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Post system
Introduction
A Post canonical system (or Post system for short) $\mathfrak{P}$ is a triple $(\Sigma,X,P)$, such that
1. $\Sigma$ is an alphabet,
2. 3. $P$ is a nonempty finite binary relation on $\Sigma^{*}$, the Kleene star of $P$, such that for every $(u,v)\in P$,

$u\in\Sigma^{*}X\Sigma^{*}$, 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 $\mathfrak{P}$, $x$ 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
$a_{1}S_{1}a_{2}S_{2}\cdots a_{n}S_{n}a_{{n+1}}\to b_{1}S_{{\phi(1)}}b_{2}S_{{% \phi(2)}}\cdots b_{m}S_{{\phi(m)}}b_{{m+1}}$  (1) 
where $a_{i}$ and $b_{j}$ are fixed words on $\Sigma$, while $S_{k}$ are variables, with $0<n$, $0\leq m$, $m\leq n$, and $\phi$ is a function (not necessarily bijective) on the set $\{1,\ldots,n\}$.
Examples. Let $\Sigma=\{a,b,c\}$ and $X=\{S,U,V,W\}$. Then $(\Sigma,X,P)$ with $P$ consisting of
$aSb^{2}\to ba,\quad cVaWaUb\to aWU,\quad a^{3}cUbSW\to SabU,\quad bVa\to aV^{2}c$ 
is a Post canonical system, while $(\Sigma,X,Y)$ with $Y$ consisting of
$ab^{2}\to ba,\quad cVaWaUb\to aWU,\quad aUbSc^{2}W\to ScaV,\quad a\to S$ 
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 $\mathfrak{P}=(\Sigma,X,P)$ 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 $\Sigma$ and $S$ is a variable.
Languages generated by a Post system
Let us fix a Post system $\mathfrak{P}=(\Sigma,X,P)$. 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
$u=a_{1}u_{1}a_{2}u_{2}\cdots a_{n}u_{n}a_{{n+1}}\quad\mbox{ and }\quad v=b_{1}% a_{{\phi(1)}}b_{2}a_{{\phi(2)}}\cdots b_{m}a_{{\phi(m)}}b_{{m+1}},$ 
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

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

$ca^{4}b=caaaab\Rightarrow a^{2}$ 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 $u_{1},\ldots,u_{n}$ such that
$u=u_{1}\Rightarrow u_{2}\Rightarrow\cdots\Rightarrow u_{n}=v.$ 
When $v$ is derivable from $u$, we write $u\stackrel{*}{\Rightarrow}v$. Again, following from the example above, we see that $c^{2}abab^{2}\stackrel{*}{\Rightarrow}ac$, since
$c^{2}abab^{2}=ccababb\Rightarrow ab^{2}\Rightarrow ba\Rightarrow ac.$ 
Given a finite subset $A$ of words on $\Sigma$, let $T_{A}$ be the set of all words derivable from words in $A$. Elements of $A$ are called axioms of $\mathfrak{P}$ and elements of $T_{A}$ theorems (of $\mathfrak{P}$ derived from axioms of $A$). Intuitively, we see that the Post system $\mathfrak{P}$ 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 $\Sigma$ is generable by a Post system if there is a Post system $\mathfrak{P}=(\Sigma_{1},X,P)$ such that $\Sigma\subseteq\Sigma_{1}$, a finite set $A$ of axioms on $\Sigma_{1}$ such that $M=T_{A}\cap\Sigma^{*}$.
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 semiThue system. In this sense, Post systems and semiThue 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:
$\left\{\begin{array}[]{c}a_{{11}}S_{{11}}a_{{12}}S_{{12}}\cdots a_{{1n_{1}}}S_% {{1n_{1}}}a_{{1,n_{1}+1}},\\ \vdots\\ a_{{p1}}S_{{p1}}a_{{p2}}S_{{p2}}\cdots a_{{pn_{p}}}S_{{pn_{p}}}a_{{p,n_{p}+1}}% \end{array}\right\}\to\left\{\begin{array}[]{c}b_{{11}}S_{{\phi_{1}(1)}}b_{{12% }}S_{{\phi_{1}(2)}}\cdots b_{{1m}}S_{{\phi_{1}(m_{1})}}b_{{1,m_{1}+1}},\\ \vdots\\ b_{{q1}}S_{{\phi_{q}(1)}}b_{{q2}}S_{{\phi_{q}(2)}}\cdots b_{{qm}}S_{{\phi_{q}(% m_{q})}}b_{{q,m+1}}\end{array}\right\}$ where each $\phi_{i}$ is a function from $\{1,\ldots,m_{i}\}$ to $\{(1,1),\ldots,(1,n_{1}),\ldots,(p,1),\ldots,(p,n_{p})\}$. 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 $c_{{ij}}\in\Sigma^{*}$, and $b$ can be expressed in at least one of the consequents by the corresponding substitutions (of $S_{{\phi_{i}(j)}}$ into $c_{{\phi_{i}(j)}}$). 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 Postgenerable iff it is recursively enumerable.
Post Computability
For any positive integer $m$, and an $m$tuple $\overline{n}:=(n_{1},\ldots,n_{m})$ of natural numbers, we may associate a word
$E(\overline{n}):=ab^{{n_{1}}}ab^{{n_{2}}}a\cdots ab^{{n_{m}}}a.$ 
Let $f:\mathbb{N}^{m}\to\mathbb{N}$ be a partial function. Define
$L(f):=\{E(\overline{n})cE(f(\overline{n}))\mid\overline{n}\in\operatorname{dom% }(f)\}.$ 
We say that $f$ is Postcomputable if $L(f)$ is Postgenerable. As expected from the last remark in the previous section, a partial function is Turingcomputable iff it is Postcomputable.
References
 1 M. Davis, Computability and Unsolvability. Dover Publications, New York (1982).
 2 N. Cutland, Computability: An Introduction to Recursive Function Theory. Cambridge University Press, (1980).
 3 M. Minsky, Computation: Finite and Infinite Machines. Prentice Hall, (1967).
Mathematics Subject Classification
68Q42 no label found03D03 no label found Forums
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