International audience

We provide a finite equational presentation of graphs of treewidth at most three, solving an instance of an open problem by Courcelle and Engelfriet. We use a syntax generalising series-parallel expressions, denoting graphs with a small interface. We introduce appropriate notions of connectivity for such graphs (components, cutvertices, separation pairs). We use those concepts to analyse the structure of graphs of treewidth at most three, showing how they can be decomposed recursively, first canonically into connected parallel components, and then non-deterministically. The main difficulty consists in showing that all non-deterministic choices can be related using only finitely many equational axioms.

We propose a syntax of regular expressions, which describes languages of tree-width 2 graphs. We show that these languages correspond exactly to those languages of tree-width 2 graphs, definable in the counting monadic second-order logic (CMSO).

International audience

In the literature, there are two ways to show that the equational theory of relations over a given signature is not finitely axiomatizable. The first-one is based on games and a construction called Rainbow construction. This method is very technical but it shows a strong result: the equational theory cannot be axiomatized by any finite set of first-order formulas. There is another method, based on a graph characterization of the equational theory of relations, which is easier to get and to understand, but proves a weaker result: the equational theory cannot be axiomatized by any finite set of equations. In this presentation, I will show how to complete the second technique to get the stronger result of non-axiomatizability by first-order formulas.

The equational theory of relations can be characterized using graphs and homomorphisms. This result, found independently by Freyd and Scedrov and by Andréka and Bredikhin, shows that the equational theory of relations is decidable. In this paper, we extend this characterization to the whole universal first-order theory of relations. Using our characterization, we show that the positive universal fragment is also decidable.

We study the equational theories of composition and intersection on binary relations, with or without their associated neutral elements (identity and full relation). Without these constants, the equational theory coincides with that of semilattice-ordered semigroups. We show that the equational theory is no longer finitely based when adding one or the other constant, refuting a conjecture from the literature. Our proofs exploit a characterisation in terms of graphs and homomorphisms, which we show how to adapt in order to capture standard equational theories over the considered signatures.

We study tree-to-tree transformations that can be defined in first-order logic or monadic second-order logic. We prove a decomposition theorem, which shows that every transformation can be obtained from prime transformations, such as tree-to-tree homomorphisms or pre-order traversal, by using combinators such as function composition.

Abstract We study the Horn theories of Kleene algebras and star continuous Kleene algebras, from the complexity point of view. While their equational theories coincide and are PSpace-complete, their Horn theories differ and are undecidable. We characterise the Horn theory of star continuous Kleene algebras in terms of downward closed languages and we show that when restricting the shape of allowed hypotheses, the problems lie in various levels of the arithmetical or analytical hierarchy. We also answer a question posed by Cohen about hypotheses of the form $$1=S$$ where S is a sum of letters: we show that it is decidable.

Pomsets constitute one of the most basic models of concurrency. A pomset is a generalisation of a word over an alphabet in that letters may be partially ordered. A term $t$ using the bi-Kleene operations $0,1, +, \cdot\, ,^*, \parallel, ^{(*)}$ defines a language $ \mathopen{[\![ } t \mathclose{]\!] } $ of pomsets in a natural way. We prove that every valid universal equality over pomset languages using these operations is a consequence of the equational theory of regular languages (in which parallel multiplication and iteration are undefined) plus that of the commutative-regular languages (in which sequential multiplication and iteration are undefined). We also show that the class of $\textit{rational}$ pomset languages (that is, those languages generated from singleton pomsets using the bi-Kleene operations) is closed under all Boolean operations. An $ \textit{ideal}$ of a pomset $p$ is a pomset using the letters of $p$, but having an ordering at least as strict as $p$. A bi-Kleene term $t$ thus defines the set $ \textbf{Id} (\mathopen{[\![ } t \mathclose{]\!] }) $ of ideals of pomsets in $ \mathopen{[\![ } t \mathclose{]\!] } $. We prove that if $t$ does not contain commutative iteration $^{(*)}$ (in our terminology, $t$ is bw-rational) then $\textbf{Id} (\mathopen{[\![ } t \mathclose{]\!] }) \cap \textbf{Pom}_{sp}$, where $ \textbf{Pom}_{sp}$ is the set of pomsets generated from singleton pomsets using sequential and parallel multiplication ($ \cdot$ and $ \parallel$) is defined by a bw-rational term, and if two such terms $t,t'$ define the same ideal language, then $t'=t$ is provable from the Kleene axioms for $0,1, +, \cdot\, ,^*$ plus the commutative idempotent semiring axioms for $0,1, +, \parallel$ plus the exchange law $ (u \parallel v)\cdot ( x \parallel y) \le v \cdot y \parallel u \cdot x $.

We give a new proof that the axioms of left-handed Kleene algebra are complete with respect to language containments. This proof is significantly simpler than both the proof of Boffa (which relies on Krob’s completeness result), and the more recent proof of Kozen and Silva. Our proof builds on a recent non-wellfounded sequent calculus which makes it possible to explicitly compute the invariants required for left-handed Kleene algebra.

We give a new proof of completeness for the linear-time μ-calculus w.r.t. Kozens's axiomatization. Our proof has the advantage of being constructive, i.e., it builds a proof for every valid formula.

Infinitary and regular proofs are commonly used in fixed point logics. Being natural intermediate devices between semantics and traditional finitary proof systems, they are commonly found in completeness arguments, automated deduction, verification, etc. However, their proof theory is surprisingly underdeveloped. In particular, very little is known about the computational behavior of such proofs through cut elimination. Taking such aspects into account has unlocked rich developments at the intersection of proof theory and programming language theory. One would hope that extending this to infinitary calculi would lead, e.g., to a better understanding of recursion and corecursion in programming languages. Structural proof theory is notably based on two fundamental properties of a proof system: cut elimination and focalization. The first one is only known to hold for restricted (purely additive) infinitary calculi, thanks to the work of Santocanale and Fortier; the second one has never been studied in infinitary systems. In this paper, we consider the infinitary proof system muMALLi for multiplicative and additive linear logic extended with least and greatest fixed points, and prove these two key results. We thus establish muMALLi as a satisfying computational proof system in itself, rather than just an intermediate device in the study of finitary proof systems.

Modal μ-calculus is one of the central languages of logic and verification, whose study involves notoriously complex objects: automata over infinite structures on the model-theoretical side; infinite proofs and proofs by (co)induction on the proof-theoretical side. Nevertheless, axiomatizations have been given for both linear and branching time μ-calculi, with quite involved completeness arguments. We come back to this central problem, considering it from a proof search viewpoint, and provide some new completeness arguments in the linear time μ-calculus. Our results only deal with restricted classes of formulas that closely correspond to (non-alternating) ω-automata but, compared to earlier proofs, our completeness arguments are direct and constructive. We first consider a natural circular proof system based on sequent calculus, and show that it is complete for inclusions of parity automata expressed as formulas, making use of Safra's construction directly in proof search. We then consider the corresponding finitary proof system, featuring (co)induction rules, and provide a partial translation result from circular to finitary proofs. This yields completeness of the finitary proof system for inclusions of sufficiently deterministic parity automata, and finally for arbitrary Büchi automata.

Various logics have been introduced in order to reason over (co)inductive specifications and, through the Curry-Howard correspondence, to study computation over inductive and coinductive data. The logic mu-MALL is one of those logics, extending multiplicative and additive linear logic with least and greatest fixed point operators. In this paper, we investigate the semantics of mu-MALL proofs in (computational) ludics. This framework is built around the notion of design, which can be seen as an analogue of the strategies of game semantics. The infinitary nature of designs makes them particularly well suited for representing computations over infinite data. We provide mu-MALL with a denotational semantics, interpreting proofs by designs and formulas by particular sets of designs called behaviours. Then we prove a completeness result for the class of "essentially finite designs", which are those designs performing a finite computation followed by a copycat. On the way to completeness, we investigate semantic inclusion, proving its decidability (given two formulas, we can decide whether the semantics of one is included in the other's) and completeness (if semantic inclusion holds, the corresponding implication is provable in mu-MALL).