Turing Machines Chapter 3.1 1

Plan Turing Machines(TMs) – Alan Turing Church-Turing Thesis – Definitions Computation Configuration Recognizable vs. Decidable – Examples – Simulator 2

Alan Turing Alan Turing was one of the founding fathers of CS. His computer model the Turing Machine(1936) was the inspiration for the electronic computer that came two decades later Was instrumental in cracking the Nazi Enigma cryptosystem in WWII Invented the “Turing Test” used in AI The Turing Award. Pre-eminent award in Theoretical CS (called the “Nobel Prize” of CS) 3

Church-Turing Thesis Thesis- Every effectively calculable function is a computable function Everything that is computable is computable by a Turing machine The thesis remains a hypothesis – Despite the fact that it cannot be formally proven the Church–Turing thesis now has near-universal acceptance. 4

Turing Machine Most powerful machine so far… – Similar to a finite automata 1.Uses infinite tape as memory 2.Can both read from and write to the tape 3.Read/write head can move left/right 4.Accept/reject take affect immediately Cannot solve all problems

Comparison with Previous Models 6 Device Separate Input? Read/Write Data Structure Deterministic by default? FAYesNoneYes PDAYesLIFO StackNo TMNo 1-way infinite tape. 1 cell access per step. Yes

Formal Definition of a TM 7

Successor Program Sample Rules: 1.If read 1, write 0, go right, repeat. 2.If read 0, write 1, HALT! 3.If read “”, write 1, HALT! Using these rules on a tape containing the reverse binary representation of 47 we obtain: 8

Successor Program If read 1, write 0, go right, repeat. If read 0, write 1, HALT! If read “”, write 1, HALT!

Successor Program If read 1, write 0, go right, repeat. If read 0, write 1, HALT! If read “”, write 1, HALT!

Successor Program If read 1, write 0, go right, repeat. If read 0, write 1, HALT! If read “”, write 1, HALT!

Successor Program If read 1, write 0, go right, repeat. If read 0, write 1, HALT! If read “”, write 1, HALT!

Successor Program If read 1, write 0, go right, repeat. If read 0, write 1, HALT! If read “”, write 1, HALT!

Successor Program If read 1, write 0, go right, repeat. If read 0, write 1, HALT! If read “”, write 1, HALT!

Successor Program So the successor’s output on was which is the reverse binary representation of 48. Similarly, the successor of 127 should be 128: 15

Successor Program If read 1, write 0, go right, repeat. If read 0, write 1, HALT! If read “”, write 1, HALT!

Successor Program If read 1, write 0, go right, repeat. If read 0, write 1, HALT! If read “”, write 1, HALT!

Successor Program If read 1, write 0, go right, repeat. If read 0, write 1, HALT! If read “”, write 1, HALT!

Successor Program If read 1, write 0, go right, repeat. If read 0, write 1, HALT! If read “”, write 1, HALT!

Successor Program If read 1, write 0, go right, repeat. If read 0, write 1, HALT! If read “”, write 1, HALT!

Successor Program If read 1, write 0, go right, repeat. If read 0, write 1, HALT! If read “”, write 1, HALT!

Successor Program If read 1, write 0, go right, repeat. If read 0, write 1, HALT! If read “”, write 1, HALT!

Successor Program If read 1, write 0, go right, repeat. If read 0, write 1, HALT! If read “”, write 1, HALT!

Successor Program If read 1, write 0, go right, repeat. If read 0, write 1, HALT! If read “”, write 1, HALT!

Drawing the machine Draw the machine – Description 1.If read 1, write 0, go right, repeat. 2.If read 0, write 1, HALT! 3.If read “”, write 1, HALT! 25

TM Dynamic Picture A string w is accepted by M if after being put on the tape then letting M run, M eventually enters the accept state. Therefore, w is an element of L(M) - the language accepted by M. We can formalize this notion as follows: 26

TM Formal Definition Dynamic Picture Suppose TM’s configuration at time t is given by uapxv where p is the current state, ua is what’s to the left of the head, x is what’s being read, and v is what’s to the right of the head. If  p,x  = (q,y,R) then write: uapxv  uayqv With resulting configuration uaypv at time t+1. If,  p,x  = (q,y,L) instead, then write: uapxv  uqayv There are also two special cases: – head is forging new ground –pad with the blank symbol  – head is stuck at left end –by def. head stays put NOTE: “  ” is read as “yields” 27

TM outcomes Three possibilities occur on a given input w : 1.The TM M eventually enters q acc and therefore halts and accepts. (w  L(M) ) 2.The TM M eventually enters q rej or crashes somewhere. M rejects w. (w  L(M) ) 3.Neither occurs! I.e., M never halts its computation and is caught up in an infinite loop, never reaching q acc or q rej. In this case w is neither accepted nor rejected. However, any string not explicitly accepted is considered to be outside the accepted language. (w  L(M) ) 28

Recognizable vs. Decidable Recognizable- The TM recognizes the language but doesn’t necessarily reach an accept or reject state (could loop FOREVER  ) Decidable- is recognizable and guaranteed to reach an accept or reject state (without the possibility for an infinite loop ) 29

Bit-shifting example 30

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3.7 Diagram 32

Fall 2010 Exam 2 Question ) (25 pts) Give the full, formal description of a Turing Machine that accepts the following language: L = { w#w R | where w consists of only 0s and 1s and has length at least 1. } The input alphabet will be {0, 1, #}. The tape alphabet will include 0, 1, #, B, and any other symbols you choose to include in it. Give a simple intuitive description of what each state in the machine represents. And draw the diagram for this machine.

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Simulator Accepts 4 or 5 tuples – (Start,input,NewState, output,Direction) – (Start,input,NewState,Di rection OR output) One tape tril/tm/ tril/tm/ 37

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