We study the expressive power of linear propositional temporal logic interpreted on finite sequences or words. We first give a transparent proof of the fact that a formal language is expressible in this logic if and only if its syntactic semigroup is finite and aperiodic. This gives an effective algorithm to decide whether a given rational language is expressible. Our main result states a similar condition for the "restricted" temporal logic (RTL), obtained by discarding the "until" operator. A formal language is RTL-expressible if and only if its syntactic semigroup is finite and satisfies a certain simple algebraic condition. This leads

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Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory relating monoidal quantifiers to NC and its subclasses. In the absence of the BIT predicate, we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's "hardest context-free language" is LOGCFL-complete under quantifier-free BIT-free projections. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with the BIT predicate than without. Considering a particular groupoidal quantifier, we prove that first-order logic with majority of pairs is strictly more expressive than first-order with major...

We investigate the complexity of algorithmic problems on finitely generated subgroups of free groups. Margolis and Meakin showed how a finite monoid Synt(H) can be canonically and e#ectively associated with such a subgroup H. We show that H is pure (that is, closed under radical) if and only if Synt(H) is aperiodic. We also show that testing for this property of H is pspace-complete. In the process, we show that certain problems about finite automata which are pspace-complete in general remain pspace-complete when restricted to injective and inverse automata (with single accept state), whereas they are known to be in NC for permutation automata (with single accept state). We are concerned with the solution and the complexity of algorithmic problems about finitely generated subgroups of free groups. Our main results are that the problem of deciding purity for a finitely generated subgroup of a free group is decidable, and that it is pspace-complete. Our techniques rely largely on autom...

This paper presents a decidable characterization of tree languages that can be defined by a boolean combination of Σ1 formulas. This is a tree extension of the Simon theorem, which says that a string language can be defined by a boolean combination of Σ1 formulas if and only if its syntactic monoid is J-trivial.

We prove some results related to the generalized star-height problem. In this problem, as opposed to the restricted star-height problem, complementation is considered as a basic operator. We first show that the class of languages of star-height n is closed under certain operations (left and right quotients, inverse alphabetic morphisms, injective star-free substitutions). It is known that languages recognized by a commutative group are of star-height 1. We extend this result to nilpotent groups of class 2 and to the groups that divide a semidirect product of a commutative group by (Z=2Z) n . In the same direction, we show that one of the languages that was conjectured to be of star height 2 during the past ten years, is in fact of star height 1. Next we show that if a rational language L is recognized by a monoid of the variety generated by wreath products of the form M (G N ), where M and N are aperiodic monoids, and G is a commutative group, then L is of star-height...

Several algorithmic problems for pseudovarieties and their relationships are studied. This includes the usual membership problem and the computability of pointlike subsets of finite semigroups. Some of these problems afford equivalent formulations involving topological separation properties in free profinite semigroups. Several examples are considered and, as an application, a decidability result for joins is proved. 1. Introduction Perhaps the three most celebrated results relating the theories of formal languages and finite semigroups are: Schutzenberger's characterization of star-free languages as those whose syntactic semigroups are finite and aperiodic [19]; Simon's characterization of piecewise testable languages as those whose syntactic semigroups are finite and J-trivial [20]; and Brzozowski and Simon / McNaughton 's characterization of locally testable languages as those whose syntactic semigroups are finite local semilattices [7, 15]. These results led Eilenberg [9] to ...

Let TL[X; F] (TL[F]) be the class of formulas of temporal logic, where as the only temporal operators X (next) and F (eventually) are allowed (only F is allowed, resp.). For a class \Phi ` TL[X; F] let L(\Phi) be the class of \Phi-definable languages. Among others, characterizations of L(TL[X; F]) and L(TL[F]) in terms of forbidden patterns in finite automata are known. Here we ask for every bound k 0 on the number of nested uses of the next operator for the expressive power of the respective fragment of TL[X; F]. Denote by TL[X(k); F] the class of formulas in TL[X; F] with nesting depth k in the next operator. Obviously, L(TL[X; F]) = S k0 L(TL[X(k); F]) and L(TL[F]) = L(TL[X(0); F]). We prove a levelwise characterization of the above syntactical hierarchy (1) in terms of a certain pattern S rev k (cf. Figure 6) that must not appear in the transition graph of deterministic finite automata, and (2) in terms of a formal language representation involving k-leftdeterministic lan...

this paper is to study the first order theory of the successor, interpreted on finite words. More specifically, we complete the study of the hierarchy based on quantifier alternations (or \Sigma n-hierarchy). It was known (Thomas, 1982) that this hierarchy collapses at level 2, but the expressive power of the lower levels was not characterized effectively. We give a semigroup theoretic description of the expressive power of \Sigma 1 , the existential formulas, and B\Sigma 1 , the boolean combinations of existential formulas. Our characterization is algebraic and makes use of the syntactic semigroup, but contrary to a number of results in this field, is not in the scope of Eilenberg's variety theorem, since B\Sigma 1 -definable languages are not closed under residuals

The primary decomposition theorem due to Krohn and Rhodes ([KR65]), which has been considered as one of the fundamental results in the theory of automata and semigroups, states that every automaton is homomorphic to a cascaded decomposition (wreath-product) of simpler automata of two kinds: reset automata and permutation automata. If the automaton is non-counting (and correspondingly its transformation semigroup is group-free) then it can be decomposed using only reset components. There exist various proofs and partial proofs for the primary decomposition theorem e.g., [HS66, Ze67a, Ze67b, Gi68, MT69, La71, We76, Ei76]. None of them give explicit bounds on the size of the decomposition. 1 In this paper we give tight exponential bounds on the size of the decomposition as a function of the size of the original automaton. For the upper-bound we give an exponential algorithm by modifying the implicit construction appearing in [Ei74]. Our algorithm is constructive enough to allow implemen...