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 ( ) in 1936 defined an abstract model for use in describing the decision problem Processor Read/Write Head... Data Tape

Diagramming a Turing Machine Halt % S0 R Y b Y % Halt S2 R S1 R S3 R X b Y b L % X Halt X b R “The Turing Machine”, Isaac Malitz, Byte, November 1987, pp

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 O1 P1 P2P3 P4 P5 P6 P7 P8 P9 P10 P11 P12