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

Simulations of Weighted Tree Automata

Simulations of weighted tree automata (wta) are considered. It is shown how such simulations can be decomposed into simpler functional and dual functional simulations also called forward and backward simulations. In addition, it is shown in several cases (fields, commutative rings, Noetherian semirings, semiring of natural numbers) that all equivalent wta M and N can be joined by a finite chain of simulations. More precisely, in all mentioned cases there exists a single wta that simulates both M and N. Those results immediately yield decidability of equivalence provided that the semiring is finitely (and effectively) presented.Comment: 17 pages, 2 figure

The Equational Theory of Fixed Points with Applications to Generalized Language Theory

We review the rudiments of the equational logic of (least) fixed points and provide some of its applications for axiomatization problems with respect to regular languages, tree languages, and synchronization trees

The Equational Theory of Fixed Points with Applications to Generalized Language Theory

We review the rudiments of the equational logic of (least) fixed points and provide some of its applications for axiomatization problems with respect to regular languages, tree languages, and synchronization trees

Axiomatizing Flat Iteration

Flat iteration is a variation on the original binary version of the Kleene star operation P*Q, obtained by restricting the first argument to be a sum of atomic actions. It generalizes prefix iteration, in which the first argument is a single action. Complete finite equational axiomatizations are given for five notions of bisimulation congruence over basic CCS with flat iteration, viz. strong congruence, branching congruence, eta-congruence, delay congruence and weak congruence. Such axiomatizations were already known for prefix iteration and are known not to exist for general iteration. The use of flat iteration has two main advantages over prefix iteration: 1.The current axiomatizations generalize to full CCS, whereas the prefix iteration approach does not allow an elimination theorem for an asynchronous parallel composition operator. 2.The greater expressiveness of flat iteration allows for much shorter completeness proofs. In the setting of prefix iteration, the most convenient way to obtain the completeness theorems for eta-, delay, and weak congruence was by reduction to the completeness theorem for branching congruence. In the case of weak congruence this turned out to be much simpler than the only direct proof found. In the setting of flat iteration on the other hand, the completeness theorems for delay and weak (but not eta-) congruence can equally well be obtained by reduction to the one for strong congruence, without using branching congruence as an intermediate step. Moreover, the completeness results for prefix iteration can be retrieved from those for flat iteration, thus obtaining a second indirect approach for proving completeness for delay and weak congruence in the setting of prefix iteration.Comment: 15 pages. LaTeX 2.09. Filename: flat.tex.gz. On A4 paper print with: dvips -t a4 -O -2.15cm,-2.22cm -x 1225 flat. For US letter with: dvips -t letter -O -0.73in,-1.27in -x 1225 flat. More info at http://theory.stanford.edu/~rvg/abstracts.html#3

Axiomatizing Omega and Omega-op Powers of Words

In 1978, Courcelle asked for a complete set of axioms and rules for the equational theory of (discrete regular) words equipped with the operations of product, omega power and omega-op power. In this paper we find a simple set of equations and prove they are complete. Moreover, we show that the equational theory is decidable in polynomial time

Cyclic Datatypes modulo Bisimulation based on Second-Order Algebraic Theories

Cyclic data structures, such as cyclic lists, in functional programming are tricky to handle because of their cyclicity. This paper presents an investigation of categorical, algebraic, and computational foundations of cyclic datatypes. Our framework of cyclic datatypes is based on second-order algebraic theories of Fiore et al., which give a uniform setting for syntax, types, and computation rules for describing and reasoning about cyclic datatypes. We extract the "fold" computation rules from the categorical semantics based on iteration categories of Bloom and Esik. Thereby, the rules are correct by construction. We prove strong normalisation using the General Schema criterion for second-order computation rules. Rather than the fixed point law, we particularly choose Bekic law for computation, which is a key to obtaining strong normalisation. We also prove the property of "Church-Rosser modulo bisimulation" for the computation rules. Combining these results, we have a remarkable decidability result of the equational theory of cyclic data and fold.Comment: 38 page

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