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A framework for eager encoding

A framework for eager encoding. Daniel Kroening ETH, Switzerland Ofer Strichman Technion, Israel. (Executive summary) (submitted to: Formal Aspects of Computing). A generic framework for reducing decidable logics to propositional logic (beyond NP).

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A framework for eager encoding

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  1. A framework for eager encoding Daniel Kroening ETH, Switzerland Ofer Strichman Technion, Israel (Executive summary) (submitted to: Formal Aspects of Computing)

  2. A generic framework for reducing decidable logics to propositional logic (beyond NP). • Instantiating the framework for a specific logic L, requires a deductive system for Lthat meets several criteria. • Linear arithmetic, EUF, arrays etc all have it.

  3. A proof rule: • A proof step: (Rule, Antecedent, Proposition) • Definition(Proof-step Constraint): let A1…Ak be the Antecedents and p the Proposition of step. Then: Boolean encoding

  4. PC(P) • A proofP =(s1,…, sn) is a set of Proof Steps, • …in which the Antecedence relation is acyclic • The ProofConstraintc(P) induced by P is the conjunction of the constraints induced by its steps:

  5. Propositional skeleton: • Theorem1: For every formula  and any sound proof P,  is satisfiable )skÆ c(P) is satisfiable.

  6. Complete proofs • Definition (Complete proofs): A proof P is called complete with respect to  if

  7. Sufficient condition for completeness #1 • Notation: A – assumption, B – a proposition. denotes: P proves B from A. • Let  be an unsatisfiable formula • Theorem 2: A proof P is complete with respect to  if for every full assignment  TL(): Theory Literals corresponding to  Not constructive!

  8. Projection of a variable x: a set of proof steps that eliminate x and maintains satisfiability. • Strong projection of a variable x: a projection of x that maintains: The projected consequences from each minimal unsatisfiable core of literals is unsatisfiable.

  9. Example– strong projection Both sub-formulas are unsatisfiable and do not contain x1. Consider the formula U2 U1 Now strongly project x1:

  10. Let C be a conjunction of ’s literals. • A proof construction procedure: eliminate all variables in C through strong projection. • Theorem 3: The constructed proof is ‘complete’ for .

  11. Goal: for a given logic L, • Find a strong projection procedure. • Construct P • Generate c(P) • Check skÆ c(P)

  12. e6 x3 + x2 < 0 e5 2x3 < 0, Example: Disjunctive Linear Arithmetic [S02] e1e2e3e4 C : x1 - x2< 0, x1 - x3< 0, -x1 + 2x3 + x2 < 0, -x3< -1 A proof P by (Strong) projection: e1 e3  e5 x1: e2 e3  e6 e4 e5  false x3: 4. Solve ’ =skÆ c(P)

  13. What now ? • It is left to show a strong projection method for each logic we are interested in integrating. • Current eager procedures are far too wasteful. Need to find better ones.

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