David Janin

On the Expressive Completeness of the Propositional mu-Calculus with Respect to Monadic Second Order Logic

Monadic second order logic (MSOL) over transition systems is considered. It is shown that every formula of MSOL which does not distinguish between bisimilar models is equivalent to a formula of the propositional Μ-calculus. This expressive completeness result implies that every logic over transition systems invariant under bisimulation and translatable into MSOL can be also translated into the Μ-calculus. This gives a precise meaning to the statement that most propositional logics of programs can be translated into the Μ-calculus.

Automata for the Modal mu-Calculus and related Results

The propositional Μ-calculus as introduced by Kozen in [4] is considered. The notion of disjunctive formula is defined and it is shown that every formula is semantically equivalent to a disjunctive formula. For these formulas many difficulties encountered in the general case may be avoided. For instance, satisfiability checking is linear for disjunctive formulas. This kind of formula gives rise to a new notion of finite automaton which characterizes the expressive power of the Μ-calculus over all transition systems.

Permissive strategies: from parity games to safety games

It is proposed to compare strategies in a parity game by comparing the sets of behaviours they allow. For such a game, there may be no winning strategy that encompasses all the behaviours of all winning strategies. It is shown, however, that there always exists a permissive strategy that encompasses all the behaviours of all memoryless strategies. An algorithm for finding such a permissive strategy is presented. Its complexity matches currently known upper bounds for the simpler problem of finding the set of winning positions in a parity game. The algorithm can be seen as a reduction of a parity to a safety game and computation of the set of winning positions in the resulting game.

Algebras, automata and logic for languages of labeled birooted trees

In this paper, we study the languages of labeled finite birooted trees: Munns birooted trees extended with vertex labeling. We define a notion of finite state birooted tree automata that is shown to capture the class of languages that are upward closed w.r.t. the natural order and definable in Monadic Second Order Logic. Then, relying on the inverse monoid structure of labeled birooted trees, we derive a notion of recognizable languages by means of (adequate) premorphisms into finite (adequately) ordered monoids. This notion is shown to capture finite boolean combinations of languages as above. We also provide a simple encoding of finite (mono-rooted) labeled trees in an antichain of labeled birooted trees that shows that classical regular languages of finite (mono-rooted) trees are also recognized by such premorphisms and finite ordered monoids.

The T-calculus: towards a structured programing of (musical) time and space

In the field of music system programming, the T-calculus is a proposal for combining space modeling and time programming into a single programming feature: spatiotemporal tiled programming. Based on a solid algebraic model, it aims at decomposing every operation on musical objects into the sequence of a synchronization operation that describes how objects are positioned one with respect the other, and a fusion operation that describes how their values are then combined. A first simple version of such a tiled calculus is presented and studied in this paper.

On Languages of One-Dimensional Overlapping Tiles

A one-dimensional tile with overlaps is a standard finite word that carries some more information that is used to say when the concatenation of two tiles is legal. Known since the mid 70’s in the rich mathematical field of inverse monoid theory, this model of tiles with the associated partial product have yet not been much studied in theoretical computer science despite some implicit appearances in studies of two-way automata in the 80’s.

Quasi-recognizable vs MSO definable languages of one-dimensional overlapping tiles

It has been shown [6] that, within the McAlister inverse monoid [10], whose elements can be seen as overlapping one-dimensional tiles, the class of languages recognizable by finite monoids collapses compared with the class of languages definable in Monadic Second Order Logic (MSO). This paper aims at capturing the expressive power of the MSO definability of languages of tiles by means of a weakening of the notion of algebraic recognizability which we shall refer to as quasi-recognizability. For that purpose, since the collapse of algebraic recognizability is intrinsically linked with the notion of monoid morphism itself, we propose instead to use premorphisms, monotonic mappings on ordered monoids that are only required to be sub-multiplicative with respect to the monoid product, i.e. mapping φ so that for all x and y, φ(xy)≤φ(x) φ(y). In doing so, we indeed obtain, with additional but relatively natural closure conditions, the expected quasi-algebraic characterization of MSO definable languages of positive tiles. This result is achieved via the axiomatic definition of an original class of well-behaved ordered monoid so that quasi-recognizability implies MSO definability. An original embedding of any (finite) monoid S into a (finite) well-behaved ordered monoid

On the (High) Undecidability of Distributed Synthesis Problems

{\mathcal Q}(S)

Relating levels of the mu-calculus hierarchy and levels of the monadic hierarchy

is then used to prove the converse.

Tiled polymorphic temporal media

The distributed synthesis problem [11] is known to be undecidable. Our purpose here is to study further this undecidability. For this, we consider distributed games [8], an infinite variant of Peterson and Reif multiplayer games with partial information [10], in which Pnueli and Rosners distributed synthesis problem can be encoded and, when decidable [11,6,7], uniformly solved [8]. We first prove that even the simple problem of solving 2-process distributed game with reachability conditions is undecidable (