description with respect to the general properties of fixed point operations. We have obtained complete axiomatizations of rational power series and tree series, and the bisimulation based behavior of finite automata. As an intermediate step of the completeness proofs, we have shown that Kleene's fundamental theorem and its generalizations follow from the equational properties of fixed point operations. We have developed an algebraic framework for describing the expressive power of branching time temporal logics and fragments of monadic second-order logic on trees. Our main results establish a bijective correspondence between these logics and certain pseudo-varieties of finite algebras and/or finitary preclones. We have characterized the lexicographic orderings of the regular and context-free languages and generalized the notion of context-free languages to infinite words and established several of their algorithmic properties