#jsDisabledContent { display:none; } My Account |  Register |  Help

# Post canonical system

Article Id: WHEBN0013995373
Reproduction Date:

 Title: Post canonical system Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 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 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.

## Definition

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

• A is a finite alphabet, and finite (possibly empty) strings on A are called words.
• I is a finite set of initial words.
• R is a finite set of string-transforming rules (called production rules), each rule being of the following form:



\begin{matrix}

      g_{10}\ $_{11}\ g_{11}\$_{12}\ g_{12} \ \dots \ $_{1m_1}\ g_{1m_1} \\ g_{20}\$_{21}\ g_{21}\ $_{22}\ g_{22} \ \dots \$_{2m_2}\ g_{2m_2} \\
\dots \ \dots   \ \dots \ \dots   \ \dots  \ \dots \ \dots   \ \dots  \\
g_{k0}\ $_{k1}\ g_{k1}\$_{k2}\ g_{k2} \ \dots \ $_{km_k}\ g_{km_k} \\ \\ \downarrow \\ \\ h_0 \$'_1 \ h_1 \ $'_2 \ h_2 \ \dots \$'_n \ h_n \\


\end{matrix}

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

$g_0 \ _1 \ g_1 \ _2 \ g_2 \ \dots \ _m \ g_m \ \rightarrow \ h_0 \ \text{'}_1 \ h_1 \ \text{'}_2 \ h_2 \ \dots \ \text{'}_n \ h_n$

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 precisely the recursively enumerable languages.

### Example (well-formed bracket expressions)

Alphabet:
Initial word: []
Production rules:
(1)       $→ [$]
(2)       $→$$(3)$1[]$2 →$1\$2

Derivation of a few words in the language of well-formed bracket expressions:

[]             initial word
[][]           by (2)
by (1)
by (2)
by (3)
[]       by (3)
...


## Normal-form theorem

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

$g \ \rightarrow \ h$

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

Given any Post canonical system on an alphabet A, a Post canonical system in normal form can be constructed from it, possibly enlarging the alphabet, such that the set of words involving only letters of A that are generated by the normal-form system is exactly the set of words generated by the original 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.)

## String rewriting systems, Type-0 formal grammars

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

$P_1 g P_2 \ \rightarrow \ P_1 h P_2$

That is, each production rule is a simple substitution rule, often written in the form gh. It has been proved that any Post canonical system is reducible to such a substitution system, which, as a formal grammar, is also called a phrase-structure grammar, or a type-0 grammar in the Chomsky hierarchy.

## References

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.