IST4_lec11 - IST 4 Information and Logic IST 4 Planned...

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IST 4 Information and Logic
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1 M1 31 6 2 6 5 26 19 M2 5 4 12 4 5 3 M2 28 3 2 21 2 1 14 M1 7 fri thr wed tue mon IST 4: Planned Schedule – Spring 2008 x= hw#x out x= hw#x due = today T Mx= MQx out Mx= MQx due T start early
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Old/Odd Machines that Compute MQ 2 Deadline Friday, 5/16/2008, 10pm Please send a pdf file named yourname-mq2.pdf to [email protected]
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So Far IST 4 is focusing on CONTEXT… Boole was never a student at a university… only a professor Leibniz got a degree in Law never held an academic position… became a Babylonian… Stone planned to go to Law School became a successful mathematician… Proofs constitute an important language understand, read, write and speak
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Boolean Functions Start with a Boolean formula with n variables assume it is over SOME Boolean algebra with a set of elements B Assign all possible elements in B to the formula A Boolean function is a mapping from {B} Æ {B} defined by a Boolean formula n
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Boolean Functions Start with a Boolean formula with n variables assume it is over a Boolean algebra with a set of elements B Assign all possible elements in B to the formula The same formula for all algebra sizes
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Boolean Functions xy XOR(x,y) 00 01 10 11 0 1 1 0 2 elements 4 elements Identical 0-1 assignments have the same formula (DNF)
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DNF D isjunctive N ormal F orm Idea: Representing a Boolean function with a formula of a specific form
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Representation of Boolean Functions Disjunctive (Additive) Normal Form (DNF): Sum of terms, each term is Normal Normal = contains all the variables or their complements
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Representation of Boolean Functions Disjunctive (Additive) Normal Form (DNF): Sum of terms, each term is Normal Normal = contains all the variables or their complements Is it a DNF? No! This term is not normal
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Every Boolean function can be expressed in DNF
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DNF Theorem DNF Theorem: Every Boolean function can be expressed in DNF. Proof: Apply DeMorgan Theorem ( T4 ) until each negation is applied to a single variable Apply distributive axiom (A4) to get a sum of terms Use self absorption (L1) to eliminate duplicate terms Augment a missing variable a to a term using (A2) multiplying by By the algorithm
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DNF Representation Theorem DNF is a representation: two Boolean functions are equal if and only if their DNFs are identical. DNF Representation Theorem: The key: Two Boolean functions with identical values for all 0-1 assignments have identical DNFs Two Boolean functions are equal if and only if they are equal for all 0-1 assignments
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Illuminating Question???
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