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

Reactive Turing machines

We propose reactive Turing machines (RTMs), extending classical Turing machines with a process-theoretical notion of interaction, and use it to define a notion of executable transition system. We show that every computable transition system with a bounded branching degree is simulated modulo divergence-preserving branching bisimilarity by an RTM, and that every effective transition system is simulated modulo the variant of branching bisimilarity that does not require divergence preservation. We conclude from these results that the parallel composition of (communicating) RTMs can be simulated by a single RTM. We prove that there exist universal RTMs modulo branching bisimilarity, but these essentially employ divergence to be able to simulate an RTM of arbitrary branching degree. We also prove that modulo divergence-preserving branching bisimilarity there are RTMs that are universal up to their own branching degree. We establish a correspondence between executability and finite definability in a simple process calculus. Finally, we establish that RTMs are at least as expressive as persistent Turing machines

Reactive Turing machines

We propose reactive Turing machines (RTMs), extending classical Turing machines with a process-theoretical notion of interaction, and use it to define a notion of executable transition system. We show that every computable transition system with a bounded branching degree is simulated modulo divergence-preserving branching bisimilarity by an RTM, and that every effective transition system is simulated modulo the variant of branching bisimilarity that does not require divergence preservation. We conclude from these results that the parallel composition of (communicating) RTMs can be simulated by a single RTM. We prove that there exist universal RTMs modulo branching bisimilarity, but these essentially employ divergence to be able to simulate an RTM of arbitrary branching degree. We also prove that modulo divergence-preserving branching bisimilarity there are RTMs that are universal up to their own branching degree. We establish a correspondence between executability and finite definability in a simple process calculus. Finally, we establish that RTMs are at least as expressive as persistent Turing machines

Computations and interaction

We enhance the notion of a computation of the classical theory of computing with the notion of interaction. In this way, we enhance a Turing machine as a model of computation to a Reactive Turing Machine that is an abstract model of a computer as it is used nowadays, always interacting with the user and the world

Expressiveness modulo bisimilarity of regular expressions with parallel composition

The languages accepted by finite automata are precisely the languages denoted by regular expressions. In contrast, finite automata may exhibit behaviours that cannot be described by regular expressions up to bisimilarity. In this paper, we consider extensions of the theory of regular expressions with various forms of parallel composition and study the effect on expressiveness. First we prove that adding pure interleaving to the theory of regular expressions strictly increases its expressiveness modulo bisimilarity. Then, we prove that replacing the operation for pure interleaving by ACP-style parallel composition gives a further increase in expressiveness, still insufficient, however, to facilitate the expression of all finite automata up to bisimilarity. Finally, we prove that the theory of regular expressions with ACP-style parallel composition and encapsulation is expressive enough to express all finite automata up to bisimilarity. Our results extend the expressiveness results obtained by Bergstra, Bethke and Ponse for process algebras with (the binary variant of) Kleene’s star operation

A Basic Parallel Process as a Parallel Pushdown Automaton

AbstractWe investigate the set of basic parallel processes, recursively defined by action prefix, interleaving, 0 and 1. Different from literature, we use the constants 0 and 1 standing for unsuccessful and successful termination in order to stay closer to the analogies in automata theory.We prove that any basic parallel process is rooted branching bisimulation equivalent to a regular process communicating with a bag (also called a parallel pushdown automaton) and therefore we can regard the bag as the prototypical basic parallel process.This result is closely related to the fact that any context-free process is either rooted branching bisimulation equivalent or contrasimulation equivalent to a regular process communicating with a stack, a result that is the analogy in process theory of the language theory result that any context-free language is the language of a pushdown automaton

A process-theoretic look at automata

Determinism is a semantic property of (a fragment of) a language that specifies that a program cannot evolve operationally in several different ways. Idempotence is a property of binary composition operators requiring that the composition of two identical specifications or programs will result in a piece of specification or program that is equivalent to the original components. In this paper, we propose two (related) meta-theorems for guaranteeing determinism and idempotence of binary operators. These meta-theorems are formulated in terms of syntactic templates for operational semantics, called rule formats. We show the applicability of our formats by applying them to various operational semantics from the literature