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.Comment: 37 page

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.Comment: 12 pages, STACS 201

The atnext/atprevious hierarchy on the starfree languages

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 define an asymmetric but natural notion of depth for the expressions of this linear temporal logic. The sequence of classes at_n of languages expressible via such depth-n expressions gives a parametrization of the starfree regular languages which we call the atnext/atprevious hierarchy, or simply at hierarchy. It turns out that the at hierarchy equals the hierarchy given by the n-fold weakly iterated block product of DA. It is shown that the at hierarchy is situated properly between the until/since and the dot-depth hierarchy

The dot-depth and the polynomial hierarchy correspond on the delta levels

It is well-known that the Sigma_k- and Pi_k-levels of the dot-depth hierarchy and the polynomial hierarchy correspond via leaf languages. In this paper this correspondence will be extended to the Delta_k-levels of these hierarchies: Leaf^P(Delta_k^L) = Delta_k^p

Genetic Association Study Identifies HSPB7 as a Risk Gene for Idiopathic Dilated Cardiomyopathy

Dilated cardiomyopathy (DCM) is a structural heart disease with strong genetic background. Monogenic forms of DCM are observed in families with mutations located mostly in genes encoding structural and sarcomeric proteins. However, strong evidence suggests that genetic factors also affect the susceptibility to idiopathic DCM. To identify risk alleles for non-familial forms of DCM, we carried out a case-control association study, genotyping 664 DCM cases and 1,874 population-based healthy controls from Germany using a 50K human cardiovascular disease bead chip covering more than 2,000 genes pre-selected for cardiovascular relevance. After quality control, 30,920 single nucleotide polymorphisms (SNP) were tested for association with the disease by logistic regression adjusted for gender, and results were genomic-control corrected. The analysis revealed a significant association between a SNP in HSPB7 gene (rs1739843, minor allele frequency 39%) and idiopathic DCM (p = 1.06×10−6, OR = 0.67 [95% CI 0.57–0.79] for the minor allele T). Three more SNPs showed p < 2.21×10−5. De novo genotyping of these four SNPs was done in three independent case-control studies of idiopathic DCM. Association between SNP rs1739843 and DCM was significant in all replication samples: Germany (n = 564, n = 981 controls, p = 2.07×10−3, OR = 0.79 [95% CI 0.67–0.92]), France 1 (n = 433 cases, n = 395 controls, p = 3.73×10−3, OR = 0.74 [95% CI 0.60–0.91]), and France 2 (n = 249 cases, n = 380 controls, p = 2.26×10−4, OR = 0.63 [95% CI 0.50–0.81]). The combined analysis of all four studies including a total of n = 1,910 cases and n = 3,630 controls showed highly significant evidence for association between rs1739843 and idiopathic DCM (p = 5.28×10−13, OR = 0.72 [95% CI 0.65–0.78]). None of the other three SNPs showed significant results in the replication stage

An algebraic approach to communication complexity

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 most ( k - 1), and has linear complexity otherwise. When M is aperiodic, we show that its 2-party communication complexity is constant if M is commutative, logarithmic if M is not commutative but is in the variety DA, and is linear otherwise. Moreover, we show that if M is in DA, there exists a k such that M has constant k-party communication complexity. We conjecture that this condition is also necessary and prove lower bounds in that direction.These results lead us to state a conjecture which would generalize Szegedy's theorem to O(1) players. They also suggest several interesting possibilities to further uncover algebraic characterizations of communication complexity classes. (Abstract shortened by UMI.

Computational complexity questions related to finite monoids and semigroups

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" model of D. Barrington and D. Therien [BT88] and set out to answer two fundamental questions: which monoids are rich enough to recognize arbitrary languages via programs of arbitrary length, and which monoids are so weak that any program over them has an equivalent of polynomial length? We find evidence that the two notions are dual and in particular prove that every monoid in DS has exactly one of these two properties. We also prove that for certain "weak" varieties of monoids, programs can only recognize those languages with a "neutral letter" that can be recognized via morphisms over that variety.We then build an algebraic approach to communication complexity, a field which has been of great importance in the study of small complexity classes. We prove that every monoid has communication complexity O(1), &THgr;(log n) or &THgr;(n) in this model. We obtain similar classifications for the communication complexity of finite monoids in the probabilistic, simultaneous, probabilistic simultaneous and MOD p-counting variants of this two-party model and thus characterize the communication complexity (in a worst-case partition sense) of every regular language in these five models. Furthermore, we study the same questions in the Chandra-Furst-Lipton multiparty extension of the classical communication model and describe the variety of monoids which have bounded 3-party communication complexity and bounded k-party communication complexity for some k. We also show how these bounds can be used to establish computational limitations of programs over certain classes of monoids.Finally, we consider the computational complexity of testing if an equation or a system of equations over some fixed finite monoid (or semigroup) has a solution

Bridges between Algebraic Automata Theory and Complexity Theory

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.