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Introduction to SAT, Predicate Logic and DPLL solving Lecture 1

Introduction to SAT, Predicate Logic and DPLL solving Lecture 1. Nikolaj Bjørner Microsoft Research DTU Winter course January 2 nd 2012 Organized by Flemming Nielson & Hanne Riis Nielson. Plan. General overview of propositional and predicate logic Refresher on SAT and modern DPLL.

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Introduction to SAT, Predicate Logic and DPLL solving Lecture 1

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  1. Introduction to SAT, Predicate Logic and DPLL solving Lecture 1 Nikolaj Bjørner Microsoft Research DTU Winter course January 2nd2012 Organized by Flemming Nielson & Hanne Riis Nielson

  2. Plan • General overview of propositional and predicate logic • Refresher on SAT and modern DPLL

  3. Takeaways: • The syntax and semantics of propositional and predicate logic. • Algorithmic principles of modern SAT solvers: • DPLL (actually DLL) algorithm • Conflict Directed Clause Learning (CDCL) • Two-watch literal indexing

  4. What is logic? • Logic is the art and science of effective reasoning. • How can we draw general and reliable conclusions from a collection of facts? • Formal logic: Precise, syntactic characterizations of well-formed expressions and valid deductions. • Formal logic makes it possible to calculate consequences at the symbolic level. • Computers can be used to automate such symbolic calculations.

  5. What is logic? • Logic studies the relationship between language, meaning, and (proof) method. • A logic consists of a language in which (well-formed) sentences are expressed. • A semantic that distinguishes the valid sentences from the refutable ones. • A proof system for constructing arguments justifying valid sentences. • Examples of logics include propositional logic, equational logic, first-order logic, higher-order logic, and modal logics.

  6. What is logical language? • A language consists of logical symbols whose interpretations are fixed, and non-logical ones whose interpretations vary. • These symbols are combined together to form well-formed formulas. • In propositional logic PL, the connectives , , and have a fixed interpretation, whereas the constants p, q, rmay be interpreted at will.

  7. SMT : Basic Architecture • Equality + UF • Arithmetic • Bit-vectors • … Case Analysis

  8. Propositional Logic Formulas:  := p | 12| 12|1 | 1 2 Examples: p  q  q  p p  q  (p  q) We say pand qare propositional variables.

  9. Interpretation An interpretation M assigns values {true, false} to propositional variables. Let F and G range over PL formulas.

  10. Satisfiability & Validity • A formula is satisfiable if it has an interpretation that makes it logically true. • In this case, we say the interpretation is a model. • A formula is unsatisfiable if it does not have any model. • A formula is valid if it is logically true in any interpretation. • A propositional formula is valid if and only if its negation is unsatisfiable.

  11. Satisfiability & Validity: examples p  q  q  p p  q  q p  q  (p  q)

  12. Satisfiability & Validity: examples p  q  q  p VALID p  q  q SATISFIABLE p  q  (p  q) UNSATISFIABLE

  13. Equivalence We say two formulas F and G are equivalent if and only if they evaluate to the same value (true or false) in every interpretation

  14. Equisatisfiable We say formulas A and B are equisatisfiable if and only if A is satisfiable if and only if B is. During this tutorial, we will describe transformations that preserve equivalence and equisatisfiability.

  15. Normal Forms

  16. Normal Forms NNF? (p  q)  (q  (r  p))

  17. Normal Forms NNF? NO (p  q)  (q  (r  p))

  18. Normal Forms NNF? NO (p  q)  (q  (r  p))

  19. Normal Forms NNF? NO (p  q)  (q  (r  p))  (p  q)  (q  (r  p))

  20. Normal Forms NNF? NO (p  q)  (q  (r  p))  (p  q)  (q  (r  p))  (p  q)  (q  (r  p))

  21. Normal Forms CNF? ((p  s) (q r))  (q  p s)  (r s)

  22. Normal Forms CNF? NO ((p  s) (q  r))  (q  p  s)  (r  s)

  23. Normal Forms CNF? NO ((p  s) (q  r))  (q  p  s)  (r  s) Distributivity 1. A(BC)  (AB)(AC) 2. A(BC)  (AB)(AC)

  24. Normal Forms CNF? NO ((p  s) (q  r))  (q  p  s)  (r  s)  ((p  s) q)) ((p  s) r))  (q  p  s)  (r  s) Distributivity 1. A(BC)  (AB)(AC) 2. A(BC)  (AB)(AC)

  25. Normal Forms CNF? NO ((p  s) (q  r))  (q  p  s)  (r  s)  ((p  s) q)) ((p  s) r))  (q  p  s)  (r  s)  (p  q)  (s q) ((p  s) r))  (q  p  s)  (r  s) Distributivity 1. A(BC)  (AB)(AC) 2. A(BC)  (AB)(AC)

  26. Normal Forms CNF? NO ((p  s) (q  r))  (q  p  s)  (r  s)  ((p  s) q)) ((p  s) r))  (q  p  s)  (r  s)  (p  q)  (s q) ((p  s) r))  (q  p  s)  (r  s)  (p  q)  (s  q)  (p  r) (s  r)  (q  p  s)  (r  s)

  27. Normal Forms DNF? p  (p  q)  (q  r)

  28. Normal Forms DNF? NO, actually this formula is in CNF p  (p  q)  (q  r)

  29. Normal Forms DNF? NO, actually this formula is in CNF p  (p  q)  (q  r) Distributivity 1. A(BC)  (AB)(AC) 2. A(BC)  (AB)(AC)

  30. Normal Forms DNF? NO, actually this formula is in CNF p  (p  q)  (q  r)  ((p  p) (p q))  (q  r) Distributivity 1. A(BC)  (AB)(AC) 2. A(BC)  (AB)(AC)

  31. Normal Forms DNF? NO, actually this formula is in CNF p  (p  q)  (q  r)  ((p  p) (p q))  (q  r)  (p  q)  (q  r) Distributivity 1. A(BC)  (AB)(AC) 2. A(BC)  (AB)(AC) Other Rules AA   A  A

  32. Normal Forms DNF? NO, actually this formula is in CNF p  (p  q)  (q  r)  ((p  p) (p q))  (q  r)  (p  q)  (q  r)  ((p  q) q) ((p  q)  r) Distributivity 1. A(BC)  (AB)(AC) 2. A(BC)  (AB)(AC) Other Rules AA   A  A

  33. Efficient CNF Translation

  34. CNF translation (example)

  35. Resolution

  36. Resolution (example)

  37. Unit & Input Resolution

  38. DPLL DPLL

  39. Pure Literals A literal is pure if only occurs positively or negatively.

  40. Pure Literals A literal is pure if only occurs positively or negatively.

  41. DPLL (as a procedure)

  42. M | F DPLL Partial model Set of clauses

  43. Guessing DPLL • p | p  q, q  r p, q | p  q, q  r

  44. Deducing DPLL • p | p  q, p  s p, s| p  q, p  s

  45. Backtracking DPLL • p, s, q | p  q, s  q, p q p, s| p  q, s  q, p q

  46. Efficient indexing (two-watch literal) Non-chronological backtracking (backjumping) Lemma learning Modern DPLL

  47. Modern DPLL in a nutshell Adapted and modified from [Nieuwenhuis, Oliveras, Tinelli J.ACM 06]

  48. Lemma learning CDCL – Conflict Directed Clause Learning • t, p, q, s | t  p  q, q  s, p s • t, p, q, s | t  p  q, q  s, p s |p s • t, p, q, s | t  p  q, q  s, p s |p q • t, p, q, s | t  p  q, q  s, p s |p t

  49. Two-watch literals • Indexing is key to efficient theorem proving. • An index is a dictionary that is tuned to search. • The literal two-watch scheme is the main index in modern SAT solvers. • Track two literals per clause, such that: • Track literals that are unassigned or assigned to true. • If one of the tracked literals is assigned to false, search for non-tracked literal that is either unassigned or true. • If there is no other unassigned or true literal to select, then the other watched literal can be assigned to true. • Unit propagation or conflict detection.

  50. Two-watch literals • Watch literals for Propagate and Conflict • Naïve: For every literal lmaintain map: • Watch(l) = {C1 … Cm} where l  Ci • If l is assigned to true, check each Cj Watch(l)for Conflict or Propagate • But most of the time, some other literal inCj is either: • Unassigned (not yet assigned) • Assigned to true.

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