Prof. Hsien-Hsin Sean Lee

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Presentation transcript:

ECE2030 Introduction to Computer Engineering Lecture 6: Canonical (Standard) Forms Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering Georgia Tech

Boolean Variables A multi-dimensional space spanned by a set of n Boolean variables is denoted by Bn A literal is an instance (e.g. A) of a variable or its complement (Ā)

SOP Form A product of literals is called a product term or a cube (e.g. Ā·B·C in B3, or B·C in B3) Sum-Of-Product (SOP) Form: OR of product terms, e.g. ĀB+AC A minterm is a product term in which every literal (or variable) appears in Bn ĀBC is a minterm in B3 but not in B4. ABCD is a minterm in B4. A canonical (or standard) SOP function: a sum of minterms corresponding to the input combination of the truth table for which the function produces a “1” output.

Minterms in B3

Canonical (Standard) SOP Function

POS form (dual of SOP form) A sum of literals is called a sum term (e.g. Ā+B+C in B3, or (B+C) in B3) Product-Of-Sum (POS) Form: AND of sum terms, e.g. (Ā+B)(A+C) A maxterm is a sum term in which every literal (or variable) appears in Bn (Ā+B+C) is a maxterm in B3 but not in B4. A+B+C+D is a maxterm in B4. A canonical (or standard) POS function: a product of maxterms corresponding to the input combination of the truth table for which the function produces a “0” output.

Maxterms in B3

Canonical (Standard) POS Function

Convert a Boolean to Canonical SOP Expand the Boolean eqn into a SOP Take each product term w/ a missing literal A, “AND” () it with (A+Ā)

Convert a Boolean to Canonical SOP F ABC 1 1 Minterms listed as 1’s 3 7

Convert a Boolean to Canonical SOP

Convert a Boolean to Canonical POS Expand Boolean eqn into a POS Use distributive property Take each sum term w/ a missing variable A and OR it with A·Ā

Convert a Boolean to Canonical POS

Convert a Boolean to Canonical POS F ABC 1 Maxterms listed as 0’s 2 4 5 6

Convert a Boolean to Canonical SOP F ABC 1 1 Minterms listed as 1’s 3 7

Convert a Boolean to Canonical POS

Convert a Boolean to Canonical SOP

Interchange Canonical SOP and POS For the same Boolean eqn Canonical SOP form is complementary to its canonical POS form Use missing terms to interchange  and  Examples F(A,B,C) =  m(0,1,4,6,7) Can be re-expressed by F(A,B,C) =  M(2,3,5) Where 2, 3, 5 are the missing minterms in the canonical SOP form