On the transition graphs of turing machines

• Published in: Theoretical computer science Vol. 296; no. 2; pp. 195 - 223
• Main Author: Caucal, Didier
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Amsterdam Elsevier B.V 08.03.2003; Elsevier
• Subjects: Applied sciences; Automata. Abstract machines. Turing machines; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Exact sciences and technology; Formal languages; Graph theory; Infinite graphs; Logic and foundations; Mathematical logic, foundations, set theory; Mathematics; Recursion theory; Sciences and techniques of general use; Theoretical computing; Formal languages; Infinite graphs; Tree(graph); Word; Configuration; Closure; Transition graph; Language family; Turing machine; Context free language; Mapping; Language; Substitution; Rational graph; Inverse; Rewriting systems; Formal language; Automaton; Rewriting; Deterministic automaton; Graph theory; Labelled graph; Binary tree; State; Infinite graph; Transition; Enumerable language
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/S0304-3975(02)00655-2

As for pushdown automata, we consider labelled Turing machines with ε-rules. With any Turing machine M and with a rational set C of configurations, we associate the restriction to C of the ϵ-closure of the transition set of M. We get the same family of graphs by using the labelled word rewriting systems. We show that this family is the set of graphs obtained from the binary tree by applying an inverse mapping into F followed by a rational restriction, where F is any family of recursively enumerable languages containing the rational closure of all linear languages. We show also that this family is obtained from the rational graphs by inverse rational mappings. Finally we show that this family is also the set of graphs recognized by (unlabelled) Turing machines with labelled final states, and even if we restrict to deterministic Turing machines.