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Class 25 Notes

Lecture Slides, Suitable For Printing

Notes

Models of Computation

What is a model?



What do we need to model computation?




Finite State Machines

FSM ::= <Alphabet, States, InitialState, HaltingStates, TransitionRules>
Alphabet ::= { Symbol* }     A set of symbols for the input.
States ::= { StateName* }
InitialState ::= StateName
    Must be one of the states in States.
HaltingStates ::= { StateName* }
    Must all be states in States.
TransitionRules ::= { TransitionRule* }
TransitionRule ::= < StateName, Symbol, StateName>
    StateName X Symbol → StateName
What does this FSM do?

Why is there no FSM powerful enough to determine if an input has balanced parentheses?








Turing Machine

Infinite Tape + Finite State Machine with tape head

One step:

TM ::= <Alphabet, Tape, TFSM>

Alphabet ::= { Symbol*}
    A set of symbols for the tape.

Tape ::= < LeftSide, OneSquare, RightSide >
OneSquare ::= Symbol | #
LeftSide ::= [ Squares* ]
    Everything to left of LeftSide is #. RightSide ::= [ Squares* ]
    Everything to right of RightSide is #. Squares ::= OneSquare, Squares
Squares ::=
TFSM ::= <States, InitialState, HaltingStates, TransitionRules>
    Like a FSM, except the transition rules write to the tape and move the tape head.

States ::= { StateName* }
InitialState ::= StateName    Must be one of the states in States.
HaltingStates ::= { StateName* }    Must all be states in States.
TransitionRules ::= { TransitionRule* }
TransitionRule ::= < StateName, OneSquare, StateName, OneSquare, Direction>
    StateName X OneSquare → StateName X OneSquare X Direction

Since we can write down a complete description or any Turing Machine using this grammar, we can also assign every TM a unique number that identifies it.

Universal Turing Machine

A Turing Machine that can simulate any other Turing Machine on an input:

TMU (P, I) = the output of running TM-P on input I

Church-Turing Thesis

Named for Alonzo Church (inventor of Lambda Calculus) and Alan Turing. First proposed by Stephen Kleene in 1943.

There are many different ways of stating the Church-Turing thesis:

A universal programming language is a programming language in which it is possible to express all computations.

How do we prove a programming language is a universal programming language?






How do we prove a programming language is not a universal programming language?





What must a programming language provide to be able to simulate a Turing machine?





Is Charme a universal programming language?





Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child's arithmetic book. In elementary arithmetic the two-dimensional character of the paper is sometimes used. But such a use is always avoidable, and I think that it will be agreed that the two-dimensional character of paper is no essential of computation. I assume then that the computation is carried out on one-dimensional paper, i.e. on a tape divided into squares. I shall also suppose that the number of symbols which may be printed is finite. If we were to allow an infinity of symbols, then there would be symbols differing to an arbitrarily small extent. The effect of this restriction of the number of symbols is not very serious. It is always possible to use sequences of symbols in the place of single symbols. Thus an Arabic numeral such as 17 or 999999999999999 is normally treated as a single symbol. Similarly in any European language words are treated as single symbols (Chinese, however, attempts to have an enumerable infinity of symbols). The differences from our point of view between the single and compound symbols is that the compound symbols, if they are too lengthy, cannot be observed at one glance. This is in accordance with experience. We cannot tell at a glance whether 9999999999999999 and 999999999999999 are the same.

The behaviour of the computer at any moment is determined by the symbols which he is observing. and his “state of mind” at that moment. We may suppose that there is a bound B to the number of symbols or squares which the computer can observe at one moment. If he wishes to observe more, he must use successive observations. We will also suppose that the number of states of mind which need be taken into account is finite. The reasons for this are of the same character as those which restrict the number of symbols. If we admitted an infinity of states of mind, some of them will be “arbitrarily close” and will be confused. Again, the restriction is not one which seriously affects computation, since the use of more complicated states of mind can be avoided by writing more symbols on the tape. ...

We may now construct a machine to do the work of this computer. To each state of mind of the computer corresponds an “m-configuration” of the machine. The machine scans B squares corresponding to the B squares observed by the computer. In any move the machine can change a symbol on a scanned square or can change anyone of the scanned squares to another square distant not more than L squares from one of the other scanned squares. The move which is done, and the succeeding configuration, are determined by the scanned symbol and the m-configuration....

Alan Turing, On computable numbers, with an application to the Entscheidungsproblem, 1936. [an error occurred while processing this directive]