9/16/2020 CS4350: Logic in computer Science Optimization Problems in Propositional Logic 1 Optimization Problems in Logic Optimization Problems: those with minimum or maximum solutions (usually in terms of an objective/utility function). • Minimally Sufficient Set of Operators • Logic Minimization • Maximum Satisfiability 2 1 9/16/2020 Minimally Sufficient • A set of logical operators is called minimally sufficient if every formula is logically equivalent to a formula involving only this set of logical operators, and it does not contain a proper subset which is also sufficient. • { , } is a sufficient set of operators. • Is this set minimally sufficient? • Yes. (How to prove it?) 3 Minimally Sufficient • { , } is minimally sufficient. • { , } is minimally sufficient. • Is { , 0, 1 } minimally sufficient? • x x 0 ? • No, x 0 0 • { ite, 0, 1 } is sufficient. • x ite(x, 0, 1) • x y ite(x, ite(y, 1, 0), 0) • Is { ite, 0, 1 } minimally sufficient? • unknown 4 2 9/16/2020 Minimally Sufficient • Is there a minimally sufficient singleton set? • YES: { } (nand), { } (nor) • x x x • x y (x y) = (x y) (x y) • x x x • x y (x y) = (x y) (x y) • Is there a minimally sufficient set of a single trinary operator? • Yes, define f(x, y, z) = x y. 5 Minimal Definability • In Boolean Algebra, the meaning of a sufficient operators can be defined by a set of equations and the minimal number of equations is sought. • 3 equations to specify : – x 0 = 1, 0 y = 1, and 1 1 = 0. • What is the minimal number of equations needed to define ? One: ((x y) z) (x ((x z) x)) = z It contains 3 variables and 14 symbols. 6 3 9/16/2020 Minimal Definability • What is the minimal number of equations needed to define ? One: ((x y) z) (x ((x z) x)) = z • From this equation, we get • (x x) (x x) = x and it means x = x. • (x y) = (y x) and it means is commutative. • ((x x) x) = ((y y) y)) and it means ( x x) = ( y y) = 1. • … (all Boolean algebra laws) 7 Minimal Definability • What is the minimal number of equations to define the properties of {, }? (((x y) z) (x ( z (z u)))) = z. It contains 4 variables and 21 symbols. 8 4 9/16/2020 Logic Minimization • Also called Combinatorial Circuit Minimalization at gate level. • A combinatorial circuit is represented by a propositional formula (Boolean function) • Finding an equivalent formula of minimal size. – Difficult to perform manually. – Can use computer-based logic synthesis tool – Karnough Map (K-map) can be used for manual design of digital circuits. 9 Define Functions by Truth Tables ab c out minterms clauses 000 0 m0=a b c M0=a+b+c 001 1 m1=a b c M1=a+b+c • i in mi and Mi is 010 0 m2=a b c M2=a+b+c the decimal 011 1 m3=a b c M3=a+b+c value of abc (in binary) 100 0 m4=a b c M4=a+b+c • m M 101 1 m5=a b c M5=a+b+c i i 110 0 m6=a b c M6=a+b+c 111 0 m7=a b c M7=a+b+c • DNF f1 = a b c + a b c + a b c = m1 + m3 + m5 • CNF f2 = (a b c)(a b c)(a b c)(a b c)(a b c) = M0M2M4M6M7 After simplification: f1 = a c + b c; f2 = (a + b) c 10 5 9/16/2020 Circuit Minimization • Transistor count reduction AREA • Gate count reduction POWER • Gate level reduction DELAY (Speed) • Area reduction, power reduction and delay reduction improve designs 11 Logic Gates A Out In 0 1 NOT: 1 0 A B Out A 00 1 A B Out NAND: B 01 1 00 0 A 10 1 AND: 01 0 11 0 B 10 0 11 1 A A B Out A B Out NOR: A B 00 1 OR: 00 0 01 0 B 01 1 10 0 10 1 11 0 11 1 A B Out XOR: e): AND, OR, NOT, etc. IFF: A A B Out A 00 0 00 1 B 01 1 B 01 0 10 1 10 0 11 0 11 1 12 6 9/16/2020 Gate Level implementation • Two different implementations of the same function: DNF and CNF (+ for OR, * for AND, ’ for negation) • Both use 4 AND/OR gates and the level is 2. • Not all gates use the same number of transistors and power 13 Logic Optimizations f1 = AB + AC + AD + AE + A BC D E F = f2 = AB + AC + AD + AF + A BCDF • Finding common f1 = A( B + C + D + E )+ ABC DE F = subexpressions. f2 = A(B + C + D + F )+ A BC DF • Substituting one expression g1 = B + C + D by a symbol (may increase G = f = A( g + E )+ A E g formula level, i.e., high of 1 1 1 f = A ( g + F + F g1 ) the formula tree). 2 1 g = B + C + D • Find smaller equivalent 1 G = f = A g + A E + A E g formulas. 1 1 1 f2 = A Final circuit: 5 AND/OR gates; 3 levels 14 7 9/16/2020 General Approaches How does one find the smallest formula? – Manipulate algebraically until…? – Use Karnaugh maps (optimize visually) – Use a software optimizer For large circuits – Decomposition & reuse of building blocks 15 The K-Map Method • The truth table representation of a function is unique. • But, not the algebraic expression – Several versions of an algebraic expression exist. – Difficult to minimize algebraic functions manually. • The K-map method is a simple graphic tool to minimize functions in DNF/CNF. – Pictorial form of a truth table. – Also called Karnough Map. 16 8 9/16/2020 Two-Variable Map • Two variables • Four squares for four minterms • Squares’ positions are indexed by the values of variables • Figure (b) shows the relationship between the squares and the values of variables x and y. 17 Two-Variable Map (Cont.) • May only be useful to represent 16 Boolean functions. nd • 2 Example: m1=m2=m3=1 m1+m2+m3 = x’y+xy’+xy = = x’y + x(y’ + y) = x’y + x = x+y (OR function) 18 9 9/16/2020 Three-Variable Map • There are 8 minterms for 3 variables. • So, there are 8 squares, mi , each is a minterm. • mi: i is the decimal value of xyz. • Minterms are arranged not in a binary sequence, but in sequence similar to Gray code: Adjacent cells differ by one bit. 19 Three-Variable Map: Example • The square for square m5 corresponds to row 1 and column 01 (101). • Another look to m5 is m5 = xy’z. y’ is the negation of y. 20 10 9/16/2020 Adjacent Squares • Two adjacent squares differs by one bit (one is negated and the other is not). – So, they can be merged – Ex: m5+m7 = xy’z+xyz = xz(y’+y) = xz • Merge: cover as many adjacent squares as possible by a rectangle of size of power of two. 21 Adjacent Squares: Example 1 • Simplify the Boolean function F(x,y,z) = Σ(2,3,4,5) = m2 + m3 + m4 + m5 F(x,y,z) = x’y + xy’ 22 11 9/16/2020 Adjacent Squares • Far right squares are adjacent to far left squares (they don’t touch each other in K-map). – m0 is adjacent to m2 and m4 is adjacent to m6. – m0+m2 = x’y’z’+x’yz’ = x’z’(y’+y) = x’z’ – m4+m6 = xy’z’+xyz’ = xz’(y’+y) = xz’ 23 Simplify function: Example 2 F(x,y,z) = Σ(3,4,6,7) m3 + m7 = yz m4 + m6 = xz’ F(x,y,z) = yz+xz’ m6 + m7 = xy (redundant) 24 12 9/16/2020 Karnaugh Minimization Tricks ab c 00 01 11 10 Minterms can overlap 0 0 1 1 1 out = bc’ + ac’ + ab 1 0 0 1 0 ab c 00 01 11 10 0 1 1 1 1 Minterms can span 2, 4, 8 1 0 0 1 0 or more cells out = c’ + ab 25 Simplify function: Example 3 F(x,y,z) = Σ(0,2,4,5,6) F(x,y,z) = z’+xy’ 26 13 9/16/2020 Simplify function: Example 4 • F=A’C+A’B+AB’C+BC – Express F as a sum of minterms. • F(A,B,C) = Σ(1,2,3,5,7) – Find the minimal DNF • F = C+A’B 27 Square Numbering is not unique s1 s2 s3 s1 s2 s3 mi: i is the decimal value of s1s2s3. 28 14 9/16/2020 Square Numbering: Example 5 After optimization: f = s1s2 + s3 29 Four-Variable Map • 16 minterms (and squares) for 4 variables. 30 15 9/16/2020 Karnaugh Minimization Tricks ab cd 00 01 11 10 00 0 00 0 • The map wraps around 01 1 00 1 – out = 11 1 00 1 b’d 10 0 00 0 ab cd 00 01 11 10 00 1 0 0 1 – out = 01 0 0 0 0 b’d’ 11 0 0 0 0 10 1 0 0 1 31 Simplify Function: Example 6 f(x1, x2, x3, x4) = Σ(4,6,12,14) f(x1, x2, x3, x4) = x2x4’ 32 16 9/16/2020 CNF Simplification • Mark the empty squares by 0’s. • Combine them (as we did for sum-of- products) to get a simplified DNF. – We obtain F’ (complement of the function) • Take the dual of F’ (a DNF) to obtain F (a CNF). 33 CNF Simplification: Example 7 F(A,B,C,D) = Σ(0,1,2,5,8,9,10) F = B’D’+B’C’+A’C’D F’=AB+CD+BD’ dual(F’) = F = (A’+B’)(C’+D’)(B’+D) 34 17 9/16/2020 CNF (The POS Expression) f(x1, x2, x3) = M1 M3 f(x1, x2, x3) = ( x1 + x2 + x3)( x1 + x2 + x3) = ( x1 + x3 + x2)( x1 + x3 + x2) = ( x1 + x3 )( x2 + x2 ) = ( x1 + x3 ) 35 CNF Simplification f’(x1, x2, x3) = m1 + m3 = x1’ x2’ x3 + x1’ x2 x3 = x1’ x3 dual(f’) = f = x1+ x3’ 36 18 9/16/2020 Alternative Solution Using K-Maps 1 1 1 1 0 0 1 1 dual(f’) = f = x1+ x3’ ( x1 + x3’ ) This is both CNF and DNF!!! 37 CMOS Implementation • CMOS: Complementary Metal– Oxide–Semiconductor, an advanced and popular technology for implementing logic circuits.