【其它】计算理论小知识点

Ch 1. Computable Functions

Definition 1.1

expression

An expression is a finite sequence (possibly empty) of symbols chosen from the list: q1,q2,q3,...;S0,S1,S2,...;R,L.

Definition 1.2

quadruple

A quadruple is an expression having one of the following forms:
(1) qi Sj Sk ql.
(2) qi Sj R ql.
(3) qi Sj L ql.
(4) qi Sj qk ql.

simple

If none of the quadruples of a Turing machine Z is of the type 4, Z is called simple.

Definition 1.3

Definition 1.4

instantaneous description

Definition 1.5

tape expression

Definition 1.6

internal configuration of Z at α
symbol scanned by Z at α
expression on the tape of Z at α

Definition 1.7

α→β

Theorem 1.1

Theorem 1.2

Definition 1.8

terminal with respect to Z

Definition 1.9

αp=ResZ(α1) and we call αp the resultant of α1 with respect to Z.

Definition 2.1

n¯=1n+1

With each number n we associate the tape expression n¯ where n¯=1n+1.

Definition 2.2

(n1,n2,...,nk¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯)=n1¯¯¯¯Bn2¯¯¯¯B...Bnk¯¯¯¯.

With each k-tuple (n1,n2,...,nk) of integers we associate the tape expression (n1,n2,...,nk¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯), where

(n1,n2,...,nk¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯)=n1¯¯¯¯Bn2¯¯¯¯B...Bnk¯¯¯¯.

Thus, (2,3,0¯¯¯¯¯¯¯¯)=111B1111B1.

Definition 2.3

Let M be any expression. Then ⟨M⟩ is the number of occurrences of 1 in M.
Thus, ⟨11BS4q3⟩=2;⟨q3q2S5⟩=0;⟨m−1¯¯¯¯¯¯¯¯¯⟩=m.

Ch 2. Operations on Computable Functions

Definition 1.1

θ(Z)

If Z is a Turing machine, we let θ(Z) be the largest number i such that qi is an internal configuration of Z.

Definition 1.2

n−regular

A Turing machine Z is called n-regular (n>0) if
(1) There is an s > 0 such that, whenever ResZA[q1(m1,...,mn¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯)] is defined, it has the form qθ(Z)(r1,...,rs¯¯¯¯¯¯¯¯¯¯¯¯¯) for suitable r1,...,rs, and
(2) No quadruple of Z begins with qθ(Z).

n-regular Turing machines present the results of a computation (“outputs”) in a form suitable for use (as “inputs”) at the beginning of a new computation by another Turing machine.

Definition 1.3

Z(n):qi→qn+i

Let Z be a Turing machine. Then Z(n) is the Turing machine obtained from Z by replacing each internal configuration qi, at all of its occurrences in quadruples of Z, by qn+i.

Lemma 1.

Z′

For every Turing machine Z, we can find a Turing machine Z′ such that, for each n, Z′ is n-regular, and, in fact,

ResZ′A[q1(m1,...,mn¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯)]=qθ(Z′)ψ(n)Z;A(m1,...,mn)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯.

This lemma enables us to rewrite the numerical result of a computation in such a form that it is available for use as the beginning of a new computation.