Reachability analysis of first-order definable pushdown systems

We study pushdown systems where control states, stack alphabet, and transition relation, instead of being finite, are first-order definable in a fixed countably-infinite structure. We show that the reachability analysis can be addressed with the well-known saturation technique for the wide class of oligomorphic structures. Moreover, for the more restrictive homogeneous structures, we are able to give concrete complexity upper bounds. We show ample applicability of our technique by presenting several concrete examples of homogeneous structures, subsuming, with optimal complexity, known results from the literature. We show that infinitely many such examples of homogeneous structures can be obtained with the classical wreath product construction.Comment: to appear in CSL'1

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 Formal Languages

We introduce formal languages over infinite alphabets where words may contain binders. We define the notions of nominal language, nominal monoid, and nominal regular expressions. Moreover, we extend history-dependent automata (HD-automata) by adding stack, and study the recognisability of nominal languages

Regularity Preserving but not Reflecting Encodings

Encodings, that is, injective functions from words to words, have been studied extensively in several settings. In computability theory the notion of encoding is crucial for defining computability on arbitrary domains, as well as for comparing the power of models of computation. In language theory much attention has been devoted to regularity preserving functions. A natural question arising in these contexts is: Is there a bijective encoding such that its image function preserves regularity of languages, but its pre-image function does not? Our main result answers this question in the affirmative: For every countable class C of languages there exists a bijective encoding f such that for every language L in C its image f[L] is regular. Our construction of such encodings has several noteworthy consequences. Firstly, anomalies arise when models of computation are compared with respect to a known concept of implementation that is based on encodings which are not required to be computable: Every countable decision model can be implemented, in this sense, by finite-state automata, even via bijective encodings. Hence deterministic finite-state automata would be equally powerful as Turing machine deciders. A second consequence concerns the recognizability of sets of natural numbers via number representations and finite automata. A set of numbers is said to be recognizable with respect to a representation if an automaton accepts the language of representations. Our result entails that there is one number representation with respect to which every recursive set is recognizable

Set Augmented Finite Automata over Infinite Alphabets

A data language is a set of finite words defined on an infinite alphabet. Data languages are used to express properties associated with data values (domain defined over a countably infinite set). In this paper, we introduce set augmented finite automata (SAFA), a new class of automata for expressing data languages. We investigate the decision problems, closure properties, and expressiveness of SAFA. We also study the deterministic variant of these automata.Comment: This is a full version of a paper with the same name accepted in DLT 2023. Other than the full proofs, this paper contains several new results concerning more closure properties, universality problem, comparison of expressiveness with register automata and class counter automata, and more results on deterministic SAF

Polishness of some topologies related to word or tree automata

We prove that the B\"uchi topology and the automatic topology are Polish. We also show that this cannot be fully extended to the case of a space of infinite labelled binary trees; in particular the B\"uchi and the Muller topologies are not Polish in this case.Comment: This paper is an extended version of a paper which appeared in the proceedings of the 26th EACSL Annual Conference on Computer Science and Logic, CSL 2017. The main addition with regard to the conference paper consists in the study of the B\"uchi topology and of the Muller topology in the case of a space of trees, which now forms Section

On the Structure and Complexity of Rational Sets of Regular Languages

In a recent thread of papers, we have introduced FQL, a precise specification language for test coverage, and developed the test case generation engine FShell for ANSI C. In essence, an FQL test specification amounts to a set of regular languages, each of which has to be matched by at least one test execution. To describe such sets of regular languages, the FQL semantics uses an automata-theoretic concept known as rational sets of regular languages (RSRLs). RSRLs are automata whose alphabet consists of regular expressions. Thus, the language accepted by the automaton is a set of regular expressions. In this paper, we study RSRLs from a theoretic point of view. More specifically, we analyze RSRL closure properties under common set theoretic operations, and the complexity of membership checking, i.e., whether a regular language is an element of a RSRL. For all questions we investigate both the general case and the case of finite sets of regular languages. Although a few properties are left as open problems, the paper provides a systematic semantic foundation for the test specification language FQL