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1 A framework for eager encoding Daniel KroeningETH, Switzerland Ofer StrichmanTechnion, Israel ( Executive summary ) )submitted to: Formal Aspects of Computing(

A framework for eager encoding

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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). - PowerPoint PPT Presentation

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Page 1: A framework for eager encoding

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A framework for eager encoding

Daniel Kroening ETH, Switzerland

Ofer Strichman Technion, Israel

(Executive summary)(submitted to: Formal Aspects of Computing)

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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.

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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

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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)

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Propositional skeleton:

Theorem 1: For every formula and any sound proof P,

is satisfiable ) sk Æ c(P) is satisfiable.

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Complete proofs

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

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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

Sufficient condition for completeness #1

Not constructive!

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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.

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Consider the formula

Example – strong projection

Both sub-formulas are unsatisfiable and do not contain x1.

Now strongly project x1:

U1U2

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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 .

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Goal: for a given logic L, Find a strong projection procedure. Construct P Generate c(P) Check sk Æ c(P)

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C : x1 - x2 < 0, x1 - x3 < 0, -x1 + 2x3 + x2 < 0, -x3 < -1

Example: Disjunctive Linear Arithmetic [S02]

A proof P by (Strong) projection:

e1 e2 e3 e4

e1 e3 e5

4. Solve ’ = sk Æ c(P)

x1:

e2 e3 e6

2x3 < 0, e5

x3 + x2 < 0 e6

e4 e5 falsex3:

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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.