Sławomir Lasota

Alternating timed automata

A notion of alternating timed automata is proposed. It is shown that such automata with only one clock have decidable emptiness problem over finite words. This gives a new class of timed languages that is closed under boolean operations and which has an effective presentation. We prove that the complexity of the emptiness problem for alternating timed automata with one clock is nonprimitive recursive. The proof gives also the same lower bound for the universality problem for nondeterministic timed automata with one clock. We investigate extension of the model with epsilon-transitions and prove that emptiness is undecidable. Over infinite words, we show undecidability of the universality problem.

Automata with Group Actions

Our motivating question is a My hill-Nerode theorem for infinite alphabets. We consider several kinds of those: alphabets whose letters can be compared only for equality, but also ones with more structure, such as a total order or a partial order. We develop a framework for studying such alphabets, where the key role is played by the automorphism group of the alphabet. This framework builds on the idea of nominal sets of Gabbay and Pitts, nominal sets are the special case of our framework where letters can be only compared for equality. We use the framework to uniformly generalize to infinite alphabets parts of automata theory, including decidability results. In the case of letters compared for equality, we obtain automata equivalent in expressive power to finite memory automata, as defined by Francez and Kaminski.

Automata theory in nominal sets

We study languages over infinite alphabets equipped with some structure that can be tested by recognizing automata. We develop a framework for studying such alphabets and the ensuing automata theory, where the key role is played by an automorphism group of the alphabet. In the process, we generalize nominal sets due to Gabbay and Pitts.

Towards nominal computation

Nominal sets are a different kind of set theory, with a more relaxed notion of finiteness. They offer an elegant formalism for describing lambda-terms modulo alpha-conversion, or automata on data words. This paper is an attempt at defining computation in nominal sets. We present a rudimentary programming language, called Nlambda. The key idea is that it includes a native type for finite sets in the nominal sense. To illustrate the power of our language, we write short programs that process automata on data words.

Logical relations for monadic types

Logical relations and their generalisations are a fundamental tool in proving properties of lambda calculi, for example, for yielding sound principles for observational equivalence. We propose a natural notion of logical relations that is able to deal with the monadic types of Moggis computational lambda calculus. The treatment is categorical, and is based on notions of subsconing, mono factorisation systems and monad morphisms. Our approach has a number of interesting applications, including cases for lambda calculi with non-determinism (where being in a logical relation means being bisimilar), dynamic name creation and probabilistic systems.

An Extension of Data Automata that Captures XPath

We define a new kind of automata recognizing properties of data words or data trees and prove that the automata capture all queries definable in Regular XPath. We show that the automata-theoretic approach may be applied to answer decidability and expressibility questions for XPath. Finally, we use the newly introduced automata as a common framework to classify existing automata on data words and trees, including data automata, register automata and alternating register automata.

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.

A machine-independent characterization of timed languages

We use a variant of Fraenkel-Mostowski sets (known also as nominal sets) as a framework suitable for stating and proving the following two results on timed automata. The first result is a machine-independent characterization of languages of deterministic timed automata. As a second result we define a class of automata, called by us timed register automata, that extends timed automata and is effectively closed under minimization.

Relating timed and register automata

Timed automata and register automata are well-known models of computation over timed and data words respectively. The former has clocks that allow to test the lapse of time between two events, whilst the latter includes registers that may store data values for later comparison. Although these two models behave in appearance differently, several decision problems have the same (un)decidability and complexity results for both models. As a prominent example, emptiness is decidable for alternating automata with one clock or register, both with non-primitive recursive complexity. This is not by chance. This work confirms that there is indeed a tight relationship between the two models. We show that a run of a timed automaton can be simulated by a register automaton, and conversely that a run of a register automaton can be simulated by a timed automaton. Our results allow to transfer complexity and decidability results back and forth between these two kinds of models. We justify the usefulness of these reductions by obtaining new results on register automata.

Fast equivalence-checking for normed context-free processes

Bisimulation equivalence is decidable in polynomial time over normed graphs generated by a context-free grammar. We present a new algorithm, working in timeO(n 5 ), thus improving the previously known complexityO(n 8 polylog(n)). It also improves the previously known complexity O(n 6 polylog(n)) of the equality problem for simple grammars.