Post on 21-Dec-2015
Computability and Complexity 3-1
Turing Machine
Computability and ComplexityAndrei Bulatov
Computability and Complexity 3-2
What is a computation?
Alan Turing considered this question in 1936
He argued that any computation can be done using the following steps:
• Concentrate on one part of the problem (for example, one piece of paper, one arithmetic operation, etc.)
• Depending on this, and present state of the computer:
- Modify this part of problem;
- Move to another part of problem;
- Change into a new state.
• Repeat this until finished.
Computability and Complexity 3-3
Implications
• If the computation can be described in a finite space, then there can can be only a finite number of different situations to deal with
• Therefore, there can be only a finite number of “states” and a finite number of “symbols”.
• But note that the computation might continue forever and never finish, and it might use (potentially) infinite storage space
I N P U T
Computability and Complexity 3-4
000
The Turing Machine
Computability and Complexity 3-5
Turing Machine Configuration
Tape contains a sequence (infinite to the right) of symbols from an alphabet almost all of the members of this sequence (except for a finite number) are the blank symbol At the first step it contains the input, i.e. the non-blank symbols on the tape constitutes the input
Head reads one cell (one symbol on the tape) at the first step it reads the leftmost symbol on the tape
Control Device is in one of the states from a finite set of states Q at the first step it is in the initial state 0q
Computability and Complexity 3-6
Turing Machine Operation
1. At each step of operation machine is in one of the states, say q (initially state is )
2. The machine is reading one symbol on tape, say s
3. To execute one step, the machine looks up the value of a transition function (q,s), which is, say, (q’,s’,m)
4. The machine:
• changes to state q’;
• overwrites the symbol s with s’;
• moves the head along the tape according to m (R=Right, L=Left, S=Static).
5. Execution stops when a final state is reached
0q
Computability and Complexity 3-7
Turing Machine Definition
Definition A Turing Machine is a 6-tuple where
1. Q is a finite set of states
2. is the input alphabet not containing the blank symbol
3. is the tape alphabet, where and
4. is a transition function
5. is the initial state
6. F Q is a set of final states
FqQ ,,,,, 0
Qq 0
},,{: SRLQQ
Computability and Complexity 3-8
Question
What do the following machines do?
Computability and Complexity 3-9
TM Example
Let T be the TM :
Q = {1,2,3}
= {a,b}
= {,a,b} = { (1,a),(2,a,R), (1,b),(2,b,R), (1, ),(2, ,R), (2,a),(3,a,S), (2,b),(2,b,R), (2,),(2,,R) } = 1 F = {3}
FqQ ,,,,, 0
0q
Computability and Complexity 3-10
TM Example
Let T be the TM :
Q = {a,b,c,d,e}
= {0,1}
= {,0,1} = { (a,0),(e,1,L), (c,0),(d,,R), (a,1),(a,0,L), (c,1),(b,,R), (a, ),(e,1,S), (c, ),(e,,S), (b,0),(d,1,R), (d,0),(d,0,R), (b,1),(b,1,R), (d,1),(b,0,R), (b,),(a,1,S), (d,),(a,0,S) } = c F = {e}
FqQ ,,,,, 0
0q
Computability and Complexity 3-11
TM Diagram
c
e
a
b d
||S 0|01|LS
0||R
1||R
|1|S
0|1|R1|1|R
1|0|R
0|0|R
|0|S
1|0|L
Computability and Complexity 3-12
Configurations
Configurations: To completely describe a TM in operation we need:
• The current state• The content of the tape• The position of the head
This information is called the configuration (or global state)
Computation: a (possibly infinite) sequence of configurations entered by a TM beginning with an initial configuration
Initial configuration: for machine T and input string x consists of:
• The initial state,• The tape starting with x, all other cells blank• The head on the leftmost cell
0q
Computability and Complexity 3-13
Acceptors and Transducers
Acceptor machines: have 2 final states
• “Accept” ( )• “Reject” ( )
These machines can be used to solve decision problems: input x is accepted by T if T(x) is finite and ends in the “Accept” state.
Language accepted: set of inputs accepted by T, denoted by L(T)
Aq
Rq
Computability and Complexity 3-14
I N P U T
A? R?
Computability and Complexity 3-15
Transducer: T computes (partial) function f if computation T(x) halts in a final state with f(x) on the output tape
Computability and Complexity 3-16
I N P U T
H?
O U T P U T
Computability and Complexity 3-17
Turing’s Thesis
“Any function which can be computed by any well-defined procedurecan be computed by a Turing Machine”
(Also called Church’s Hypothesis, Sequential Computing Hypothesis)
Computability and Complexity 3-18
k-tape TM
Transition function:
times times times
},,{},,{:kkk
SRLSRLQQ
),,;',,';'(),,;( 111 kkk mmssqssq
Initial configuration:
• the first tape contains an input
• the other tapes contain only blank symbols
Computability and Complexity 3-19
Nondeterministic TM
We can relax the definition of a TM to allow more than one possible transition for any pair q,s of state and symbol
This gives a nondeterministic TM (i.e. it has a choice of actions)
Nondeterministic computation: a tree of possible configurations resulting from a start configuration For machine NT and input x this is denoted NT(x)
Computability and Complexity 3-20
Nondeterministic Acceptor
Computation path: is any path (finite or infinite) from the root of NT(x)
Accepting computation: is a finite computation path ending in the “Accept” state
Language accepted: is the set of inputs x having some accepting computation in NT(x), denoted L(NT).
Computability and Complexity 3-21
R R
R
R
R A R
Tim
e
Computability and Complexity 3-22
Random Access Machines
Another model of computation is the Random Access Machine,which has
• an infinite series of memory locations, where each can contain an arbitrary integer
• a processor which can perform basic operations, including addition, subtraction, integer division by 2, indirect addressing, conditional goto, and halt
• a finite program specifying a sequence of basic operations involving the contents of memory locations
,, ,21 MM
iM
Theorem Given a RAM, it is possible to derive an equivalent 3-tape deterministic TM
Computability and Complexity 3-23
H?
1. READ 12. STORE 13. STORE 54. READ 25. STORE 26. HALF7. STORE 38. ADD 39. SUB 210. JZERO 1411. LOAD 412. ADD 513. STORE 414. LOAD 515. ADD 516. STORE 517. LOAD 318. JZERO 2019. JUMP 520. LOAD 421. HALT
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3556
21
Computability and Complexity 3-24
Simulations
The following machines can all be simulated by a TM with one tapeand alphabet {0,1}
• TMs with tape infinite in both directions
• TMs with more than one tape
• TMs with 2-dimensional tape (or more!)
• TMs with several read/write heads on each tape
• Nondeterministic TMs
• RAM Machines
Computability and Complexity 3-25
Other Models
The following people also proposed models of general computation*:
• Post (1936) — “Post’s normal systems”
• Kleene (1936) — “recursive functions”
• Church (1941) — “lambda calculus”
• Markov (1954) — “Markov’s rewriting systems”
All of these have been shown to be equivalent to Turing Machines
*see “Machine Models and Simulations” in the Handbook of Theoretical ComputerScience, MIT Press, 1990