Boolean Functions and their Representations ECE 697B (667) Spring 2006 Synthesis and Verification of Digital Circuits Boolean Functions and their Representations Slides adopted (with permission) from A. Kuehlmann, UC Berkeley 2003 ECE 667

The Boolean Space Bn B = { 0,1}, B2 = {0,1} X {0,1} = {00, 01, 10, 11} Karnaugh Maps: Boolean Cubes: B0 B1 B2 B3 B4 ECE 667 - Synthesis & Verification - Lecture 9a

Boolean Functions x2 x1 ECE 667 - Synthesis & Verification - Lecture 9a

Boolean Functions Literal x1 represents the logic function f, where f = {x| x1 = 1} Literal x1 represents the logic function g where g = {x| x1 = 0} x1 x3 x2 f = x1 x1 x2 x3 f = x1 Notation: x’ = x ECE 667 - Synthesis & Verification - Lecture 9a

Set of Boolean Functions Truth Table or Function Table: x1x2x3 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0  1 1 0 1 0 1 1 0 1 1 1 1 0 x3 x2 x1 There are 2n vertices in input space Bn There are 22n distinct logic functions. Each subset of vertices is a distinct logic function: f  Bn ECE 667 - Synthesis & Verification - Lecture 9a

Boolean Operations - AND, OR, COMPLEMENT Given two Boolean functions: f : Bn  B g : Bn  B AND operation f × g = {x | f(x)=1 Ù g(x)=1} The OR operation f + g = {x | f(x)=1 Ú g(x)=1} The COMPLEMENT operation (^f or f’ ) f’ = {x | f(x) = 0} ECE 667 - Synthesis & Verification - Lecture 9a

Cofactor and Quantification Given a Boolean function: f : Bn  B, with the input variables (x1,x2,…,xi,…,xn) Positive Cofactor of function f w.r.t variable xi fxi = {x | f(x1,x2,…,1,…,xn)=1} Negative Cofactor of f w.r.t variable xi fxi’ = {x | f(x1,x2,…,0,…,xn)=1} Existential Quantification of function f w.r.t variable xi, $xi f = {x | f(x1,x2,…,0,…,xn)=1 Ú f(x1,x2,…,1,…,xn)=1} Universal Quantification of function f w.r.t variable xi, "xi f = {x | f(x1,x2,…,0,…,xn)=1 Ù f(x1,x2,…,1,…,xn)=1} ECE 667 - Synthesis & Verification - Lecture 9a

Representations of Boolean Functions We need representations for Boolean Functions for two reasons: to represent and manipulate the actual circuit we are “synthesizing” as mechanism to do efficient Boolean reasoning Forms to represent Boolean Functions Truth table List of cubes: Sum of Products, Disjunctive Normal Form (DNF) List of conjuncts: Product of Sums, Conjunctive Normal Form (CNF) Boolean formula Binary Decision Tree, Binary Decision Diagram Circuit (network of Boolean primitives) Canonicity – which forms are canonical? ECE 667 - Synthesis & Verification - Lecture 9a

Truth Table Truth table (Function Table): ab’c’d + ab’cd + abc’d + The truth table of a function f : Bn  B is a tabulation of its values at each of the 2n vertices of Bn. (all mintems) Example: f = a’b’c’d + a’b’cd + a’bc’d + ab’c’d + ab’cd + abc’d + abcd’ + abcd (Notation for complement: a’ = a ) The truth table representation is - intractable for large n - canonical Canonical means that if two functions are the same, then the canonical representations of each are isomorphic (identical). abcd f 0 0000 0 1 0001 1 2 0010 0 3 0011 1 4 0100 0 5 0101 1 6 0110 0 7 0111 0 8 1000 0 9 1001 1 10 1010 0 11 1011 1 12 1100 0 13 1101 1 14 1110 1 15 1111 1 ECE 667 - Synthesis & Verification - Lecture 9a

Boolean Formula A Boolean formula is defined as an expression with the following syntax: formula ::= ‘(‘ formula ‘)’ | <variable> | formula “+” formula (OR operator) | formula “×” formula (AND operator) | ^ formula (complement) Example: f = (x1×x2) + (x3) + ^^(x4 × (^x1)) typically the “×” is omitted and the ‘(‘ and ‘^’ are simply reduced by priority, e.g. f = x1x2 + x3 + x4^x1 ECE 667 - Synthesis & Verification - Lecture 9a

Cubes A cube is defined as the product (AND) of a set of literal functions (“conjunction” of literals). Example: C = x1x’2x3 represents the following function f = (x1=1)(x2=0)(x3=1) x1 x2 x3 c = x1 f = x1x2 f = x1x2x3 ECE 667 - Synthesis & Verification - Lecture 9a

Cubes If C  f, C a cube, then C is an implicant of f. If C  Bn, and C has k literals, then |C| covers 2n-k vertices. Example: C = xy  B3 k = 2 , n = 3 => |C| = 2 = 23-2. C = {100, 101} In an n-dimensional Boolean space Bn, an implicant with n literals is a minterm. ECE 667 - Synthesis & Verification - Lecture 9a

List of Cubes Sum of Products (SOP) F = {ab, ac, bc} A function can be represented by a sum of cubes (products): f = ab + ac + bc Since each cube is a product of literals, this is a “sum of products” (SOP) representation A SOP can be thought of as a set of cubes F F = {ab, ac, bc} A set of cubes that represents f is called a cover of f. F1={ab, ac, bc} and F2={abc, abc, abc, abc} are covers of f = ab + ac + bc. ECE 667 - Synthesis & Verification - Lecture 9a

Binary Decision Diagram (BDD) f = ab+a’c+a’bd Graph representation of a Boolean function - vertices represent decision nodes for variables - two children represent the two subfunctions f(x = 0) and f(x = 1) (cofactors) - restrictions on ordering and reduction rules can make a BDD representation canonical c a b d 1 c+bd root node c+d 1 ECE 667 - Synthesis & Verification - Lecture 9a

Boolean Networks Used for two main purposes as representation for Boolean reasoning engine as target structure for logic implementation which gets restructured in a series of logic synthesis steps until result is acceptable Efficient representation for most Boolean problems memory complexity is same as the size of circuits we are actually building Close to input representation and output representation in logic synthesis ECE 667 - Synthesis & Verification - Lecture 9a

Definitions – fanin, fanout, support A Boolean circuit is a directed graph C(G,N) where G are the gates and N Í G´G is the set of directed edges (nets) connecting the gates. Some of the vertices are designated: Inputs: I Í G Outputs: O Í G, I Ç O = Æ Each gate g is assigned a Boolean function fg which computes the output of the gate in terms of its inputs. ECE 667 - Synthesis & Verification - Lecture 9a

Definitions – fanin, fanout, support The fanin FI(g) of a gate g are all predecessor vertices of g: FI(g) = {g’ | (g’,g) Î N} The fanout FO(g) of a gate g are all successor vertices of g: FO(g) = {g’ | (g,g’) Î N} The cone CONE(g) of a gate g is the transitive fanin of g and g itself. The support SUPPORT(g) of a gate g are all inputs in its cone: SUPPORT(g) = CONE(g) Ç I ECE 667 - Synthesis & Verification - Lecture 9a

Example – Boolean Network 6 1 5 3 4 7 8 9 2 O I FI(6) = {2,4} FO(6) = {7,9} CONE(6) = {1,2,4,6} SUPPORT(6) = {1,2} Nodes = logic functions of arbitrary complexity ECE 667 - Synthesis & Verification - Lecture 9a

Boolean Network Representations Vertices can have arbitrary number of inputs and outputs typically single-output functions are used Vertices can represent any Boolean function stored in different ways, such as: other circuits (hierarchical representation) truth tables or cube representation Boolean expressions read from a library description BDDs, AIGs, etc. Data structure allow very general mechanisms for insertion and deletion of vertices, pins (connections to vertices), and nets ECE 667 - Synthesis & Verification - Lecture 9a

AND-INVERTER Circuits Base data structure uses two-input AND function for vertices and INVERTER attributes at the edges (individual bit) use De’Morgan’s law to convert OR operation etc. Hash table to identify and reuse structurally isomorphic circuits f g g f Means complement ECE 667 - Synthesis & Verification - Lecture 9a

Data Representation Vertex: Edge: pointers (integer indices) to left and right child and fanout vertices collision chain pointer other data Edge: pointer or index into array one bit to represent inversion Global hash table holds each vertex to identify isomorphic structures Garbage collection to regularly free un-referenced vertices ECE 667 - Synthesis & Verification - Lecture 9a

Data Representation Hash Table one 8456 6423 …. ... …. 0455 0456 0456 Constant One Vertex zero left 0457 1 right ... 1345 …. next 7463 …. fanout hash value left pointer complement bits right pointer 0456 left next in collision chain right array of fanout pointers next fanout ECE 667 - Synthesis & Verification - Lecture 9a