Turing Machines: Turing Machines Zeph Grunschlag
Agenda: Agenda Turing Machines
Alan Turing
Motivation
Church-Turing Thesis
Definitions
Computation
TM Configuration
Recognizers vs. Deciders
Alan Turing: Alan Turing Alan Turing was one of the founding fathers of CS.
His computer model –the Turing Machine– was inspiration/premonition of 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
Legacy: The Turing Award. Pre-eminent award in Theoretical CS
A Thinking Machine: A Thinking Machine First Goal of Turing’s Machine: A model that can compute anything that a human can compute. Before invention of electronic computers the term “computer” actually referred to a person who’s line of work is to calculate numerical quantities!
As this is a philosophical endeavor, it can’t really be proved.
Turing’s Thesis: Any “algorithm” can be carried out by one of his machines
A Thinking Machine: A Thinking Machine Second Goal of Turing’s Machine: A model that’s so simple, that can actually prove interesting epistemological results. Eyed Hilbert’s 10th problem, as well as a computational analog of Gödel’s Incompleteness Theorem in Logic.
Philosophy notwithstanding, Turing’s programs for cracking the Enigma cryptosystem prove that he really was a true hacker! Turing’s machine is actually easily programmable, if you really get into it. Not practically useful, though…
A Thinking Machine: A Thinking Machine Imagine a super-organized, obsessive-compulsive human computer. The computer wants to avoid mistakes so everything written down is completely specified one letter/number at a time. The computer follows a finite set of rules which are referred to every time another symbol is written down. Rules are such that at any given time, only one rule is active so no ambiguity can arise. Each rule activates another rule depending on what letter/number is currently read, EG:
A Thinking MachineEG Successor Program: A Thinking Machine EG Successor Program Sample Rules:
If read 1, write 0, go right, repeat.
If read 0, write 1, HALT!
If read , write 1, HALT!
Let’s see how they are carried out on a piece of paper that contains the reverse binary representation of 47:
A Thinking MachineEG Successor Program: A Thinking Machine EG Successor Program If read 1, write 0, go right, repeat.
If read 0, write 1, HALT!
If read , write 1, HALT!
A Thinking MachineEG Successor Program: A Thinking Machine EG Successor Program If read 1, write 0, go right, repeat.
If read 0, write 1, HALT!
If read , write 1, HALT!
A Thinking MachineEG Successor Program: A Thinking Machine EG Successor Program If read 1, write 0, go right, repeat.
If read 0, write 1, HALT!
If read , write 1, HALT!
A Thinking MachineEG Successor Program: A Thinking Machine EG Successor Program If read 1, write 0, go right, repeat.
If read 0, write 1, HALT!
If read , write 1, HALT!
A Thinking MachineEG Successor Program: A Thinking Machine EG Successor Program If read 1, write 0, go right, repeat.
If read 0, write 1, HALT!
If read , write 1, HALT!
A Thinking MachineEG Successor Program: A Thinking Machine EG Successor Program If read 1, write 0, go right, repeat.
If read 0, write 1, HALT!
If read , write 1, HALT!
A Thinking MachineEG Successor Program: A Thinking Machine EG Successor Program So the successor’s output on 111101 was 000011 which is the reverse binary representation of 48.
Similarly, the successor of 127 should be 128:
A Thinking MachineEG Successor Program: A Thinking Machine EG Successor Program If read 1, write 0, go right, repeat.
If read 0, write 1, HALT!
If read , write 1, HALT!
A Thinking MachineEG Successor Program: A Thinking Machine EG Successor Program If read 1, write 0, go right, repeat.
If read 0, write 1, HALT!
If read , write 1, HALT!
A Thinking MachineEG Successor Program: A Thinking Machine EG Successor Program If read 1, write 0, go right, repeat.
If read 0, write 1, HALT!
If read , write 1, HALT!
A Thinking MachineEG Successor Program: A Thinking Machine EG Successor Program If read 1, write 0, go right, repeat.
If read 0, write 1, HALT!
If read , write 1, HALT!
A Thinking MachineEG Successor Program: A Thinking Machine EG Successor Program If read 1, write 0, go right, repeat.
If read 0, write 1, HALT!
If read , write 1, HALT!
A Thinking MachineEG Successor Program: A Thinking Machine EG Successor Program If read 1, write 0, go right, repeat.
If read 0, write 1, HALT!
If read , write 1, HALT!
A Thinking MachineEG Successor Program: A Thinking Machine EG Successor Program If read 1, write 0, go right, repeat.
If read 0, write 1, HALT!
If read , write 1, HALT!
A Thinking MachineEG Successor Program: A Thinking Machine EG Successor Program If read 1, write 0, go right, repeat.
If read 0, write 1, HALT!
If read , write 1, HALT!
A Thinking MachineEG Successor Program: A Thinking Machine EG Successor Program If read 1, write 0, go right, repeat.
If read 0, write 1, HALT!
If read , write 1, HALT!
A Thinking Machine: A Thinking Machine It was hard for the ancients to believe that any algorithm could be carried out on such a device. For us, it’s much easier to believe, especially if you’ve programmed in assembly!
However, ancients did finally believe Turing when Church’s lambda-calculus paradigm (on which lisp programming is based) proved equivalent!
Turing Machines: Turing Machines A Turing Machine (TM) is a device with a finite amount of read-only “hard” memory (states), and an unbounded1 amount of read/write tape-memory. There is no separate input. Rather, the input is assumed to reside on the tape at the time when the TM starts running.
Just as with Automata, TM’s can either be input/output machines (compare with Finite State Transducers), or yes/no decision machines. Start with yes/no machines.
Comparison with Previous Models: Comparison with Previous Models
Comparison with Previous Models: Comparison with Previous Models
Comparison with Previous Models: Comparison with Previous Models
Comparison with Previous Models: Comparison with Previous Models
Turing MachineDecision Machine Example: Turing Machine Decision Machine Example First example (adding 1 bit to reverse binary string) was basically something that a Finite Transducer could have achieved (except when there’s overflow). Let’s give an example from next step up in language hierarchy.
{bit-strings with same number of 0’s as 1’s}
–a context free language:
Turing MachineDecision Machine Example: Turing Machine Decision Machine Example This is a “true” Turing machine as:
Tape is semi-infinite (indicated by torn cell):
Input is prepared at beginning of tape
No intrinsic way to detect left tape end
similar to empty stack detection problem for PDA’s
similar trick used –introduce $ as the end symbol
All rules must include a move direction (R/L)
Situations that can’t happen aren’t dealt with (technically under-deterministic)
Turing MachineDecision Machine Example: Turing Machine Decision Machine Example {bit-strings with same number of 0’s as 1’s}:
Pseudocode:
while (there is a 0 and a 1)
cross these out
if (everything crossed out)
accept
else
reject
TM ExampleInstructions Set: TM Example Instructions Set 0. if read , go right (dummy move), ACCEPT
if read 0, write $, go right, goto 1 // $ detects start of tape
if read 1, write $, go right, goto 2
1. if read , go right, REJECT
if read 0 or X, go right, repeat (= goto 1) // look for a 1
if read 1, write X, go left, goto 3
2. if read , go right, REJECT
if read 1 or X, go right, repeat // look for a 0
if read 0, write X, go left, goto 3
3. if read $, go right, goto 4 // look for start of tape
else, go left, repeat
4. if read 0, write X, go right, goto 1 // similar to step 0
if read 1, write X, go right, goto 2
if read X, go right, repeat
if read , go right, ACCEPT
TM ExampleState Diagram: TM Example State Diagram These instructions can be expressed by a familiar looking flow diagram:
0 1 rej 0$,R acc R 2 1$,R 0|XR 1|XR 3 R 0X,L 1X,L 0|1|XL 4 $R XR 0X,R 1X,R R
TM Transition Notation: TM Transition Notation An edge from the state p to the state q labeled by …
ab,D means if in state p and tape head reading a, replace a by b and move in the direction D, and into state q
aD means if in state p and tape head reading a, don’t change a and move in the direction D, and into state q
a|b|…|z … means that given that the tape head is reading any of the pipe separated symbols, take same action on any of the symbols
TM Configuration Notation: TM Configuration Notation A TM’s next action is completely determined by current state and symbol read, so can predict all of future actions if know:
current state
current tape contents
current position of TM’s reading “head”
Handy notation lists all of these in a single string. A symbol representing current state, is sandwiched between content of tape to left of head, and content of tape to right (including tape head). The part of tape which is blank ad-infinitum is ignored.
TM Configuration Notation: TM Configuration Notation For example
Is denoted by:
$xxx1q3010 Reading rule 3
TM ExampleCrazy Web-Page: TM Example Crazy Web-Page The following link shows how the example machine accepts 01101010 and how the tape configuration notation changes step by step.
TM Formal DefinitionStatic Picture: TM Formal Definition Static Picture DEF: A Turing machine (TM) consists of a 7-tuple M = (Q, S, G, d, q0, qacc, qrej). Q, S, and q0, are the same as for an FA. G is the tape alphabet which necessarily contains the blank symbol , as well as the input alphabet S. d is as follows:
Therefore given a non-halt state p, and a tape symbol x, d(p,x) = (q,y,D) means that TM goes into state q, replaces x by y, and the tape head moves in direction D.
TM Dynamic Picture: TM Dynamic Picture A string x is accepted by M if after being put on the tape with the Turing machine head set to the left-most position, and letting M run, M eventually enters the accept state. In this case w is an element of L(M) –the language accepted by M.
We can formalize this notion as follows:
TM Formal DefinitionDynamic Picture: 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 d(p,x) = (q,y,R) then write:
uapxv uaypv
With resulting configuration uaypv at time t+1. If, d(p,x) = (q,y,L) instead, then write:
uapxv upayv
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 (only case)
“” is read as “yields”
TM Formal DefinitionDynamic Picture: TM Formal Definition Dynamic Picture As with context free grammars, one can consider the reflexive, transitive closure “*” of “”. I.e. this is the relation between strings recursively defined by:
if u = v then u * v
if u v then u * v
if u *v and v * w, then u *w
“*” is read as “computes to”
A string x is said to be accepted by M if the start configuration q0 x computes to some accepting configuration y –i.e., a configuration containing qacc.
The language accepted by M is the set of all accepted strings. I.e:
L(M) = { x S* | accepting config. y, q0 x * y }
TM Acceptors vs. Deciders: TM Acceptors vs. Deciders Three possibilities occur on a given input w :
The TM M eventually enters qacc and therefore halts and accepts. (w L(M) )
The TM M eventually enters qrej or crashes somewhere. M rejects w . (w L(M) )
Neither occurs! I.e., M never halts its computation and is caught up in an infinite loop, never reaching qacc or qrej. 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) )
TM Acceptors vs. Deciders: TM Acceptors vs. Deciders Any Turing Machines is said to be a recognizer and recognizes L(M); if in addition, M never enters an infinite loop, M is called a decider and is said to decide L(M).
Q: Is the above M an recognizer? A decider? What is L(M)? 0 1 rej acc 2 R 1R 0R 1R 0R 0R
1L
TM Acceptors vs. Deciders: TM Acceptors vs. Deciders A: M is an recognizer but not a decider because 101 causes an infinite loop.
L(M) = 1+ 0+
Q: Is L(M ) decidable ? 0 1 rej acc 2 R 1R 0R 1R 0R 0R
1L
TM Acceptors vs. Deciders: TM Acceptors vs. Deciders A: Yes. All regular languages are decidable because can always convert a DFA into a TM without infinite loops.
Constructive Example: Constructive Example Here’s a document showing how modular design can help you write down a TM decider for {anbncn}. The example is non-context free.