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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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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.