Computation Engines: BDDs and SAT (part 2)

Computation Engines: BDDs and SAT(part 2) 290N: The Unknown Component Problem Lecture 8

Outline • Characteristic function • Representation of MV relations, automata, and FSMs • Case study: Equivalence checking • Combinational and sequential • BDDs vs SAT • Combining the two • Other representations • Truth table • Sums-of-products (SOPs) • Historical perspective

Characteristic Function • Definition • Manipulation of sets • MV relations as sets of I/O combinations • Examples • representing an FSM • representing an automaton

Characteristic Function • Boolean function F: {0,1}n {0,1} represents a set SF of minterms x  {0,1}n, such that F(x)=1 • Conversely, elements of any set S can be encoded using minterms x  {0,1}nand represented by a Boolean function, which takes value 1 in these minterms, and 0 otherwise. • This Boolean function is called characteristic function of the set S. s: {0,1}n {0,1}, such that s(x) = 1 iff x represents some element s  S. • the codes should be disjoint for all elements of S • the size of Boolean space of x cannot be smaller than n=log2|S|

Set Manipulation • Operations on sets are reduced to Boolean operations on the characteristic functions: • Empty set: = 0 • Union of sets:S  T= S+ T • Intersection of sets:S  T= S& T • Set different:S - T= S& T’ • Containment of sets: (ST)= 1

Multi-Valued Relations is an MV relation, where are finite sets of values. The variables are multi-valued variables which can take on any value in is a multi-output MV relation if R is binary and the are treated as outputs. This relation is between a vector of inputs and a vector of outputs:

1 1 2 0 0 2 2 0 0 2 0 2 1 3 1 Normal Simulation (NS) POs • PI/PO relation contains • 3 1 1 / 2 1 0 node with a non-deterministic relation {0,2} 0 2 fanins

Relations as Sets of I/O Combinations • As sets, relations can be represented and manipulated using their characteristic functions • One way of encoding the I/O combinations belonging to a relation, is to encode each variable and then to combine the individual variable codes into a single code of a combination • For example, in the previous example, assuming all the variables to be 4-valued and natural binary encoding: Combinations: 3 1 1 / 2 1 0 3 1 1 / 0 1 2 Codes: 11 01 01 / 10 01 00 11 01 01 / 00 01 10

Relations Representing an FSM • FSM is { I, O, S, S0, , } • Transition relation is T(i, s, s’) : I  S  S {0,1} T(i, s, s’) = 1 iff state s’ is reached in one transition from state s under input i • Output relation is F(i, s, o): I  S  O {0,1} F(i, s, o) = 1 iff output o is produced in state s under input i • Total transition/output relation of the FSM is TF(i, s, s’, o): I  S  S  O  {0,1} TF(i, s, s’, o) = 1 iff state s’ and output o can be produced in state s under input i

10 x y 00 11 DFF 0 1 01 Example of an FSM .i 2 .o 0 .s 2 .p 4 .ilb x y .ob .accepting 0 1 .names x y cs ns 00 0 0 10 0 1 01 1 0 11 1 1 .e

Relation Representing an Automaton • Automaton is { I, S, S0, , Q} • Transition relation is T(i, s, s’) : I  S  S {0,1} T(i, s, s’) = 1 iff state s’ is reached in one transition from state s under input i • Output relation (acceptance function) is F(s): S  {0,1} F(s) = 1 iff state s is accepting

Ins CS code NS code 0 A 00 B 10 0,1 A 00 A 00 0 B 10 B 10 1 B 10 A 00 0 C 01 B 10 1 C 01 A 00 Example of an Automaton C 1 0 0 B A 1 0 0,1

Example (continued) Transition relation T(I,a1,a2,b1,b2) = i'a1‘a2‘b1b2‘+ a1‘a2'b1'b2' + i'a1a2'b1b2' + ia1a2'b1'b2' + i'a1a2'b1b2' + ia1a2'b1'b2' i a1 b1 a2 b2 0 1

Case Study: Equivalence Checking • Combinational • Sequential • Equivalence checking as search • BDDs vs SAT • Combination of the two

a C1 O1 b c O a C2 b c O2 Combinational Equivalence Checking • Given two combinational circuits, C1 and C2 • Construct the Miter circuit • The output of the Miter circuit is 1 iff the two circuits are different • Prove that the output of the Miter circuit is always 0

a M1 O1 b c O a M2 b c O2 Sequential Equivalence Checking • Given two sequential circuits, C1 and C2 • Construct the product circuit (FSM) • The output of the product is 1 iff the two are different • Find the reachable states of the product circuit (FSM) • If there exist a state with the output 1, they are not equivalent

Verification as Search • Verification problem is reduced to a search problem • find an assignment of input variables (or a sequence of such assignments), which leads to a 1 at the output • If we finished exploring the search space and cannot find such an assignment, the circuits are equivalent

BDD package Builds the canonical representation of all branches up to a point Tends to run out of memory SAT solver Explores one branch at a time Tends to run out of time BDDs vs SAT for Search • Both exhaustively explore the search space

Search Problem

Different Ways of Exploring Search Space BDD approach SAT approach    

Combining BDDs and SAT • Build the BDDs for the nodes until the size reaches a limit • Use SAT to prove equivalence of pairs of cut-points using a timeout • Iterate the above two steps, while increasing the size limit and the timeout until the problem is solved • BDDs and SAT attack the problem using their comlementary strengths in a balanced manner   A. Kuehlmann, V. Paruthi, F. Krohm, M. K. Ganai, “Robust Boolean reasoning for equivalence checking and functional property verification”, IEEE Trans. CAD, Vol. 21, No. 12, Dec. 2002, pp. 1377-1394.

Other Representations • Truth table • Implemented using bit strings • Convenient for functions up to 5 variables • Useful for functions up to 8 variables • Sums-of-products • Cubes are represented in positional notation • Implemented using bit strings • The main data structure to represent SOPs in Espresso and SIS • Common features • Are explicit in nature • Exploit bit parallelism of modern computers • Have been traditionally used in many applications

Use of Functional RepresentationsHistorical Perspective Problem Size +CNF Truth table +BDD +SOP 1950-1970 1980 1990 2000 Time Period

Computation Engines: BDDs and SAT (part 2)