Decision methods for arithmetic Third summer school on formal methods Leonardo de Moura Microsoft Research.

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Presentation transcript:

Decision methods for arithmetic Third summer school on formal methods Leonardo de Moura Microsoft Research

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

Saturation x Search Proof-findingModel-finding Models Proofs Conflict Resolution Conflict Resolution

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

Two procedures ResolutionDPLL Proof-finderModel-finder SaturationSearch

Resolution

Resolution: Example

Resolution: Problem Exponential time and space

Unit Resolution

DPLL DPLL = Unit Resolution + Split rule

DPLL

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

MCSat

Propagations

MCSat Propagations

MCSat Propagations

MCSat Decisions

MCSat Model Assignments

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

MCSat Conflict

MCSat Conflict

MCSat

Conflict

MCSat Learned by resolution

MCSat

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

MCSat – Finite Basis

Every “finite” theory has a finite basis

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.

MCSat: Termination Propagations Decisions Model Assignments

MCSat Propagations Decisions Model Assignments

MCSat Propagations Decisions Model Assignments

MCSat … Maximal Elements

Conflict

MCSat

Conflict (evaluates to false)

MCSat New clause

MCSat New clause

MCSat New clause

MCSat: Architecture Arithmetic Boolean Lists Arrays

MCSat: development