Pushdown Automata and Context-Free Grammars in Bisimulation Semantics

TL;DR

This paper extends classical automata theory by adapting pushdown automata and context-free grammars within a bisimulation semantics framework, introducing a reactive model with enhanced correspondence properties.

Contribution

It introduces a reactive Turing machine model with bisimilarity, modifies process operators for better automata-grammar correspondence, and highlights the importance of state awareness.

Findings

Bisimilarity replaces language equivalence in process graphs.

Modified process operators improve automata-grammar correspondence.

State awareness is crucial for full correspondence.

Abstract

The Turing machine models an old-fashioned computer, that does not interact with the user or with other computers, and only does batch processing. Therefore, we came up with a Reactive Turing Machine that does not have these shortcomings. In the Reactive Turing Machine, transitions have labels to give a notion of interactivity. In the resulting process graph, we use bisimilarity instead of language equivalence. Subsequently, we considered other classical theorems and notions from automata theory and formal languages theory. In this paper, we consider the classical theorem of the correspondence between pushdown automata and context-free grammars. By changing the process operator of sequential composition to a sequencing operator with intermediate acceptance, we get a better correspondence in our setting. We find that the missing ingredient to recover the full correspondence is the addition of a notion of state awareness.