Turing Machine Model• Are there computations that no “reasonable” computing machine can perform?

– the machine should not store the answer to all possible problems

– it should process information (execute instructions) at a finite speed

– it is capable of performing a particular computation only if it can generate the answer in a finite number of steps

• Alan M. Turing (1912-1954) in 1936 defined an abstract model for use in describing the decision problem

Processor

Read/Write Head

... ...Data Tape

Diagramming a Turing MachineHalt

%

S0

R

Y

b

Y

%

Halt

S2

R

S1

R

S3

R

X b

Y

bL

%

XHalt

X b

L

@

R

“The Turing Machine”, Isaac Malitz, Byte, November 1987, pp 345-357.

Finite State Automatons

• A Finite-State Automaton (FSA) consists of:– a set I, the input alphabet;– a set S, the states;– an initial state;– a subset of S called accepting states;– a state transition function N: S x I S

• N(s,m) is the state to which the FSA goes if m is the input when the FSA is in state s.

Petri Nets• A Petri Net is a bipartite directed graph and consists of:

– a set P of places (a state in which the system could be observed);– a set T of transitions (the rules “fire” causing state changes);– an input function I:T P*; a mapping from transitions to

“bags” of places– an output function O:T P*; a mapping from transitions to

“bags” of places;– a marking M:P {0,1,2,…} which assigns tokens to places:

• M’(p) = M(p) + 1 if p is a member of O(t) and p is not a member of I(t),

• M’(p) = M(p) - 1 if p is a member of I(t) and p is not a member of O(t),

• M’(p) = M(p) otherwise.– a transition is enabled if M(p) > 0 for all p members of I(t)

A Marked Petri Net Example

T7

T2

T4

T3

T5

T6

T8

T10

T9

T1

I1 O1P1 P2 P3P4

P5

P6

P7

P8

P9

P10

P11

P12