Pascal Tesson

Logic Meets Algebra: the Case of Regular Languages

The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Buchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point of view on automata is an essential complement of this classification: by providing alternative, algebraic characterizations for the classes, it often yields the only opportunity for the design of algorithms that decide expressibility in some logical fragment. We survey the existing results relating the expressibility of regular languages in logical fragments of MSO[S] with algebraic properties of their minimal automata. In particular, we show that many of the best known results in this area share the same underlying mechanics and rely on a very strong relation between logical substitutions and block-products of pseudovarieties of monoid. We also explain the impact of these connections on circuit complexity theory.

Symmetric Datalog and Constraint Satisfaction Problems in Logspace

We introduce symmetric Datalog, a syntactic restriction of linear Datalog and show that its expressive power is exactly that of restricted symmetric Krom monotone SNP. The deep result of Reingold [17] on the complexity of undirected connectivity suffices to show that symmetric Datalog queries can be evaluated in logarithmic space. We show that for a number of constraint languages Gamma, the complement of the constraint satisfaction problem CSP(Gamma) can be expressed in symmetric Datalog. In particular, we show that if CSP(Gamma) is first-order definable and Lambda is a finite subset of the relational clone generated by Gamma then notCSP(Lambda) is definable in symmetric Datalog. Over the two-element domain and under standard complexity-theoretic assumptions, expressibility of notCSP(Gamma) in symmetric Datalog corresponds exactly to the class of CSPs computable in logarithmic space. Finally, we describe a fairly general subclass of implicational (or 0/1/all) constraints for which the complement of the corresponding CSP is also definable in symmetric Datalog. Our results provide preliminary evidence that symmetric Datalog may be a unifying explanation for families of CSPs lying in L.

Universal algebra and hardness results for constraint satisfaction problems

We present algebraic conditions on constraint languages Γ that ensure the hardness of the constraint satisfaction problem CSP(Γ) for complexity classes L, NL, P, NP and ModpL. These criteria also give non-expressibility results for various restrictions of Datalog. Furthermore, we show that if CSP(Γ) is not first-order definable then it is L-hard. Our proofs rely on tame congruence theory and on a fine-grain analysis of the complexity of reductions used in the algebraic study of CSPs. The results pave the way for a refinement of the dichotomy conjecture stating that each CSP(Γ) lies in P or is NP-complete and they match the recent classification of [1] for Boolean CSP. We also infer a partial classification theorem for the complexity of CSP(Γ) when the associated algebra of Γ is the idempotent reduct of a preprimal algebra.

Complete Classifications for the Communication Complexityof Regular Languages

Abstract We show that every regular language L has either constant, logarithmic or linear two-party communication complexity (in a worst-case partition sense). We prove similar classifications for the communication complexity of regular languages for the simultaneous, probabilistic, simultaneous probabilistic and Modp-counting models of communication.

Equation Satisfiability and Program Satisfiability for Finite Monoids

We study the computational complexity of solving equations and of determining the satisfiability of programs over a fixed finite monoid. We partially answer an open problem of [4] by exhibiting quasi-polynomial time algorithms for a subclass of solvable nonnilpotent groups and relate this question to a natural circuit complexity conjecture. In the special case when M is aperiodic, we show that PROGRAM SATISFIABILITY is in P when the monoid belongs to the variety DA and is NP-complete otherwise. In contrast, we give an example of an aperiodic outside DA for which EQUATION SATISFIABILITY is computable in polynomial time and discuss the relative complexity of the two problems. We also study the closure properties of classes for which these problems belong to P and the extent to which these fail to form algebraic varieties.

An Algebraic Approach to Communication Complexity

Let M be a finite monoid: define C(k)(M) to be the maximum number of bits that need to be exchanged in the k-party communication game to decide membership in any language recognized by M. We prove the following: a) If M is a group then, for any k, C(k)(M) = O(1) if M is nilpotent of class k − 1 and C(k)(M) = θ(n) otherwise. b) If M is aperiodic, then C(2)(M) = O(1) if M is commutative, C(2)(M) = θ(log n) if M belongs to the variety DA but is not commutative and C(2)(M) = θ(n) otherwise.

The Complexity of the List Homomorphism Problem for Graphs

We completely classify the computational complexity of the list H-colouring problem for graphs (with possible loops) in combinatorial and algebraic terms: for every graph H, the problem is either NP-complete, NL-complete, L-complete or is first-order definable; descriptive complexity equivalents are given as well via Datalog and its fragments. Our algebraic characterisations match important conjectures in the study of constraint satisfaction problems.

MONOIDS AND COMPUTATIONS

This contribution wishes to argue in favor of increased interaction between experts on finite monoids and specialists of theory of computation. Developing the algebraic approach to formal computations as well as the computational point of view on monoids will prove to be beneficial to both communities. We give examples of this two-way relationship coming from temporal logic, communication complexity and Boolean circuits. Although mostly expository in nature, our paper proves some new results along the way.

Restricted two-variable FO + MOD sentences, circuits and communication complexity

We obtain a logical characterization of an important class of regular languages, denoted

Tractable clones of polynomials over semigroups

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