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Advanced Decision Methods for Symbolic Reasoning: A Summer School Overview

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This overview discusses decision methods for symbolic reasoning, emphasizing the importance of logic in computer science, as highlighted in the summer school conducted by Leonardo de Moura at Microsoft Research. Key concepts like SAT, CNF, clauses, literals, and resolution procedures are explored, showcasing techniques for conflict resolution, model finding, and proof generation. Additionally, the significance of finite basis theories, including those related to uninterpreted functions and arrays, is addressed. The complexities of conflict-driven clause learning (CDCL) and DPLL algorithms are also examined, along with their application in software analysis and verification.

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Advanced Decision Methods for Symbolic Reasoning: A Summer School Overview

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  1. Decision methods for arithmeticThird summer school on formal methods Leonardo de Moura Microsoft Research

  2. Symbolic Reasoning Software analysis/verification tools need some form of symbolic reasoning Logic is “The Calculus of Computer Science” Zohar Manna

  3. Saturation x Search Proof-finding Model-finding Proofs Conflict Resolution Models

  4. SAT CNF is a set (conjunction) set of clauses Clause is a disjunction of literals Literal is an atom or the negation of an atom

  5. Two procedures

  6. Resolution Improvements Delete tautologies Ordered Resolution Subsumption (delete redundant clauses) …

  7. Resolution: Example

  8. Resolution: Example

  9. Resolution: Example

  10. Resolution: Example

  11. Resolution: Example

  12. Resolution: Problem Exponential time and space

  13. Unit Resolution subsumes Complete for Horn Clauses

  14. DPLL Split rule DPLL = Unit Resolution + Split rule

  15. DPLL

  16. DPLL

  17. DPLL

  18. DPLL

  19. DPLL

  20. DPLL

  21. CDCL: Conflict Driven Clause Learning DPLL Conflict Resolution Resolution Model Proof

  22. MCSat

  23. MCSat Propagations

  24. MCSat Propagations

  25. MCSat Propagations

  26. MCSat Decisions

  27. MCSat Model Assignments

  28. MCSat Model Assignments We can’t falsify any fact in the trail.

  29. MCSat Conflict We can’t find a value of s.t.

  30. MCSat Conflict Learning that = 2) is not productive We can’t find a value of s.t.

  31. Conflict

  32. MCSat

  33. MCSat Conflict

  34. MCSat Learned by resolution

  35. MCSat

  36. MCSat – Finite Basis Every theory that admits quantifier elimination has a finite basis (given a fixed assignment order)

  37. MCSat – Finite Basis

  38. MCSat – Finite Basis

  39. MCSat – Finite Basis

  40. MCSat – Finite Basis

  41. MCSat – Finite Basis Every “finite” theory has a finite basis

  42. MCSat – Finite Basis Theory of uninterpreted functions has a finite basis Theory of arrays has a finite basis [Brummayer- Biere 2009] In both cases the Finite Basis is essentially composed of equalities between existing terms.

  43. MCSat: Termination Propagations Decisions Model Assignments

  44. MCSat Propagations Decisions Model Assignments

  45. MCSat Propagations Decisions Model Assignments

  46. MCSat Maximal Elements …

  47. Conflict

  48. Conflict

  49. Conflict

  50. MCSat

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