Post system

Introduction

A Post canonical system (or Post system for short) $𝔓$ is a triple $(\mathrm{\Sigma },X,P)$, such that

1. $\mathrm{\Sigma }$ is an alphabet,

2. $X$ is an alphabet, disjoint from $\mathrm{\Sigma }$, whose elements we call variables, and

3. $P$ is a non-empty finite binary relation on ${\mathrm{\Sigma }}^{*}$, the Kleene star of $P$, such that for every $(u,v)\in P$,

   • $u\in {\mathrm{\Sigma }}^{*}X{\mathrm{\Sigma }}^{*}$, and

   • if $S$ is a variable occurring in $v$, then it occurs in $u$.

   An element $(x,y)\in P$ is called a production of $𝔓$, $x$ is its antecedent, and $y$ the consequent. $(x,y)\in P$ is often written $x\to y$.

The last condition basically says that in a production $x\to y$, $x$ must contain at least one variable, and $y$ can not contain any variables that are not already occurring in $x$. Put it more concretely, a production in a Post canonical system has the form

$$a_1{S}_1{a}_2{S}_2\mathrm{\cdots }{a}_n{S}_n{a}_{n+1}\to b_1{S}_{\varphi (1)}{b}_2{S}_{\varphi (2)}\mathrm{\cdots }{b}_m{S}_{\varphi (m)}{b}_{m+1}$$ (1)

where $a_i$ and $b_j$ are fixed words on $\mathrm{\Sigma }$, while $S_k$ are variables, with , $0\le m$, $m\le n$, and $\varphi$ is a function (not necessarily bijective) on the set $\{1,\mathrm{\dots },n\}$.

Examples. Let $\mathrm{\Sigma }=\{a,b,c\}$ and $X=\{S,U,V,W\}$. Then $(\mathrm{\Sigma },X,P)$ with $P$ consisting of

$$aS{b}^2\to ba,cVaWaUb\to aWU,a^{3}cUbSW\to SabU,bVa\to a{V}^{2}c$$

is a Post canonical system, while $(\mathrm{\Sigma },X,Y)$ with $Y$ consisting of

$$a{b}^2\to ba,cVaWaUb\to aWU,aUbS{c}^{2}W\to ScaV,a\to S$$

is not, for the following reasons:

• the antecedents in the first and fourth productions do not contain a variable

• the consequents in the third and fourth productions contain variables ($V$ in the third, and $S$ in the fourth) which do not occur in the corresponding antecedents.

Normal systems. A Post canonical system $𝔓=(\mathrm{\Sigma },X,P)$ is called a Post normal system, or normal system for short, if each production has the form $aS\to Sb$ (called a normal production), where $a,b$ are words on $\mathrm{\Sigma }$ and $S$ is a variable.

Languages generated by a Post system

Let us fix a Post system $𝔓=(\mathrm{\Sigma },X,P)$. A word $v$ is said to be immediately derivable from a word $u$ if there is a production of the form (1) above, such that

$$u=a_1{u}_1{a}_2{u}_2\mathrm{\cdots }{a}_n{u}_n{a}_{n+1}\mathit{\hspace{1em}}\text{ and }\mathit{\hspace{1em}}v=b_1{a}_{\varphi (1)}{b}_2{a}_{\varphi (2)}\mathrm{\cdots }{b}_m{a}_{\varphi (m)}{b}_{m+1},$$

where $a_i$ are words (not variables). This means that if we can write a word $u$ in the form of an antecedent of a production by replacing all the variables with words, then we can “produce”, or “derive” a word $v$ in the form of the corresponding consequent, replacing the corresponding variables with the corresponding words. When $v$ is immediately derivable from $u$, we write $u\Rightarrow v$. Using the example above, with the production $cVaWaUb\to aWU$, we see that

• $c{a}^{4}b=caaaab\Rightarrow a^3$ if we set $V=\lambda$ and $W=U=a$, or

• $c{a}^{4}b=caaaab\Rightarrow a^2$ if we set $V=a$ and exactly one of $W,U=a$ and the other $\lambda$.

A word $v$ is derivable from a word $u$ if there is a finite sequence of words $u_1,\mathrm{\dots },u_n$ such that

$$u=u_1\Rightarrow u_2\Rightarrow \mathrm{\cdots }\Rightarrow u_n=v.$$

When $v$ is derivable from $u$, we write $u\stackrel{*}{\Rightarrow }v$. Again, following from the example above, we see that $c^{2}aba{b}^2\stackrel{*}{\Rightarrow }ac$, since

$$c^{2}aba{b}^2=ccababb\Rightarrow a{b}^2\Rightarrow ba\Rightarrow ac.$$

Given a finite subset $A$ of words on $\mathrm{\Sigma }$, let $T_A$ be the set of all words derivable from words in $A$. Elements of $A$ are called axioms of $𝔓$ and elements of $T_A$ theorems (of $𝔓$ derived from axioms of $A$). Intuitively, we see that the Post system $𝔓$ is a language generating machine that creates the language $T_A$ via a set $A$ of axioms. In general, we say that a language $M$ over an alphabet $\mathrm{\Sigma }$ is generable by a Post system if there is a Post system $𝔓=({\mathrm{\Sigma }}_1,X,P)$ such that $\mathrm{\Sigma }\subseteq {\mathrm{\Sigma }}_1$, a finite set $A$ of axioms on ${\mathrm{\Sigma }}_1$ such that $M=T_A\cap {\mathrm{\Sigma }}^{*}$.

Remarks.

• If a language is generable by a Post system, it is generable by a normal system.

• A language is generable by a Post system iff it is generable by a semi-Thue system. In this sense, Post systems and semi-Thue systems are “equivalent”.

• Instead of allowing for one antecedent and one consequent in any production, one can have a more generalized system where one production involves a finite number of antecedents as well as a finite number of consequents:

  $$\left\{\begin{array}{c}\hfill a_{11}{S}_{11}{a}_{12}{S}_{12}\mathrm{\cdots }{a}_{1n_1}{S}_{1n_1}{a}_{1,n_1+1},\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill a_{p1}{S}_{p1}{a}_{p2}{S}_{p2}\mathrm{\cdots }{a}_{p{n}_p}{S}_{p{n}_p}{a}_{p,n_p+1}\hfill \end{array}\right\}\to \left\{\begin{array}{c}\hfill b_{11}{S}_{{\varphi }_1(1)}{b}_{12}{S}_{{\varphi }_1(2)}\mathrm{\cdots }{b}_{1m}{S}_{{\varphi }_1(m_1)}{b}_{1,m_1+1},\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill b_{q1}{S}_{{\varphi }_q(1)}{b}_{q2}{S}_{{\varphi }_q(2)}\mathrm{\cdots }{b}_{qm}{S}_{{\varphi }_q(m_q)}{b}_{q,m+1}\hfill \end{array}\right\}$$

  where each ${\varphi }_i$ is a function from $\{1,\mathrm{\dots },m_i\}$ to $\{(1,1),\mathrm{\dots },(1,n_1),\mathrm{\dots },(p,1),\mathrm{\dots },(p,n_p)\}$. We may define $b$ to be immediately derivable from $a$ if $a$ can be expressed using each of the antecedents by substituting the variables $S_{ij}$ by words $c_{ij}\in {\mathrm{\Sigma }}^{*}$, and $b$ can be expressed in at least one of the consequents by the corresponding substitutions (of $S_{{\varphi }_i(j)}$ into $c_{{\varphi }_i(j)}$). It can be shown that any language generated by this more general system is in fact Post generable!

• It can be shown that a language is Post-generable iff it is recursively enumerable.

Post Computability

For any positive integer $m$, and an $m$-tuple $\overline{n}:=(n_1,\mathrm{\dots },n_m)$ of natural numbers, we may associate a word

$$E(\overline{n}):=a{b}^{n_1}a{b}^{n_2}a\mathrm{\cdots }a{b}^{n_m}a.$$

Let $f:{\mathbb{N}}^m\to \mathbb{N}$ be a partial function. Define

$$L(f):=\{E(\overline{n})cE(f(\overline{n}))\mid \overline{n}\in \mathrm{dom}(f)\}.$$

We say that $f$ is Post-computable if $L(f)$ is Post-generable. As expected from the last remark in the previous section, a partial function is Turing-computable iff it is Post-computable.

Title	Post system
Canonical name	PostSystem
Date of creation	2013-03-22 17:33:28
Last modified on	2013-03-22 17:33:28
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	19
Author	CWoo (3771)
Entry type	Definition
Classification	msc 68Q42
Classification	msc 03D03
Synonym	normal system
Synonym	Post generable
Synonym	Post computable
Related topic	FormalGrammar
Defines	Post canonical system
Defines	Post normal system
Defines	normal production
Defines	generable by a Post system
Defines	Post-computable