Canonical Computation without Canonical Data Structure Alan Mishchenko Department of EECS, UC Berkeley

Overview Motivation Definition of canonicity Fundamentals of SAT solving Canonicity in SAT-based computations for satisfiable calls (LEXSAT) for unsatisfiable calls (LEXUNSAT) Experiments Conclusion 2

Motivation Complementary data structures: BDDs vs SAT BDDs SAT trading space for time BDDs canonical, easy to use but difficult to construct (can mem out) SAT non-canonical, easy to construct but difficult to use (can time out) However, SAT is “better” in most cases Can we improve SAT by borrowing from BDDs? How about canonicity? - Yes, we can!

What Is Canonicity? Canonical representation Canonical computation Representing Boolean functions in a unique way for a given Boolean function with a given variable order, only one representation is possible Canonical computation Computing Boolean functions in a unique way for a given Boolean function with a given variable order, only one result of computation is possible In the BDD world, we did not distinguish the two In the SAT world, we cannot have both However, we can have canonical computation!

SAT in Practical Applications Netlist Answer: “SAT” or “UNSAT” CNF SAT solver CNF generator CNF Design constraints If SAT, a counter-example If UNSAT, a core User cost functions Both counter-examples and cores are useful in SAT-based applications. In practice, cores are often represented as subsets of assumptions that make the problem UNSAT.

Incremental SAT Initial CNF Round 1: SAT solver Initial assumptions Additional CNF Round 2: SAT solver New assumptions Additional CNF Round 3: SAT solver New assumptions Assumptions are CNF clauses used only in the current round – they are handled differently from the rest of CNF clauses.

Proposed Modification to SAT Traditional: Input: CNF, assumptions Output: (1) satisfying assignment or (2) UNSAT core, that is, a subset of literals that make the instance unsatisfiable Proposed: Input: CNF, assumptions, variable order (1) canonical satisfying assignment or (2) canonical UNSAT core, that is, subset of literals that make the instance unsatisfiable

The Main Idea Canonicity is achieved by adopting a variable order Now all satisfying assignments can be compared, and the smallest one can be returned Similarly, all UNSAT cores can be compared, and the smallest one can be returned

Choosing The Smallest Since we have a variable order, we order all assignments (cores) using this order, and take the first one in the list All satisfying assignments: 0000101101  the smallest one 0001111001 … 0111100111 Fortunately, there is no need to compute all assignments in order to find the minimum one

LEXSAT Input: cnf F Output: the smallest satisfying assignment as literals in array A array LEXSAT( cnf F ) { Initialize array A to have all negative literals in the given order; for ( i = 0; i <|A|; i++ ) { if ( F is UNSAT under assumptions A[0] through A[i] ) invert the polarity of literal A[i] to be positive; } return A;

LEXUNSAT Input: cnf F Output: the smallest satisfying assignment as literals in array A array LEXUNSAT( cnf F ) { Initialize array A to have all positive literals in the given order; for ( i = 0; i <|A|; i++ ) { if ( F is UNSAT under assumptions in A without A[i] ) invert the polarity of literal A[i] to be negative; } return A;

Applications of LEX{SAT,UNSAT} Canonical SAT-based ISOP (similar to BDD-based ISOP) useful in collapsing/refactoring, timing-driven optimization, etc Computing minimal supports in node minimization, resubstitution, Boolean decomposition, ECO Diversifying SAT assignments useful in both logic synthesis and formal verification Approximate computing, SMT solving, etc

ISOP Computations Compared BDDovo BDD-based ISOP for the original variable order without dynamic variable reordering BDDdvr BDD-based ISOP for the original variable order with dynamic variable reordering SATovo SAT-based ISOP with LEXSAT and LEXUNSAT for the original variable order SATrvo SAT-based ISOP with LEXSAT and LEXUNSAT for a random variable order

Computing Canonical ISOP

Counter-Example Minimization

Conclusion Reviewed BDDs and SAT distinguished canonical representation and canonical computation Showed that SAT is capable of canonical computations discussed two algorithms (LEXSAT and LEXUNSAT) Listed several practical applications most of them in logic synthesis

Abstract A computation is canonical if the result depends only on the Boolean function and a selected variable order, and does not depend on how the function is represented and how the computation is implemented. In the context of Boolean satisfiability (SAT), canonicity implies that the result (a satisfying assignment for satisfiable instances and a UNSAT core for unsatisfiable ones) does not depend on the circuit structure, CNF generation algorithm, and the SAT solver used. The main highlight of this paper is that all SAT-based computations can be made canonical without building a canonical data-structure. The runtime overhead for inducing canonicity is relatively small and is often justifies by the uniqueness and the improved quality of results.