Playing Games in the Baire Space

We solve a generalized version of Church's Synthesis Problem where a play is given by a sequence of natural numbers rather than a sequence of bits; so a play is an element of the Baire space rather than of the Cantor space. Two players Input and Output choose natural numbers in alternation to generate a play. We present a natural model of automata ("N-memory automata") equipped with the parity acceptance condition, and we introduce also the corresponding model of "N-memory transducers". We show that solvability of games specified by N-memory automata (i.e., existence of a winning strategy for player Output) is decidable, and that in this case an N-memory transducer can be constructed that implements a winning strategy for player Output.Comment: In Proceedings Cassting'16/SynCoP'16, arXiv:1608.0017

Solving Infinite Games in the Baire Space

Infinite games (in the form of Gale-Stewart games) are studied where a play is a sequence of natural numbers chosen by two players in alternation, the winning condition being a subset of the Baire space $\omega^\omega$. We consider such games defined by a natural kind of parity automata over the alphabet $\mathbb{N}$, called $\mathbb{N}$-MSO-automata, where transitions are specified by monadic second-order formulas over the successor structure of the natural numbers. We show that the classical B\"uchi-Landweber Theorem (for finite-state games in the Cantor space $2^\omega$) holds again for the present games: A game defined by a deterministic parity $\mathbb{N}$-MSO-automaton is determined, the winner can be computed, and an $\mathbb{N}$-MSO-transducer realizing a winning strategy for the winner can be constructed.Comment: Minor revision. 26 pages, 1 figur

Synthesizing Structured Reactive Programs via Deterministic Tree Automata

Existing approaches to the synthesis of reactive systems typically involve the construction of transition systems such as Mealy automata. However, in order to obtain a succinct representation of the desired system, structured programs can be a more suitable model. In 2011, Madhusudan proposed an algorithm to construct a structured reactive program for a given ω-regular specification without synthesizing a transition system first. His procedure is based on two-way alternating ω-automata on finite trees that recognize the set of ”correct ” programs. We present a more elementary and direct approach using only deterministic bottom-up tree automata that compute so-called signatures for a given program. In doing so, we extend Madhusudan’s results to the wider class of programs with bounded delay, which may read several input symbols before producing an output symbol (or vice versa). As a formal foundation, we inductively define a semantics for such programs.

Strategies in infinite games : structured reactive programs and transducers over infinite alphabets

The subject of this thesis is the construction of winning strategies in games of the following kind: Two players alternately choose symbols from some fixed alphabet. They play forever, thus composing an infinite sequence of symbols. The winning condition is a set L of such sequences - the second player wins if the resulting sequence is in the set L, otherwise the first player wins. Such games can be used to model the interaction between a so-called reactive system and its environment: The system continually reads input symbols provided by the environment and produces output symbols in response. A winning condition can be regarded as a specification for the system, and a winning strategy for the system constitutes an implementation of that specification. Church's Synthesis Problem is to determine which player has a winning strategy and to present such a strategy in the form of a finite automaton, given an ω-regular winning condition L ⊆ Σ^ω over a finite alphabet Σ. We consider two variants of this problem. The first variant is a refinement of the task, demanding a strategy in the form of a structured reactive program. It was introduced and solved in 2011 by Madhusudan. His solution involves building an alternating two-way co-Büchi tree automaton recognizing the programs that are winning strategies. We present a direct construction of a deterministic bottom-up tree automaton recognizing these programs and give a lower bound for the size of any such automaton. In both approaches, the number of (Boolean) variables available to the programs is crucial, as the time required by the synthesis algorithm is doubly exponential in that number. We show that for certain winning conditions defined in linear temporal logic (LTL), the required number of variables is exponential in the formula size, matching the known upper bound. The second variant of the synthesis problem is a generalization where the players choose symbols from an infinite alphabet instead of a finite one. More precisely, the alphabet is of the form Σ*, for a finite set Σ, so each symbol is a finite word. To represent winning conditions, we define automata over such infinite alphabets, namely ℕ-memory automata. We study closure properties and decidability questions for these automata, both on finite words and on ω-words. Analogously, we introduce ℕ-memory transducers to represent strategies. We show that the synthesis problem is solvable in this setting: Given a winning condition L ⊆ (Σ*)^ω defined by a deterministic ℕ-memory parity automaton, we can determine which player has a winning strategy and construct such a strategy in the form of a deterministic ℕ-memory transducer

Round-Bounded Control of Parameterized Systems

International audienceWe consider systems with unboundedly many processes that communicate through shared memory. In that context, simple verification questions have a high complexity or, in the case of pushdown processes , are even undecidable. Good algorithmic properties are recovered under round-bounded verification, which restricts the system behavior to a bounded number of round-robin schedules. In this paper, we extend this approach to a game-based setting. This allows one to solve synthesis and control problems and constitutes a further step towards a theory of languages over infinite alphabets