Equivalent Variations of Turing Machines
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1 Equivalent Variations of Turing Machines Nondeterministic TM = deterministic TM npda = pushdown automata with n stacks 2PDA = npda = TM for all n 2 Turing machines with n tapes (n 2) and n tape heads has the same expressive power as a Turing machine with one tape head. Turing machines with a tape that is infinite to the left and right has the same expressive power as a Turing machine with a tape that is finite to the left and infinite to the right. (Chapters 21 and 22) 1
2 Chapter 23: Turing Machine Languages Definition. A language L over the alphabet Σ is recursively enumerable if there exists a Turing machine such that for every word w L, w is accepted, and for every word w L, either w is rejected (crashes) or w causes the machine to go into an infinite loop. Definition. A language L over an alphabet Σ is recursive if there exists a Turing machine such that for every word w L, w is accepted, and for every word w L, w is rejected (crashes). 2
3 Theorem. If there exist two Turing machines T 1 and T 2 that accept languages L 1 and L 2, then there exists a Turing machine that accepts L 1 + L 2. Proof: (abbreviated) It is possible to build a Turing machine T 3 that simulates running the input on the two machines alternately. First, transform T 1 which accepts L 1 to a Turing machine such that for every word that is not in L 1, the machine loops forever. Call the new machine T 1. Do the same for L 2. For each state, if a letter x has no outgoing transition, add a transition (x, x, R) to a new state NOWHERESVILLE and in this state add transitions back to itself that together implement an infinite loop, e.g., (x,x,r) for every x ϵ Σ Γ { }. Similarly, transform T 2 to T 2. 3
4 SET-UP: the initial steps of the new Turing machine must make 2 copies of the input word on the tape, one for T 1 and one for T 2. T 3 will simulate the two machines T 1 et T 2 executing at the same time. We add a SIMULATE-T 1 state and a SIMULATE-T 2 state to T 3. For each state x i of T 1, we add a set of states to T 3 to perform the simulation of a step of execution starting from x i. Similarly, for each state y i of T 2, we add a set of states to T 3 to perform the simulation of a step of execution starting from y i. A state FIND-Y pushes the tape head right until it finds a state in T 2, and a FIND-X state pushes the tape head left until it finds a state in T 1. 4
5 The SET-UP Subprogram a b b # x 1 a b b * y 1 a b b state from T 1 state from T 2 symbol marking symbol separating the 2 copies the first cell Neither of the two machines will crash if the word is in L 1 + L 2. (If the word in in L 1, then T 2 will either accept the word or go into an infinite loop and T 1 will have time to accept the word. So T 3 will accept the word. Similarly, for when the word is in L 2.) 5
6 x 5 (b,c,l) x 3 b a x 5 b a a b a b a a b a c a a b x 3 a c a a 6
7 START SET-UP (x 1,x 1,R) SIMULATE T 1 SIM-x 1 (x 2,x 2,R) SIM-x 2 (x 3,x 3,R) SIM-x 3 FIND Y (y 1,y 1,R) SIMULATE T 2 SIM-y 1 (y 2,y 2,R) SIM-y 2 (y 3,y 3,R) SIM-y 3 FIND X 7
8 Theorem. If a language L and its complement L are both recursively enumerable then L is recursive.. Proof: (abbreviated) T 1 is aturing machine for L. T 2 is a Turing machine for L. We transform both machines into new Turing machines T 1 et T 2 such that: T 2 crashes on every word in L and goes into an infinite loop for every word in L. T 1 accepts every word in L and goes into an infinite loop for every word in L. 8
9 The machine T 3 will simulate both machines T 1 and T 2 (as was done for the union). If the word is in L, then T 2 will go into an infinite loop and T 1 will have time to accept the word. So T 3 will accept the word. If the word is not in L, then T 1 will go into an infinite loop and it will give time for T 2 to crash. So T 3 will reject the word. The Turing machine T 3 will not go into an infinite loop and this makes the language L recursive. 9
10 Theorem. If there exist two Turing machines T 1 and T 2 that accept languages L 1 and L 2, then there exists a Turing machine that accepts L 1 L 2. Theorem: The complement of a recursively enumerable language is not necessarily recursively enumerable. 10
11 The Encoding of Turing machines Example (b,b,r) (a,b,l) (a,b,r) 3 START 1 HALT 2 (,b,l) X 1 X 2 X 3 X 4 X 5 From To Read Write Move 1 1 b b R 1 3 a b R 3 3 a b L 3 2 b L 11
12 Encoding a state = encoding a positive integer Σ={a,b} 1: ab 2: aab 3: aaab Σ Γ = {a,b,#} Read/Write Code a aa b ab ba # bb Move L R Code a b 12
13 From To Read Write Move Code 1 1 b b R ababababb 1 3 a b R abaaabaaabb 3 3 a b L aaabaaabaaaba 3 2 b L aaabaabbaaba Code of the machine: ababababbabaaabaaabbaaabaaabaaabaaaabaabbaaba ababababb abaaabaaabb aaabaaabaaaba aaabaabbaaba Code Word Language: CWL = language((a + ba + b(a+b) 5 )*) Remark: It is possible to determine if a word in CWL is the code of a Turing machine. 13
14 ALAN = all words w CWL that are not accepted by the Turing machines that they represent. ALAN CWL Example: (b,b,r) START 1 HALT 2 From To Read Write Move 1 2 b b R Language L defined by this machine: words starting with b Code word for the machine that accepts L: abaabababb abaabababb L Thus abaabababb ALAN 14
15 Other Examples Example: The code word for a machine that accepts language((a+b)*) is not in ALAN. Example: The code word for a machine that accepts the empty language is in ALAN. Example: The code word for a machine that accepts L=language((a+b)*aa(a+b)*) contains aa, and thus is in L. Thus the code word is not in ALAN. 15
16 Theorem. There does not exist any Turing machine that accepts ALAN. Proof: Assume there is a Turing machine T that accepts ALAN. We denote the code word for T as code(t). Either code(t) ALAN, or code(t) ALAN. Case 1. If T accepts code(t), then code(t) ALAN. (ALAN is the language accepted by T.) But by definition, ALAN cannot contain a code word accepted by its own machine. Thus code(t) ALAN. A contradiction. Case 2. If T does not accept code(t), then code(t) ALAN. But ALAN is the language defined by T. A contradiction. Thus there is no Turing machine that accepts ALAN. 16
17 Theorem. Not all languages are recursively enumerable. Example: ALAN 17
18 Universal Turing Machine Definition. A universal Turing machine is a Turing machine MTU such that: Input words to MTU have the form: #w#x where w is the code word that represents a Turing machine T and x is a word containing letters of T s input alphabet. MTU will operate on the data #w#x exactly the same as T would operate on x. (MTU crashes, accepts, or loops if and only if T does the same.) 18
19 Theorem: Universal Turing machines exist. A universal Turing machine is a model of a computer. For program w and input x, the Turing machine performs the execution of program w on input x. A programming language is Turing complete if it can be used to simulate a universal Turing machine. 19
20 Decidability: The Halting Problem Does there exist a Turing machine such that given an input word x and a code word w for a Turing machine T that can decide whether or not T halts (enters a HALT state) on input x? A Turing machine solves a problem that is decidable if it can produce a yes or no answer in a finite number of steps. It does not decide the problem if it enters an infinite loop on some inputs. We assume that input to a Turing machine HP that solves the halting problem has the form #code(t)#x, and HP prints yes on the tape and then halts whenever T halts, and prints no on the tape and then halts whenever T crashes or enters an infinite loop. 20
21 Theorem. No Turing machine exists that can decide the halting problem. In other words, there is no Turing machine that for any string w and any code for T, can determines whether or not T halts on w. Proof idea: If we assume such a machine exists, we can build a Turing machine that accepts ALAN. Note: the halting problem is the membership problem for recursively enumerable languages: Is w in the language accepted by T? A Turing machine cannot decide this problem. 21
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