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Formal verification : SAT

SAT applied in equivalence checking , but also in property checking. Formal verification : SAT. Satisfiability aka SAT. SAT: Boolean function is satisfiable if there exists a variable assignment for which the function is TRUE. Satisfiability aka SAT. SAT is transformed to CNF

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Formal verification : SAT

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  1. SAT applied in equivalence checking, but also in property checking Formal verification: SAT Digitaalsüsteemide verifitseerimise kursus

  2. Satisfiability aka SAT • SAT: Boolean function is satisfiable if there exists a variable assignment for which the function is TRUE Digitaalsüsteemide verifitseerimise kursus

  3. Satisfiability aka SAT • SATis transformed to CNF (i.e. product of sums). • Sums are called terms. • If terms have max 2 literals then 2-SAT  2-SAT solved in a polynomial time • 3-SAT is an NP complete task Digitaalsüsteemide verifitseerimise kursus

  4. Satisfiability aka SAT • Is this CNF satisfiable? • Yes: a = 1, b = 0, c = 0! • Worst case: 2ncombinations to try Digitaalsüsteemide verifitseerimise kursus

  5. Some terminology • If x in the formula alwaysin one phase (i.e. always inverted or always noninverted then x is unate. • If x in the formula in both phases then x is binate. • Term having just one literal called unit term. Digitaalsüsteemide verifitseerimise kursus

  6. Resolvent-algorithm • Resolvent: f = (x+A)(¬x+B) = (x +A)(¬x+B)(A+B) • Consensus: f = xC + ¬xD = xC + ¬xD + CD • Since SAT is in CNF we use resolvent. Digitaalsüsteemide verifitseerimise kursus

  7. Resolvent-algorithm • Choose another variable x. • If xis unate, apply unate rule. • If xis unit term, apply unit term rule. • If xis binate, solve resolvent of x. • Repeat the steps until all resolvents solved. • If the result is 1, then function satisfiable; otherwise not satisfiable (unit term). Digitaalsüsteemide verifitseerimise kursus

  8. a binate terms resolvent solved Resolvent-algorithm example Digitaalsüsteemide verifitseerimise kursus

  9. Resolvent-algorithm: summary • Resolvent-algorithm mathematically elegant but... • ... Designed for small SAT problems • In the worst case 2n resolvents to solve • In order to solve complex SAT instances, search based algorithms needed Digitaalsüsteemide verifitseerimise kursus

  10. Search-based SAT Digitaalsüsteemide verifitseerimise kursus

  11. Implication Graph • Directed acyclic graph: • Nodes labeled by variable names, followed by the rank of the decision • Variables preceded by minus were assigned 0, not preceded by minus were assigned 1 • Directed arcs show from which assignments what new assignments imply • Decision nodes (grey) and implication nodes (white) Digitaalsüsteemide verifitseerimise kursus

  12. Implication Graph • decisions: k = 1, j = 1, a = 0, b = 1. • reach a conflict: x = 1 ja x = 0! • learning: add a new term (¬e + h + ¬d) Digitaalsüsteemide verifitseerimise kursus

  13. Implication Graph Example Since e is a unit term then e = 1; first decision: a = 1 It implies that c = 0; the function is simplified: If we choose b=1, then conflict! Two possibilities to handle this: 1) Invert the last decision (backtrack) 2) Add a new term (learning): Digitaalsüsteemide verifitseerimise kursus

  14. Equivalence Checking • Two principal approaches: • Transform implementation and spec (reference implementation) to a canonical form • Search for an input assignment that would distinguish the responses of the implementation and reference implementation. • SATapplied for the latter Digitaalsüsteemide verifitseerimise kursus

  15. Equivalence checking with SAT • Equivalence Checking can be reduced to SAT: d = f g • Iff dis satisfiable then fand gare NOT equivalent. Digitaalsüsteemide verifitseerimise kursus

  16. Specification (reference implementation) + + 1 Implementation Miter circuit SAT? Digitaalsüsteemide verifitseerimise kursus

  17. SATfor schematics: characteristic equation • Build CNFs corresponding to logic gates using logic implication: • ab = ¬a + b Digitaalsüsteemide verifitseerimise kursus

  18. a c & b SATfor schematics: characteristic equation • Implications for describing the AND gate: ¬a¬c & ¬b ¬c & ¬c  ¬a  ¬b • Characteristic equation for AND in CNF: (a+ ¬c) (b+ ¬c) (c+ ¬a+ ¬b) Digitaalsüsteemide verifitseerimise kursus

  19. a c 1 b SATfor schematics: characteristic equation • Implications for describing the OR-gate: ac & b c & c  a  b • Characteristic equation for OR in CNF: (¬a + c) (¬b + c) (¬c + a + b) Digitaalsüsteemide verifitseerimise kursus

  20. a d & b f 1 e c SATfor schematics: characteristic equation Characteristic equation for a schematic: (a+¬d)(b+¬d)(d+¬a+¬b)(¬c+¬e)(c+e)(¬d+f)(¬e+f)(¬f+d+e) Digitaalsüsteemide verifitseerimise kursus

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