Pascal Tesson

• Université Laval

The notion of recognition of a language by a finite semigroup can be generalized to recognition by finite groupoids, i.e. sets equipped with a binary operation ‘·’ which is not necessarily associative. It is well known that L can be recognized by a groupoid iff L is context-free. But it is also known that some subclasses of groupoids can only recog...

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...

Unlike the wreath product, the block product is not associative at the level of varieties. All decomposition theorems involving block products, such as the bilateral version of Krohn–Rhodes' theorem, have always assumed a right-to-left bracketing of the operands. We consider here the left-to-right bracketing, which is generally weaker. More precise...

We present algebraic conditions on constraint languages Γ that ensure the hardness of the constraint satisfaction problem 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 is not first-order definable then it is L-hard. Our proofs rely...

We show that the directed st-connectivity problem cannot be expressed in symmetric Datalog, a fragment of Datalog introduced in [5]. It was shown there that symmetric,Datalog programs can be eval- uated in logarithmic space and that this fragment of Datalog captures logspace when augmented with negation, and an auxiliary successor re- lation S toge...

We consider the problem of testing whether a given system of equations over a fixed finite semigroup S has a solution. For the case where S is a monoid, we prove that the problem is computable in polynomial time when S is commutative and is the union of its subgroups but is NP-complete otherwise. When S is a monoid or a regular semigroup, we obtain...

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 class...

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...

We study languages with bounded communication complexity in the multiparty "input on the forehead model" with worst-case partition. In the two-party case, languages with bounded complexity are exactly those recognized by programs over commutative monoids [19]. This can be used to show that these languages all lie in shallow ACC0. In contrast, we us...

A letter e ∈Σ is said to be neutral for a language L if it can be inserted and deleted at will in a word without affecting membership in L. The Crane Beach Conjecture, which was recently disproved, stated that any language containing a neutral letter and definable in first-order with arbitrary numerical predicates (\({\bf FO}[\mathit{Arb}]\)) is in...

We study the complexity of counting the number of solutions to a system of equations over a fixed finite semigroup. We show that this problem is always either in FP or #P-complete and describe the borderline precisely. We use these results to convey some intuition about the conjectured dichotomy for the complexity of counting the number of solution...

We study the problem of learning an unknown function represented as an expression or a program over a known finite monoid. As in other areas of computational complexity where programs over algebras have been used, the goal is to relate the computational complexity of the learning problem with the algebraic complexity of the finite monoid. Indeed, o...

We study languages with bounded communication complexity in the multiparty “input on the forehead model” with worst-case partition. In the two-party case, languages with bounded complexity are exactly those recognized by programs over commutative monoids [19]. This can be used to show that these languages all lie in shallow ACC0. In contrast, we us...

The algebraic theory of finite automata has been one of the most successful tools to study and classify regular languages. These very same tools can in fact be used to understand more powerful models of computation and we discuss here the impact that semigroup theory can have in computational complexity.

We contribute to the algebraic study of the complexity of constraint satisfaction problems. We give a new sufficient condition on a set of relations Γ over a domain S for the tractability of CSP(Γ): if S is a block-group (a particular class of semigroups) of exponent ω and Γ is a set of relations over S preserved by the operation defined by the pol...

It is well-known that the Σk- and Πk-levels of the dot-depth hierarchy and the polynomial hierarchy correspond via leaf languages. We extend this correspondence to the Δk-levels of these hierarchies: ${\rm Leaf}^{\rm P} (\Delta_k^L) = \Delta_k^p$. The same methods are used to give evidence against an earlier conjecture of Straubing and Thérien abou...

We obtain a logical characterization of an important class of regular languages, denoted \({\mathcal DO}\), and of its most important subclasses in terms of two-variable sentences with ordinary and modular quantifiers but in which all modular quantifiers lie outside the scope of ordinary quantifiers. The result stems from a new decomposition of the...

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...

The leaf-language mechanism associates a complexity class to a class of regular languages. It is well-known that the Σ k - and Π k -levels of the dot-depth hierarchy and the polynomial hierarchy correspond in this formalism. We extend this correspondence to the Δ k -levels of these hierarchies: LeafP(ΔkL_{k}^{L}) = Δkp_{k}^{p}. These results are ob...

We show that every regular language L has either constant, logarithmic or linear two-party communication complexity (in a worstcase partition sense). We prove a similar trichotomy for simultaneous communication complexity and a “quadrichotomy” for probabilistic communication complexity.

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.

In this thesis, we address a number of issues pertaining to the computational power of monoids and semigroups as machines and to the computational complexity of problems whose difficulty is parametrized by an underlying semigroup or monoid and find that these two axes of research are deeply intertwined. We first consider the "program over monoid" m...

We study the relationship between the complexity of languages, in Yao’s two-party communication game and its extensions, and the algebraic properties of finite monoids that can recognize them. For a finite monoid M, we define C (k) (M) to be the maximum number of bits of communication that players need to exchange, in the k-party game of Chandra, F...

We survey different characterizations (algebraic, combinatorial,...

We study the computational complexity of determining whether a systems of equations over a fixed finite monoid has a solution. In [6], it was shown that in the restricted case of groups the problem is tractable if the group is Abelian and NP-complete otherwise. We prove that in the case of an arbitrary finite monoid, the problem is in P if the mono...

The formalism of programs over monoids has been studied for its close connection to parallel complexity classes defined by small-depth boolean circuits. We investigate two basic questions about this model. When is a monoid rich enough that it can recognize arbitrary languages (provided no restriction on length is imposed)? When is a monoid weak eno...

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 non-nilpotent groups and relate this question to a natural circuit complexity conjecture. In the...

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 sub-class of solvable non-nilpotent groups and relate this question to a natural circuit complexity conjecture.

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 aperiod...

In this work, we define the communication complexity of a monoid M as the maximum complexity of any language recognized by M, and show that monoid classes defined by, communication complexity classes form varieties. Then we prove that a group G has constant communication complexity for k players if and only if G is a nilpotent group of class at mos...

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 Mod p L. 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...

A letter e ∈ Σ is said to be neutral for a language L if it can be inserted and deleted at will in a word without affecting member-ship in L. The Crane-Beach Conjecture, which was recently disproved, stated that any language containing a neutral letter and definable in FO is in fact FO[<] definable and is thus a regular, star-free language. More ge...

Abstract The temporal logic operators atnext and atprevious are alternatives for the operators until and since. P atnext Q has the meaning: at the next position in the future where Q holds it holds P . We dene,an asymmetric but natural notion of depth for the expressions of this linear temporal logic. The sequence of classes atn of languages expres...

We show, using the Hales-Jewett Theorem, that testing whether k subsets of [n] form a partition requires non-constant communication complexity in the k-party "input on the forehead" model. This allows us in turn to answer an open problem of [7] by obtaining a lower bound for the multiparty communication complexity of the regular language (c * ac *...