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MATH 224 – Discrete Mathematics

MATH 224 – Discrete Mathematics. Deterministic Finite Automata Finite State Machines. Set of States Q Start State q0  Q Accepting States qf  Q Alphabet  e.g, {0, 1} Transitions  (, qi)  qj. q1. q2. q3. q0. qf1. qf2. . qi. qj. MATH 224 – Discrete Mathematics.

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MATH 224 – Discrete Mathematics

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  1. MATH 224 – Discrete Mathematics Deterministic Finite AutomataFinite State Machines Set of StatesQ Start Stateq0  Q Accepting Statesqf Q Alphabet e.g, {0, 1} Transitions  (, qi)  qj q1 q2 q3 q0 qf1 qf2  qi qj

  2. MATH 224 – Discrete Mathematics DFA Represented by a Directed Graph Multitasking in an operating System Time slice expired Start Ready Queue Running Dequeued Finished with resource and ready to run Blocked waiting for a resource, e.g. data from a disk Blocked Waiting

  3. MATH 224 – Discrete Mathematics DFA for Multiples of 3 in binary 0 Start 0 q0 Example 1 46 remainder 1 string 1 0 1 1 1 0 States0 1 2 2 2 2 1 Example 2 51 remainder 0 string1 1 0 0 1 1 States 0 1 3 3 3 1 3 qf-3 1 1 1 1 q1 0 q2 0

  4. MATH 224 – Discrete Mathematics DFA to Recognize Floating Point 0 - 9 Start 0 - 9 . q0 q2 qf-0 +, – 0 - 9 0 - 9 e, E q5 q1 q3 +, – 0 - 9 q4 0 - 9 Qf-1 0 - 9

  5. MATH 224 – Discrete Mathematics Crossing the Bridge in Minimum Time1, 2, 5 and 10 minutes q6 5 &10 Start Only 2 of the final states are shown here, one of the two shortest and one of the next shortest. q0 2 &10 q5 1 & 2 2 & 5 1&10 What would the longest time be? 1 & 5 2 minutes q1 q4 q3 1 q2 2 5 minutes 3 minutes 1 19 minutes q8 q7 5 6 minutes q10 5 &10 qf17 q9 1 &10 q15 q11 q12 16 minutes 13 minutes 1 & 2 1 2 q16 17 minutes 1 & 2 q13 qf14 17 minutes 15 minutes

  6. MATH 224 – Discrete Mathematics Other DFAs Construct a DFA that recognizes legal identifier names in C++ using α to represent letters and δ to represent the digits 0 through 9. Construct a DFA to recognize even numbers in binary. Construct a DFA to recognize multiples of four in binary. Construct a DFA to recognize multiples of 5 in binary. Construct a DFA to recognize multiples of 5 in decimal.

  7. MATH 224 – Discrete Mathematics Representing a DFA in a Computer Program Probably the simplest way to represent a DFA in a program is to use a 2-dimensional array, where each column corresponds to a character in the alphabet and each row corresponds to a state. So for example here is the DFA that recognizes multiples of 3 in binary. qf-3

  8. MATH 224 – Discrete Mathematics 0 DFA for Multiples of 3 in binary Start 0 q0 0 qf-6 q3 1 1 1 1 1 q1 0 q2 0

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