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String rewriting system
In theoretical computer science and mathematical logic a string rewriting system (SRS), historically called a semi Thue system, is a rewriting system over strings from a (usually finite) alphabet. Given a binary relation between fixed strings over the alphabet, called rewrite rules, denoted by , an SRS extends the rewriting relation to all strings in which the left and right hand side of the rules appear as substrings, that is , where , , , and are strings. The notion of a semi Thue system essentially coincides with the presentation of a monoid. Thus they constitute a natural framework for solving the word problem for monoids and groups. An SRS can be defined directly as an abstract rewriting system. It can also be seen as a restricted kind of a term rewriting system, in which all function symbols have an arity of at most 1. As a formalism, string rewriting systems are Turing complete. The semi Thue name comes from the Norwegian mathematician Axel Thue, who introduced systematic treatment of string rewriting systems in a 1914 paper. Thue introduced this notion hoping to solve the word problem for finitely presented semigroups. Only in 1947 was the problem shown to be undecidable— this result was obtained independently by Emil Post and A. A. Markov Jr.

In general, the set of strings on an alphabet forms a free monoid together with the binary operation of string concatenation (denoted as and written multiplicatively by dropping the symbol). In a SRS, the reduction relation is compatible with the monoid operation, meaning that implies for all strings Since is by definition a preorder, forms a monoidal preorder. Similarly, the reflexive transitive symmetric closure of , denoted (see abstract rewriting system#Basic notions), is a congruence, meaning it is an equivalence relation (by definition) and it is also compatible with string concatenation. The relation is called the Thue congruence generated by R. In a Thue system, i. e. if R is symmetric, the rewrite relation coincides with the Thue congruence .

A semi Thue system is also a term rewriting system—one that has monadic words (functions) ending in the same variable as the left and right hand side terms, e. g. a term rule is equivalent to the string rule
A semi Thue system is also a special type of Post canonical system, but every Post canonical system can also be reduced to an SRS. Both formalisms are Turing complete, and thus equivalent to Noam Chomsky's unrestricted grammars, which are sometimes called semi Thue grammars. A formal grammar only differs from a semi Thue system by the separation of the alphabet into terminals and non terminals, and the fixation of a starting symbol amongst non terminals. A minority of authors actually define a semi Thue system as a triple , where is called the set of axioms. Under this "generative" definition of semi Thue system, an unrestricted grammar is just a semi Thue system with a single axiom in which one partitions the alphabet into terminals and non terminals, and makes the axiom a nonterminal. The simple artifice of partitioning the alphabet into terminals and non terminals is a powerful one; it allows the definition of the Chomsky hierarchy based on what combination of terminals and non terminals the rules contain. This was a crucial development in the theory of formal languages. In quantum computing, the notion of a quantum Thue system can be developed. Since quantum computation is intrinsically reversible, the rewriting rules over the alphabet are required to be bidirectional (i. e. the underlying system is a Thue system, not a semi Thue system). On a subset of alphabet characters one can attach a Hilbert space , and a rewriting rule taking a substring to another one can carry out a unitary operation on the tensor product of the Hilbert space attached to the strings; this implies that they preserve the number of characters from the set Similar to the classical case one can show that a quantum Thue system is a universal computational model for quantum computation, in the sense that the executed quantum operations correspond to uniform circuit classes (such as those in BQP when e. g. guaranteeing termination of the string rewriting rules within polynomially many steps in the input size), or equivalently a Quantum Turing machine.

Semi Thue systems were developed as part of a program to add additional constructs to logic, so as to create systems such as propositional logic, that would allow general mathematical theorems to be expressed in a formal language, and then proven and verified in an automatic, mechanical fashion. The hope was that the act of theorem proving could then be reduced to a set of defined manipulations on a set of strings. It was subsequently realized that semi Thue systems are isomorphic to unrestricted grammars, which in turn are known to be isomorphic to Turing machines. This method of research succeeded and now computers can be used to verify the proofs of mathematic and logical theorems. At the suggestion of Alonzo Church, Emil Post in a paper published in 1947, first proved "a certain Problem of Thue" to be unsolvable, what Martin Davis states as " the first unsolvability proof for a problem from classical mathematics in this case the word problem for semigroups." Davis also asserts that the proof was offered independently by A. A. Markov.

Ronald V. Book and Friedrich Otto, String rewriting Systems, Springer, 1993, Matthias Jantzen, Confluent string rewriting, Birkhäuser, 1988, .

Martin Davis, Ron Sigal, Elaine J. Weyuker, Computability, complexity, and languages: fundamentals of theoretical computer science, 2nd ed., Academic Press, 1994, , chapter 7
Elaine Rich, Automata, computability and complexity: theory and applications, Prentice Hall, 2007, , chapter 23.5.

Samson Abramsky, Dov M. Gabbay, Thomas S. E. Maibaum (ed. ), Handbook of Logic in Computer Science: Semantic modelling, Oxford University Press, 1995, Grzegorz Rozenberg, Arto Salomaa (ed. ), Handbook of Formal Languages: Word, language, grammar, Springer, 1997, .