Highly Undecidable Problems For Infinite Computations

We show that many classical decision problems about 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are $\Pi_2^1$-complete, hence located at the second level of the analytical hierarchy, and "highly undecidable". In particular, the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, and the unambiguity problem are all $\Pi_2^1$-complete for context-free omega-languages or for infinitary rational relations. Topological and arithmetical properties of 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are also highly undecidable. These very surprising results provide the first examples of highly undecidable problems about the behaviour of very simple finite machines like 1-counter automata or 2-tape automata.Comment: to appear in RAIRO-Theoretical Informatics and Application

Algebraic properties of structured context-free languages: old approaches and novel developments

The historical research line on the algebraic properties of structured CF languages initiated by McNaughton's Parenthesis Languages has recently attracted much renewed interest with the Balanced Languages, the Visibly Pushdown Automata languages (VPDA), the Synchronized Languages, and the Height-deterministic ones. Such families preserve to a varying degree the basic algebraic properties of Regular languages: boolean closure, closure under reversal, under concatenation, and Kleene star. We prove that the VPDA family is strictly contained within the Floyd Grammars (FG) family historically known as operator precedence. Languages over the same precedence matrix are known to be closed under boolean operations, and are recognized by a machine whose pop or push operations on the stack are purely determined by terminal letters. We characterize VPDA's as the subclass of FG having a peculiarly structured set of precedence relations, and balanced grammars as a further restricted case. The non-counting invariance property of FG has a direct implication for VPDA too.Comment: Extended version of paper presented at WORDS2009, Salerno,Italy, September 200

Generalizing input-driven languages: theoretical and practical benefits

Regular languages (RL) are the simplest family in Chomsky's hierarchy. Thanks to their simplicity they enjoy various nice algebraic and logic properties that have been successfully exploited in many application fields. Practically all of their related problems are decidable, so that they support automatic verification algorithms. Also, they can be recognized in real-time. Context-free languages (CFL) are another major family well-suited to formalize programming, natural, and many other classes of languages; their increased generative power w.r.t. RL, however, causes the loss of several closure properties and of the decidability of important problems; furthermore they need complex parsing algorithms. Thus, various subclasses thereof have been defined with different goals, spanning from efficient, deterministic parsing to closure properties, logic characterization and automatic verification techniques. Among CFL subclasses, so-called structured ones, i.e., those where the typical tree-structure is visible in the sentences, exhibit many of the algebraic and logic properties of RL, whereas deterministic CFL have been thoroughly exploited in compiler construction and other application fields. After surveying and comparing the main properties of those various language families, we go back to operator precedence languages (OPL), an old family through which R. Floyd pioneered deterministic parsing, and we show that they offer unexpected properties in two fields so far investigated in totally independent ways: they enable parsing parallelization in a more effective way than traditional sequential parsers, and exhibit the same algebraic and logic properties so far obtained only for less expressive language families

Decision Problems for Subclasses of Rational Relations over Finite and Infinite Words

We consider decision problems for relations over finite and infinite words defined by finite automata. We prove that the equivalence problem for binary deterministic rational relations over infinite words is undecidable in contrast to the case of finite words, where the problem is decidable. Furthermore, we show that it is decidable in doubly exponential time for an automatic relation over infinite words whether it is a recognizable relation. We also revisit this problem in the context of finite words and improve the complexity of the decision procedure to single exponential time. The procedure is based on a polynomial time regularity test for deterministic visibly pushdown automata, which is a result of independent interest.Comment: v1: 31 pages, submitted to DMTCS, extended version of the paper with the same title published in the conference proceedings of FCT 2017; v2: 32 pages, minor revision of v1 (DMTCS review process), results unchanged; v3: 32 pages, enabled hyperref for Figure 1; v4: 32 pages, add reference for known complexity results for the slenderness problem; v5: 32 pages, added DMTCS metadat

Equivalence-Checking on Infinite-State Systems: Techniques and Results

The paper presents a selection of recently developed and/or used techniques for equivalence-checking on infinite-state systems, and an up-to-date overview of existing results (as of September 2004)

From computability to executability : a process-theoretic view on automata theory

The theory of automata and formal language was devised in the 1930s to provide models for and to reason about computation. Here we mean by computation a procedure that transforms input into output, which was the sole mode of operation of computers at the time. Nowadays, computers are systems that interact with us and also each other; they are non-deterministic, reactive systems. Concurrency theory, split off from classical automata theory a few decades ago, provides a model of computation similar to the model given by the theory of automata and formal language, but focuses on concurrent, reactive and interactive systems. This thesis investigates the integration of the two theories, exposing the differences and similarities between them. Where automata and formal language theory focuses on computations and languages, concurrency theory focuses on behaviour. To achieve integration, we look for process-theoretic analogies of classic results from automata theory. The most prominent difference is that we use an interpretation of automata as labelled transition systems modulo (divergence-preserving) branching bisimilarity instead of treating automata as language acceptors. We also consider similarities such as grammars as recursive specifications and finite automata as labelled finite transition systems. We investigate whether the classical results still hold and, if not, what extra conditions are sufficient to make them hold. We especially look into three levels of Chomsky's hierarchy: we study the notions of finite-state systems, pushdown systems, and computable systems. Additionally we investigate the notion of parallel pushdown systems. For each class we define the central notion of automaton and its behaviour by associating a transition system with it. Then we introduce a suitable specification language and investigate the correspondence with the respective automaton (via its associated transition system). Because we not only want to study interaction with the environment, but also the interaction within the automaton, we make it explicit by means of communicating parallel components: one component representing the finite control of the automaton and one component representing the memory. First, we study finite-state systems by reinvestigating the relation between finite-state automata, left- and right-linear grammars, and regular expressions, but now up to (divergence-preserving) branching bisimilarity. For pushdown systems we augment the finite-state systems with stack memory to obtain the pushdown automata and consider different termination styles: termination on empty stack, on final state, and on final state and empty stack. Unlike for language equivalence, up to (divergence-preserving) branching bisimilarity the associated transition systems for the different termination styles fall into different classes. We obtain (under some restrictions) the correspondence between context-free grammars and pushdown automata for termination on final state and empty stack. We show how for contrasimulation, a weaker equivalence than branching bisimilarity, we can obtain the correspondence result without some of the restrictions. Finally, we make the interaction within a pushdown automaton explicit, but in a different way depending on the termination style. By analogy of pushdown systems we investigate the parallel pushdown systems, obtained by augmenting finite-state systems with bag memory, and consider analogous termination styles. We investigate the correspondence between context-free grammars that use parallel composition instead of sequential composition and parallel pushdown automata. While the correspondence itself is rather tight, it unfortunately only covers a small subset of the parallel pushdown automata, i.e. the single-state parallel pushdown automata. When making the interaction within parallel pushdown automata explicit, we obtain a rather uniform result for all termination styles. Finally, we study computable systems and the relation with exective and computable transition systems and Turing machines. For this we present the reactive Turing machine, a classical Turing machine augmented with capabilities for interaction. Again, we make the interaction in the reactive Turing machine between its finite control and the tape memory explicit