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A framework for eager encoding Daniel Kroening Ofer Strichman ETH, Switzerland Technion, Israel (Executive summary) (submitted to: Formal Aspects of Computing) 1 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 L that meets several criteria. Linear arithmetic, EUF, arrays etc all have it. 2 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 3 A proof P =(s1,…, sn) is a set of Proof Steps, …in which the Antecedence relation is acyclic The Proof Constraint c(P) induced by P is the conjunction of the constraints induced by its steps: P C(P) 4 Propositional skeleton: Theorem 1: For every formula and any sound proof P, is satisfiable ) sk Æ c(P) is satisfiable. 5 Complete proofs Definition (Complete proofs): A proof P is called complete with respect to if 6 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! 7 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. 8 Example – strong projection Consider the formula U2 U1 Now strongly project x1: Both sub-formulas are unsatisfiable and do not contain x1. 9 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 . 10 Goal: for a given logic L, Find a strong projection procedure. Construct P Generate c(P) Check sk Æ c(P) 11 Example: Disjunctive Linear Arithmetic [S02] e1 e2 e3 e4 C : x1 - x2 < 0, x1 - x3 < 0, -x1 + 2x3 + x2 < 0, -x3 < -1 e5 2x3 < 0, e6 x3 + x2 < 0 A proof P by (Strong) projection: x1: e1 e3 e5 e2 e3 e6 x3: e4 e5 false 4. Solve ’ = sk Æ c(P) 12 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. 13