Related papers: Zero-One Law for Regular Languages and Semigroups …
A zero-one language L is a regular language whose asymptotic probability converges to either zero or one. In this case, we say that L obeys the zero-one law. We prove that a regular language obeys the zero-one law if and only if its…
Zero-one laws state that probabilistic events of a certain type must occur with probability either $0$ or $1$, and nothing in between. We formulate a syntactic zero-one law, which enjoys good logical properties while being broadly…
A class of languages C is perfect if it is closed under Boolean operations and the emptiness problem is decidable. Perfect language classes are the basis for the automata-theoretic approach to model checking: a system is correct if the…
We introduce an operator on classes of regular languages, the star-free closure. Our motivation is to generalize standard results of automata theory within a unified framework. Given an arbitrary input class $C$, the star-free closure…
Let A be a finite alphabet and let L contained in (A*)^n be an n-variable language over A. We say that L is regular if it is the language accepted by a synchronous n-tape finite state automaton, it is quasi-regular if it is accepted by an…
It has been shown in the late 1960s that each formula of first-order logic without constants and function symbols obeys a zero-one law: As the number of elements of finite models increases, every formula holds either in almost all or in…
We consider some questions about formal languages that arise when inverses of letters, words and languages are defined. The reduced representation of a language over the free monoid is its unique equivalent representation in the free group.…
Over finite words, languages of dot-depth one are expressively complete for alternation-free first-order logic. This fragment is also known as the Boolean closure of existential first-order logic. Here, the atomic formulas comprise order,…
Each relational structure X has an associated Gaifman graph, which endows X with the properties of a graph. Suppose that X is infinite, connected and of bounded degree. A first-order sentence in the language of X is almost surely true…
We study zero-one laws for random graphs. We focus on the following question that was asked by many: Given a graph property P, is there a language of graphs able to express P while obeying the zero-one law? Our results show that on the one…
We consider extensions of monadic second order logic over $\omega$-words, which are obtained by adding one language that is not $\omega$-regular. We show that if the added language $L$ has a neutral letter, then the resulting logic is…
We study the notion of sparseness for regular languages over finite trees and infinite words. A language of trees is called sparse if the relative number of $n$-node trees in the language tends to zero, and a language of infinite words is…
We investigate two operators on classes of regular languages: polynomial closure (Pol) and Boolean closure (Bool). We apply these operators to classes of group languages G and to their well-suited extensions G+, which is the least Boolean…
We investigate the star-free closure, which associates to a class of languages its closure under Boolean operations and marked concatenation. We prove that the star-free closure of any finite class and of any class of groups languages with…
Let $\mathcal G$ be an addable, minor-closed class of graphs. We prove that the zero-one law holds in monadic second-order logic (MSO) for the random graph drawn uniformly at random from all {\em connected} graphs in $\mathcal G$ on $n$…
It is well-known that: (i) every context-free language over a singleton terminal alphabet is regular, and (ii) the class of languages that satisfy the Pumping Lemma is a proper super-class of the context-free languages. We show that any…
A regular language $L$ is union-free if it can be represented by a regular expression without the union operation. A union-free language is deterministic if it can be accepted by a deterministic one-cycle-free-path finite automaton; this is…
For an $n\times n$ random image with independent pixels, black with probability $p(n)$ and white with probability $1-p(n)$, the probability of satisfying any given first-order sentence tends to 0 or 1, provided both $p(n)n^{\frac{2}{k}}$…
We relate two measures of complexity of regular languages. The first is syntactic complexity, that is, the cardinality of the syntactic semigroup of the language. That semigroup is isomorphic to the semigroup of transformations of states…
We prove two completeness results for Kleene algebra with tests and a top element, with respect to guarded string languages and binary relations. While the equational theories of those two classes of models coincide over the signature of…