Fachgebiet Rechnersysteme 1. Boolean Algebra 1 1 Boolean Algebra 1. Boolean Algebra Verification Technology Content 1.1 Boolean algebra basics (recap) 1 2 Reasoning about Boolean expressions 1.2 Reasoning about Boolean expressions
1. Boolean Algebra 2 The problem of logic verification: Show that two circuits implement the same boolean function, or: Show that a p , circuit correctly implements a specification g a =1 1 b a & g g a a & & b & & b
1. Boolean Algebra 3 1.1 Boolean algebra basics 1.1 Boolean Algebra Basics 1.1 Boolean Algebra Basics AND-operation (conjunction) Gate representation a b a b a German (old) 0 0 0 0 0 0 b b 0 1 0 1 0 1 0 0 0 a & IEEE- 1 1 1 b standard a 0 d 0-dominance i US-standard b (old)
1. Boolean Algebra 4 1.1 Boolean algebra basics Gate representation OR-operation (disjunction) a a b a b German (old) ( ) b b 0 0 0 0 1 1 1 a IEEE- 1 0 1 b standard 1 1 1 1 1 1 a 1-dominance US-standard b (old)
1. Boolean Algebra 5 1.1 Boolean algebra basics NOT-operation (negation, complement) Gate representation Gate representation a German (old) ( ) a a 0 1 IEEE- 1 a 1 0 Standard a US-standard (old)
1. Boolean Algebra 6 1.1 Boolean algebra basics More notations ... a b a • b a & b a b a b a + b a + b a | b a | b a a a' Propositional Calculus Propositional Calculus
1. Boolean Algebra 7 1.1 Boolean algebra basics More operations: (E)XOR (exclusive-or unequal addition modulo 2) (E)XOR (exclusive-or, unequal, addition modulo 2) Gate representation Definition: a b a b a b a a a b b a a b b German (old) German (old) b b 0 0 0 0 0 1 1 1 1 a =1 IEEE- 1 0 1 standard b 1 1 1 1 0 0 a US-standard b (old)
1. Boolean Algebra 8 1.1 Boolean algebra basics NAND G t Gate representation t ti a a b (a b) German (old) b 0 0 1 0 1 1 a & 1 1 0 0 1 1 IEEE- IEEE- b standard 1 1 0 a US-standard b b (old)
1. Boolean Algebra 9 1.1 Boolean algebra basics NOR Gate representation Gate representation a a b (a b) German (old) b 0 0 1 0 1 0 1 a 1 1 0 0 0 0 IEEE- IEEE standard b 1 1 0 a US-standard b b ( ld) (old)
1. Boolean Algebra 10 1.1 Boolean algebra basics Implikation : a b a b Equivalence (equality): Equivalence (equality): a b a b a b — The equivalence-function equals one iff the arguments have equal values — The exor-function equals one iff the arguments Th f ti l iff th t have unequal values — Generally: a b = (a b) Generally: a b = (a b)
1. Boolean Algebra 11 1.1 Boolean algebra basics Boolean functions — Traffic-light checker: inputs: r red, g green, e Traffic-light checker: inputs: r red g green e yellow r e g p? 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 2 3 = 8 cases 1 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 r Traffic- e p Light Checker g
1. Boolean Algebra 12 1.1 Boolean algebra basics r, e, g are Boolean variables B = {0 1} is the set of Boolean values B = {0, 1} is the set of Boolean values A Boolean function in n variables is a mapping F: B n B Such a function models a circuit with n inputs and one output, in the example we have B 3 B B 3 B r Traffic- e p Light Checker g
1. Boolean Algebra 13 1.1 Boolean algebra basics 3 2 1 0 2 2 2 2 Representation of Boolean functions by means of x x x x 0 f 3 2 1 function tables and Veitch- function tables and Veitch- 0 0 0 0 0 0 (Karnaugh-) maps 0 0 0 1 1 1 x 3 0 0 1 0 0 0 1 0 0 0 2 2 0 0 1 1 0 3 0 1 1 0 0 1 0 0 1 4 12 4 8 0 0 1 0 1 0 1 0 1 1 1 5 5 1 1 1 0 1 1 0 0 1 6 1 5 13 9 x 0 0 1 1 1 1 7 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 8 8 15 7 3 11 1 0 0 1 1 9 x 1 1 0 1 0 0 0 10 0 0 0 1 0 1 1 0 1 0 1 1 0 2 2 6 6 10 10 14 11 1 1 0 0 1 12 x 2 1 1 0 1 1 13 2 1 1 1 0 0 14 1 1 1 1 1 15
1. Boolean Algebra 14 1.1 Boolean algebra basics Boolean terms are textual representations of Boolean functions, e.g., , g , a b c + c a (e + b) The syntax of Boolean terms: The constants 0 and 1 are Boolean terms Literals (variables and complemented variables) are Boolean terms, for instance, a and a If a and b are Boolean terms then so are If a and b are Boolean terms then so are (a b), (a + b), a
1. Boolean Algebra 15 1.1 Boolean algebra basics Product-terms (products, cubes) are conjunctions of literals Each variable occurs only once — Example: x•y•z or xyz for short Example: x y z or xyz for short Special case: Minterm (product-term in all variables) Sum-terms (clauses) are disjunctions of literals Sum terms (clauses) are disjunctions of literals Special case: Maxterm (sum-term in all variables)
1. Boolean Algebra 16 1.1 Boolean algebra basics A Sum-of-Products (sop) (also called a disjunctive normal form, dnf) is a disjunction of products , ) j p — Example: cyz + d + ax A Product-of-Sums (pos) (also called a conjunctive normal form, cnf) is a conjunction of sums — Example: (c + x + b)(s + x)(a + x)
1. Boolean Algebra 17 1.1 Boolean algebra basics The Rules of Boolean Algebra: (T1) x + 0 = x (T1) x + 0 = x (T1') x • 1 = x (T1 ) x • 1 = x Identity Identity (T2) x + 1 = 1 (T2') x • 0 = 0 0/1-Element (T3) x + x = x (T3') x • x = x Idempotence (T4) x = x (T4') 1 = 0 Involution (T5) x + x = 1 (T5) x + x = 1 (T5') x • x = 0 (T5') x • x = 0 Complement Complement
1. Boolean Algebra 18 1.1 Boolean algebra basics (T6) x + y = y + x ( ) y y (T6') x • y = y • x ( ) y y Commutativity y (T7) (x + y) + z = x + (y + z) = x + y + z Associativity (T7') (x • y) • z = x • (y • z) = x • y • z (T7 ) (x • y) • z = x • (y • z) = x • y • z (T8) x•y + x•z = x•(y + z) Distributivity (T8') (x + y)•(x + z) = x + y•z (T9) (x + y) x y (T9) (x + y) = x • y (T9 ) (x y) x + y (T9') (x • y) = x + y De Morgan's De Morgan s Law (T10) (x 1 +...+ x n )= x 1 •...• x n (T10) ( + + ) G Generalized li d De Morgan's (T10') (x 1 •...• x n )= x 1 +...+ x n ( ) ( n ) Law 1 1 n
1. Boolean Algebra 19 1.1 Boolean algebra basics (T11) x + x•y = x + y ( ) y y (T11') x•(x + y) = x•y ( ) ( y) y Absorption (T12) x + x•y = x (T12') x•(x + y) = x Very important for the simplification of terms and circuits! Example: x 1 x 1 x & & y y
1. Boolean Algebra 20 1.1 Boolean algebra basics Boole's Expansion Theorem (1849) also attributed to Shannon also attributed to Shannon
1. Boolean Algebra 21 1.1 Boolean algebra basics x x x x f 1 2 3 4 Idea: Decompose a function into two sub- 0 0 0 0 0 0 0 0 0 0 functions which do not 0 0 0 1 1 depend on some variable, 0 0 1 0 0 e g e.g., x 1 x 0 0 1 1 0 0 1 1 0 0 f(0, x , x , x ) 0 1 0 0 1 2 3 4 0 1 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 1 0 0 f(1, x , x , x ) 1 1 0 0 1 2 3 4 1 1 0 1 1 1 1 1 0 0 1 1 1 1 1
1. Boolean Algebra 22 1.1 Boolean algebra basics Expansion Theorem: a Boolean Function f can be decomposed according to a variable x as follows p g f f x x f f x x f f x x x x f , f , f f are called the Positive (1-) and Negative (0-) are called the Positive (1 ) and Negative (0 ) x x Cofactors of f with respect to x The cofactors can be calculated by replacing the y g variable x by the constants 0 and 1, respectively
1. Boolean Algebra 23 1.1 Boolean algebra basics — Illustration in the Veitch-diagram: b f a 1 1 1 f a f 1 1 1 1 1 1 a a f f c c f f c c a a b b ab ab
1. Boolean Algebra 24 1.1 Boolean algebra basics — Illustration in the Veitch-diagram: b f a 1 1 1 f a f 1 1 1 1 1 1 a a f f c c f f c c a a b b ab ab f a f a f a a
1. Boolean Algebra 25 1.1 Boolean algebra basics — Circuit realization: b f & a a f a 1 1 1 1 f f a f 1 1 1 1 1 1 a a & f a f f c c a f f c c a a b b ab ab f a f a f a a
1. Boolean Algebra 26 1.1 Boolean algebra basics The cofactors are calculated by replacing a by 0 and 1, respectively: respectively: b f & a a f a 1 1 1 1 f f f a 1 1 1 1 1 1 a a & f a f f c c a f f = = c c + + a a b b + + ab ab, f = c + 1 b + 1b = c + b, a f a = c + 0 b + 0b = c + b , f = a (c + b ) + a (c + b )
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