Lecture 5: Turning Machine. 虞台文. 大同大學資工所 智慧型多媒體研究室. Content. An Overview on Finite State Machine The Definition of Turing Machine Computing with Turing Machine Turing-Machine Programming Some Examples of Powerful TMs Extensions of the TM Nondeterministic Turing Machine.
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Content • An Overview on Finite State Machine • The Definition of Turing Machine • Computing with Turing Machine • Turing-Machine Programming • Some Examples of Powerful TMs • Extensions of the TM • Nondeterministic Turing Machine
Example • Input a 0/1 sting. • If the numbers of 0 and 1 in the string are botheven, then it is legal; otherwise, it is illegal. • 0011001001(legal) • 11001001001(illegal) • Writing a C program to do so.
Implementation (I) void main() { // Ask user to input a 0/1 string // Store the string into str state = q0; //initialization while(state!=fini){ ToEvent(*str++); EventHandler(event); } } Event (or Message) Loop
Implementation (I) void EventHandler(int event) { int next_state = parser[state][event]; switch(next_state){ case fini: printf(”%s\n”, state==q0 ? ”ok” : ”err”); default: state = next_state; //change state } } Event Handler
symbol (event) 0 0 1 <CR> q0 q1 0 q0 1 1 1 1 q1 0 state q2 q2 q3 0 q3 Implementation (II) pq1 pq2 ok pq0 pq3 err pq3 pq0 err err pq2 pq1 The Parser
symbol (event) 0 1 <CR> q0 pq1 pq2 ok q1 pq0 pq3 err state q2 pq3 pq0 err err q3 pq2 pq1 The Parser Implementation (II) #define q0 0 #define q1 1 #define q2 2 #define q3 3 #define fini 4 void pq0(), pq1(), pq2(), pq3(); void ok(), err();
symbol (event) 0 1 <CR> q0 pq1 pq2 ok q1 pq0 pq3 err state q2 pq3 pq0 err err q3 pq2 pq1 The Parser Implementation (II) void pq0() { state = q0; } void pq1() { state = q1; } void pq2() { state = q2; } void pq3() { state = q3; }
symbol (event) 0 1 <CR> q0 pq1 pq2 ok q1 pq0 pq3 err state q2 pq3 pq0 err err q3 pq2 pq1 The Parser Implementation (II) void ok() { printf(”ok\n”); state = fini; } void err() { printf(”error\n”); state = fini; }
symbol (event) 0 1 <CR> q0 pq1 pq2 ok q1 pq0 pq3 err state q2 pq3 pq0 err err q3 pq2 pq1 The Parser typedef void (*FUNCTION)(); FUNCTION parser[4][3]={ pq1, pq2, ok, pq0, pq3, err, pq3, pq0, err, pq2, pq1, err }; Implementation (II)
Implementation (II) void main() { // Ask user to input a 0/1 string // Store the string into str state = q0; //initialization while(state!=fini){ ToEvent(*str++); (*parser[state][event])(); } } Event (or Message) Loop
Hang Head # # Definition A Turing machine is a quadruple K: finite set of states, hK. : alphabet, #, L, R. : transition function s: sK, initial state.
The Transition Function • Change state from q to p. • b printb; • b{L, R} Move head in the direction of b.
a . . . . . . . . . . . . . . . . . . . q K . . . . . The Transition Function
Head # w a u # Memory Configuration or The configuration of a Turing machine is a member of
Head # w a u # Halt Configuration or The configuration of a Turing machine is a member of
├M w1 a1 u1 w1=w2a2 a1 u1 w1 a1 u1=a2u2 w2=w1 a2 u2=u1 w2 a2 u2=a1u1 w2=w1a1 a2 u2 Used to trace the computation sequence. Yields in One Step├M Let Then, if and only if , where such that
├ Let is the reflexive, transitive closure of ├M , i.e., if, for some n 0, ├ ├ ├M ├M ├M (1) Yields where Ci denotes the configuration of M. We say that computation sequence (1) is of length n or has n steps.
Let 0, 1 {#}, and let . f is said to be a Turing computable function if such that, for any , ├ Turing Computable Functions M is, then, said to compute f.
Head w # # Head Head u # # Turing Computable Functions
h R / q5 Y / N / q6 q4 R / # R / # R / # > L / # / 1 L / # / 1 L / # / 1 q0 q1 q11 q2 q21 q3 q31 0 / 0 0 / 0 0 / 0 q7 0 / 0 L / Example
h R / q5 Y / N / q6 q4 R / # R / # R / # > L / # / 1 L / # / 1 L / # / 1 q0 q1 q11 q2 q21 q3 q31 0 / 0 0 / 0 0 / 0 q7 0 / 0 L / Write the state transition table. Exercise
Discussion • Three main usages of machines: • Compute • Decision • Accept • See textbooks for the definitions of • Turing decidability • Turing acceptability
├ Head # a b c # Head # # a b c # a b c Example: (Copier)
├ >L# R #R# R#sL# L#s # R# Example: (Copier) # a b c # # a b c # # a b c # # # b c # # # b c # # # # b c # a # # b c # a # a b c # a # a b c # a
├ Head # a b c # Head a b c # Example: (Left-Shift)
├ >L# R LsR # L# Example: (Left-Shift) # a b c # # a b c # # a b c # # a b c # a a b c # a a b c # a a b c # a b b c # …
Head ├ # a b b # a Head # Y # Example: (Palindrome)
Head ├ # a b a # a Head # N # Example: (Palindrome)
├ >SRL# LaRR #R# L #L# # # #L L# #RNR L# #RYR Example: (Palindrome) # a b b a # # # a b b a # (SR) # # a b b a # a # a b b a # a # a b b a # a # a b b a # a # # b b a # a # # b b a # a # # b b a # a # # b b # # a # # b b # # …
