Post canonical system

A Post canonical system, as created by Emil Post, is a string-manipulation system that starts with finitely-many strings and repeatedly transforms them by applying a finite set j of specified rules of a certain form, thus generating a formal language. Today they are mainly of historical relevance because every Post canonical system can be reduced to a string rewriting system (semi-Thue system), which is a simpler formulation. Both formalisms are Turing complete.

A Post canonical system is a triplet (A,I,R), where

where each g and h is a specified fixed word, and each $ and $' is a variable standing for an arbitrary word. The strings before and after the arrow in a production rule are called the rule's antecedents and consequent, respectively. It is required that each $' in the consequent be one of the $s in the antecedents of that rule, and that each antecedent and consequent contain at least one variable.

In many contexts, each production rule has only one antecedent, thus taking the simpler form

The formal language generated by a Post canonical system is the set whose elements are the initial words together with all words obtainable from them by repeated application of the production rules. Such sets are recursively enumerable languages and every recursively enumerable language is the restriction of some such set to a sub-alphabet of A.

A Post canonical system is said to be in normal form if it has only one initial word and every production rule is of the simple form

Post 1943 proved the remarkable Normal-form Theorem, which applies to the most-general type of Post canonical system:

Tag systems, which comprise a universal computational model, are notable examples of Post normal-form system, being also monogenic. (A canonical system is said to be monogenic if, given any string, at most one new string can be produced from it in one step — i.e., the system is deterministic.)

A string rewriting system is a special type of Post canonical system with a single initial word, and the productions are each of the form

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