CSCI 3130: Automata theory and formal languages Andrej Bogdanov The Chinese University of Hong Kong Variants of Turing Machines Fall 2010

The Church-Turing Thesis “On Computable Numbers, with an Application to the Entscheidungsproblem” 1936: Section 9. The extent of the computable numbers All arguments [for the CT Thesis] which can be given are bound to be, fundamentally, appeals to intuition, and for this reason rather unsatisfactory mathematically. The arguments which I shall use are of three kinds: 1. A direct appeal to intuition 2. A proof of the equivalence of two definitions (In case the new definition has greater intuitive appeal) 3. Giving examples of large classes of numbers [languages] which are computable.

The multitape Turing Machine The transition may depend on the contents of all the cells Different tape heads can be moved independently control tape 1 … 010 tape 2 … 01 tape 3 … 100

The multitape Turing Machine Multiple tapes are convenient, e.g. one can serve as temporary storage … 010 … 01 … 100 … 110 … 100 q3q3 q7q7 0/1R☐/1R0/0L0/1R☐/1R0/0L … 011

The multitape Turing Machine Multitape Turing Machines are equivalent to single-tape Turing Machines M … 010 … 01 … 100  = {0, 1, ☐ } S … 01010##0#10   ’ = {0, 1, ☐, 0, 1, ☐, #}   #

Simulating a multitape TM We show how to simulate a multitape TM on an ordinary (single tape TM) To be specific, let’s do a 3-tape TM … x1x1 x2x2 xaxa xixi …… … y1y1 y2y2 ybyb yjyj …… z1z1 zczc zkzk … … Multitape TM Single-tape TM #x 1 x 2...x a...x i #y 1 y 2...y b...y j #z 1 z 2...z c...z k #  

Simulating a multitape TM Single-tape TM: Initialization … 010###  #  … 010 #w 1 w 2...w n # ☐ # ☐ #    w 1 w 2...w n S:S: On input w 1...w n : Replace tape contents by #w 1 w 2...w n # ☐ # ☐ #    1. Remember that M is in state q 0

Simulating a multitape TM Single-tape TM: Simulating Multitape TM moves Suppose Multitape TM wants to move like this: q3q3 0/1R ☐ /1R 0/0L q7q7 … … … … … … We simulate move on single-tape TM like this: 01010##0#10  # 01010##0#10  #

Simulating a multitape TM S:S: On input w 1...w n : Replace tape contents by #w 1 w 2...w n # ☐ # ☐ #    1. To simulate a step of M : 2. qiqi x/x’A y/y’B z/z’C qjqj Remember (in state) that M is in state q 0 #x 1 x 2...x...x i #y 1 y 2...y...y j #z 1 z 2...z...z k #   Make a pass over tape to find x, y, z   If M is in state q i and has transition update state / tape accordingly. If M reaches accept (reject) state, accept (reject). 3.

Doing simulations To simulate a model M by another model N : 1. Say how the state and storage of N is used to represent the state and storage of M 2. Say what should be initially done to convert the input of N (if anything) 3. Say how each transition of M can be implemented by a sequence of transitions of N

CPU registers R0R0 R1R1 R2R2 R3R3 arithmetic logical unit 0000 PC What does a computer look like? data memory instruction memory

What does a computer look like? CPU registers R0R0 R1R1 R2R2 R3R3 data memory instruction memory arithmetic logical unit PC 0000 load write R store R add R jpos

Instruction set load x load R k store R k read R k write R k jump n jpos n Put the value x into R 0 add R k Copy the value of R k into R 0 Copy the value of R 0 into R k Copy the value at memory location R k into R 0 Copy the value of R 0 into memory location R k Add R 0 and R k, and put result in R 0 Set PC to n Set PC to n, if R 0 is positive jzero n Set PC to n, if R 0 is zero

Random access machines load -7 R 0 := -7 write R2 M[R 2 ] := R 0 store R1 R 1 := R 0 add R1 R 0 := R 0 + R 5 jzero 3 if R 0 = 0 then PC := 3 accept instructionmeaning PC 0 R0R0 0 R1R1 0 R2R2 2 M program counter registers memory … It has registers that can store integer values, a program counter, and a random-access memory

Random access machines load 7 R 0 := 7 write R2 M[R 2 ] := R 0 store R1 R 1 := R 0 add R1 R 0 := R 0 + R 1 jzero 3 if R 0 = 0 then PC := 3 accept instructionmeaning PC 0 R0R0 0 R1R1 0 R2R2 0 M … The instructions are indexed by the program counter Initially, the input is in the first k memory cells, all registers and PC are set to 0

Random access machines Simulating a Turing Machine on a RAM: Random access machines are equivalent to Turing Machines PC 2 R0R0 1 M 2100 … M … 211  = {0, 1, 2,..., k} head tape blank

Simulating a TM on a RAM store R1 handle for state q 0 0 store R1 handle for state q 1 … … accept handle for state q acc M q0q0 q1q1 1/2R … 211 store R1 save head position read R1 read tape contents x add -1 jzero 6 if x = 1 goto line 6 load 2 new value of cell write R1 write in memory load R1 recall head position add 1 move head to right jump 30 go to state q 1 program 0 PC 0 R0R0 1 M 21 0 R1R1

Simulating a TM on a RAM store R1 handle for state q 0 0 store R1 handle for state q 1 … 30 M q0q0 q1q1 1/2R … R0R0 1 M 21 store R1 save head position read R1 read tape contents x add -1 jzero 6 if x = 1 goto line 6 load 2 new value of cell write R1 write in memory load R1 recall head position add 1 move head to right jump 30 go to state q 1 program 0 R1R (head) (tape) … 2

Simulating a RAM on a Turing Machine The configuration of a RAM consists of –Program counter –Contents of registers –Indices and contents of all nonempty memory cells 14 PC 3 R0R0 17 R1R1 5 R2R2 2 M … configuration = (14, 3, 17, 5, (0, 2), (2, 1), (3, 2)) 0

Simulating a RAM on a 2-tape TM The TM has a simulation tape and a scratch tape The simulation tape stores RAM configuration The TM has a set of states corresponding to each program instruction of the RAM The TM tape is updates according to RAM instruction (14,3,17,5,(0,2),(2,1),(3,2)) M

Simulating a RAM on a 2-tape TM Initialization S:S: On input w 1...w n : Replace tape contents by (0, 0, 0,..., 0, (0, w 1 ), (1, w 2 ),..., (n-1, w n )) 1. TM input: RAM initial state: 0 PC 0 R0R0 1 M

Simulating a RAM on a 2-tape TM Example: load R1 (14,3,17,5,(0,2),(2,1),(3,2)) 2. Write R 1 to conf tape c 1. Copy R 1 to scratch tape 17 s (14,1,17,5,(0,2),(2,1),(3,2)) 17 c s (14,1,17,5,(0,2),(2,1),(3,2) ) c c.. Make more space as needed (14,17,17,5,(0,2),(2,1),(3,2)) c 3. Erase scratch tape 4. Update PC (15,17,17,5,(0,2),(2,1),(3,2)) c

Simulating a RAM on a 2-tape TM S:S: On input w 1...w n : Replace tape contents by (0, 0, 0,..., 0, (0, w 1 ), (1, w 2 ),..., (n-1, w n )) Simulate instruction in RAM program by a sequence of TM transitions See notes for details. 3. If RAM instruction is accept, go to accept state. If RAM instruction is reject, go to reject state.