Szymon Toruńczyk

Energy and mean-payoff games with imperfect information

We consider two-player games with imperfect information and quantitative objective. The game is played on a weighted graph with a state space partitioned into classes of indistinguishable states, giving players partial knowledge of the state. In an energy game, the weights represent resource consumption and the objective of the game is to maintain the sum of weights always nonnegative. In a mean-payoff game, the objective is to optimize the limit-average usage of the resource. We show that the problem of determining if an energy game with imperfect information with fixed initial credit has a winning strategy is decidable, while the question of the existence of some initial credit such that the game has a winning strategy is undecidable. This undecidability result carries over to meanpayoff games with imperfect information. On the positive side, using a simple restriction on the game graph (namely, that the weights are visible), we show that these problems become EXPTIME-complete.

Turing Machines with Atoms

We study Turing machines over sets with atoms, also known as nominal sets. Our main result is that deterministic machines are weaker than nondeterministic ones; in particular, P≠NP in sets with atoms. Our main construction is closely related to the Cai-Furer-Immerman graphs used in descriptive complexity theory.

Weak MSO+U over infinite trees

We prove that, over infinite trees, satisfiability is decidable for Weak Monadic Second-Order Logic extended by the unbounding quantifier U. We develop an automaton model, prove that it is effectively equivalent to the logic, and that the automaton model has decidable emptiness.

Automata based verification over linearly ordered data domains

In this paper we work over linearly ordered data domains equipped with finitely many unary predicates and constants. We consider nondeterministic automata processing words and storing finitely many variables ranging over the domain. During a transition, these automata can compare the data values of the current configuration with those of the previous configuration using the linear order, the unary predicates and the constants. We show that emptiness for such automata is decidable, both over finite and infinite words, under reasonable computability assumptions on the linear order. Finally, we show how our automata model can be used for verifying properties of workflow specifications in the presence of an underlying database.

The MSO+U Theory of (N, <) Is Undecidable

We consider the logic MSO+U, which is monadic second-order logic extended with the unbounding quantifier. The unbounding quantifier is used to say that a property of finite sets holds for sets of arbitrarily large size. We prove that the logic is undecidable on infinite words, i.e. the MSO+U theory of (N,<) is undecidable. This settles an open problem about the logic, and improves a previous undecidability result, which used infinite trees and additional axioms from set theory.

Deterministic Automata and Extensions of Weak MSO

We introduce a new class of automata on infinite words, called min-automata. We prove that min-automata have the same expressive power as weak monadic second-order logic (weak MSO) extended with a new quantifier, the recurrence quantifier. These results are dual to a framework presented in \cite{max-automata}, where max-automata were proved equivalent to weak MSO extended with an unbounding quantifier. We also present a general framework, which tries to explain which types of automata on infinite words correspond to extensions of weak MSO. As another example for the usefulness framework, apart from min- and max-automata, we define an extension of weak MSO with a quantifier that talks about ultimately periodic sets.

Languages of profinite words and the limitedness problem

We present a new framework for the limitedness problem. The key novelty is a description using profinite words, which unifies and simplifies the previous approaches, allowing a seamless extension of the theory of regular languages. We also define a logic over profinite words, called MSO+inf and show that the satisfiability problem of MSO+

Imperative Programming in Sets with Atoms

\mathbb B

On the topological complexity of MSO+U and related automata models

reduces to the satisfiability problem of our logic.

Locally Finite Constraint Satisfaction Problems

We define an imperative programming language, which extends while programs with a type for storing atoms or hereditarily orbit-finite sets. To deal with an orbit-finite set, the language has a loop construction, which is executed in parallel for all elements of an orbit-finite set. We show examples of programs in this language, e.g. a program for minimising deterministic orbit-finite automata.