Efficient Algorithms for Morphisms over Omega-Regular Languages

Morphisms to finite semigroups can be used for recognizing omega-regular languages. The so-called strongly recognizing morphisms can be seen as a deterministic computation model which provides minimal objects (known as the syntactic morphism) and a trivial complementation procedure. We give a quadratic-time algorithm for computing the syntactic morphism from any given strongly recognizing morphism, thereby showing that minimization is easy as well. In addition, we give algorithms for efficiently solving various decision problems for weakly recognizing morphisms. Weakly recognizing morphism are often smaller than their strongly recognizing counterparts. Finally, we describe the language operations needed for converting formulas in monadic second-order logic (MSO) into strongly recognizing morphisms, and we give some experimental results.Comment: Full version of a paper accepted to FSTTCS 201

On the Complexity of the Cayley Semigroup Membership Problem

We investigate the complexity of deciding, given a multiplication table representing a semigroup S, a subset X of S and an element t of S, whether t can be expressed as a product of elements of X. It is well-known that this problem is {NL}-complete and that the more general Cayley groupoid membership problem, where the multiplication table is not required to be associative, is {P}-complete. For groups, the problem can be solved in deterministic log-space which raised the question of determining the exact complexity of this variant. Barrington, Kadau, Lange and McKenzie showed that for Abelian groups and for certain solvable groups, the problem is contained in the complexity class {FOLL} and they concluded that these variants are not hard for any complexity class containing {Parity}. The more general case of arbitrary groups remained open. In this work, we show that for both groups and for commutative semigroups, the problem is solvable in {qAC}^0 (quasi-polynomial size circuits of constant depth with unbounded fan-in) and conclude that these variants are also not hard for any class containing {Parity}. Moreover, we prove that {NL}-completeness already holds for the classes of 0-simple semigroups and nilpotent semigroups. Together with our results on groups and commutative semigroups, we prove the existence of a natural class of finite semigroups which generates a variety of finite semigroups with {NL}-complete Cayley semigroup membership, while the Cayley semigroup membership problem for the class itself is not {NL}-hard. We also discuss applications of our technique to {FOLL}

The Intersection Problem for Finite Monoids

We investigate the intersection problem for finite monoids, which asks for a given set of regular languages, represented by recognizing morphisms to finite monoids from a variety V, whether there exists a word contained in their intersection. Our main result is that the problem is PSPACE-complete if V is contained in DS and NP-complete if V is non-trivial and contained in DO. Our NP-algorithm for the case that V is contained in DO uses novel methods, based on compression techniques and combinatorial properties of DO. We also show that the problem is log-space reducible to the intersection problem for deterministic finite automata (DFA) and that a variant of the problem is log-space reducible to the membership problem for transformation monoids. In light of these reductions, our hardness results can be seen as a generalization of both a classical result by Kozen and a theorem by Beaudry, McKenzie and Therien.Comment: Extended version of a paper accepted to STACS 201

Testing Simon's congruence

Piecewise testable languages are a subclass of the regular languages. There are many equivalent ways of defining them; Simon's congruence $\sim_k$ is one of the most classical approaches. Two words are $\sim_k$-equivalent if they have the same set of (scattered) subwords of length at most k. A language L is piecewise testable if there exists some k such that L is a union of $\sim_k$-classes. For each equivalence class of $\sim_k$, one can define a canonical representative in shortlex normal form, that is, the minimal word with respect to the lexicographic order among the shortest words in $\sim_k$. We present an algorithm for computing the canonical representative of the $\sim_k$-class of a given word $w \in A^*$ of length n. The running time of our algorithm is in O(|A|n) even if $k \le n$ is part of the input. This is surprising since the number of possible subwords grows exponentially in k. The case $k > n$ is not interesting since then, the equivalence class of w is a singleton. If the alphabet is fixed, the running time of our algorithm is linear in the size of the input word. Moreover, for fixed alphabet, we show that the computation of shortlex normal forms for $\sim_k$ is possible in deterministic logarithmic space. One of the consequences of our algorithm is that one can check with the same complexity whether two words are $\sim_k$-equivalent (with k being part of the input)