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# Module04 - Module 4 Formal Definition of Solvability...

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1 Module 4: Formal Definition of Solvability Analysis of decision problems Two types of inputs:yes inputs and no inputs Language recognition problem Analysis of programs which solve decision problems Four types of inputs: yes, no, crash, loop inputs Solving and not solving decision problems Classifying Decision Problems Formal definition of solvable and unsolvable decision problems

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2 Analyzing Decision Problems Can be defined by two sets
3 Decision Problems and Sets Decision problems consist of 3 sets The set of legal input instances (or universe of input instances) The set of “yes” input instances The set of “no” input instances Yes Inputs No Inputs Set of All Legal Inputs

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4 Redundancy * Only two of these sets are needed; the third is redundant Given The set of legal input instances (or universe of input instances) This is given by the description of a typical input instance The set of “yes” input instances This is given by the yes/no question We can compute The set of “no” input instances
5 Typical Input Universes Σ *: The set of all finite length strings over finite alphabet Σ Examples {a}*: {/\, a, aa, aaa, aaaa, aaaaa, … } {a,b}*: {/\, a, b, aa, ab, ba, bb, aaa, aab, aba, abb, … } {0,1}*: {/\, 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, … } The set of all integers If the input universe is understood, a decision problem can be specified by just giving the set of yes input instances

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6 Language Recognition Problem Input Universe Σ * for some finite alphabet Σ Yes input instances Some set L subset of Σ * No input instances Σ * - L When Σ is understood, a language recognition problem can be specified by just stating what L is.
7 Language Recognition Problem * Traditional Formulation Input A string x over some finite alphabet Σ Task Is x in some language L subset of Σ ?

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Module04 - Module 4 Formal Definition of Solvability...

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