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Decision methods for arithmeticThird summer school on formal methods
Leonardo de Moura Microsoft Research
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Software analysis/verification tools need some form of symbolic reasoning
Symbolic Reasoning
Logic is βThe Calculus of Computer ScienceβZohar Manna
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Saturation x Search
Proof-finding Model-finding
Models
Proo
fsConflict
Resolution
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SAT
CNF is a set (conjunction) set of clausesClause is a disjunction of literalsLiteral is an atom or the negation of an atom
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Two procedures
Resolution DPLLProof-finder Model-finderSaturation Search
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Resolution
ImprovementsDelete tautologies Ordered ResolutionSubsumption (delete redundant clauses)
β¦
unsat
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Resolution: Example
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Resolution: Example
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Resolution: Example
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Resolution: Example
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Resolution: Example
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Resolution: Problem
Exponential time and space
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Unit Resolution
Complete for Horn Clauses
subsumes
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DPLL
DPLL = Unit Resolution + Split rule
Split rule
π ,π π ,Β¬π
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DPLLπ₯β¨ π¦ ,Β¬π₯β¨π¦ , π₯β¨Β¬ π¦ ,Β¬π₯β¨Β¬ π¦
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DPLLπ₯β¨ π¦ ,Β¬π₯β¨π¦ , π₯β¨Β¬ π¦ ,Β¬π₯β¨Β¬ π¦
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DPLLπ₯β¨ π¦ ,Β¬π₯β¨π¦ , π₯β¨Β¬ π¦ ,Β¬π₯β¨Β¬ π¦
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DPLLπ₯β¨ π¦ ,Β¬π₯β¨π¦ , π₯β¨Β¬ π¦ ,Β¬π₯β¨Β¬ π¦
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DPLLπ₯β¨ π¦ ,Β¬π₯β¨π¦ , π₯β¨Β¬ π¦ ,Β¬π₯β¨Β¬ π¦
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DPLLπ₯β¨ π¦ ,Β¬π₯β¨π¦ , π₯β¨Β¬ π¦ ,Β¬π₯β¨Β¬ π¦
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CDCL: Conflict Driven Clause Learning
Resolution
DPLLConflict
Resolution
Proof
Model
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MCSat
π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
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MCSat
π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
π₯β₯2
Propagations
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MCSat
π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
π₯β₯2
Propagations
π₯β₯1
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MCSat
π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
π₯β₯2
Propagations
π₯β₯1 π¦ β₯1
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MCSat
π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
π₯β₯2
Decisions
π₯β₯1 π¦ β₯1π₯2+ π¦2β€1
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MCSat
π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
π₯β₯2
Model Assignments
π₯β₯1 π¦ β₯1π₯2+ π¦2β€1π₯β2
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MCSat
π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
π₯β₯2
Model Assignments
π₯β₯1 π¦ β₯1π₯2+ π¦2β€1π₯β2
We canβt falsify any fact in the trail.
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MCSat
π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
π₯β₯2
Conflict
π₯β₯1 π¦ β₯1π₯2+ π¦2β€1π₯β2
We canβt find a value of s.t.
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MCSat
π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
π₯β₯2
Conflict
π₯β₯1 π¦ β₯1π₯2+ π¦2β€1π₯β2
We canβt find a value of s.t.
Learning that = 2)is not productive
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π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
π₯β₯2
Conflict
π₯β₯1 π¦ β₯1π₯2+ π¦2β€1π₯β2
π¦
π₯
π₯2+ π¦2β€1 π₯β2
β1β€ π₯ , π₯β€1
Β¬(π₯2+π¦2β€1)β¨π₯β€1
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MCSat
π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
π₯β₯2 π₯β₯1 π¦ β₯1π₯2+ π¦2β€1 π₯β€1
Β¬(π₯2+π¦2β€1)β¨π₯β€1
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MCSat
π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
π₯β₯2 π₯β₯1 π¦ β₯1π₯2+ π¦2β€1 π₯β€1
Β¬(π₯2+π¦2β€1)β¨π₯β€1
ConflictΒ¬ (π₯β₯2 )β¨Β¬(π₯β€1)
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MCSat
π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
π₯β₯2 π₯β₯1 π¦ β₯1π₯2+ π¦2β€1
Β¬(π₯2+π¦2β€1)β¨π₯β€1
Learned by resolution
Β¬ (π₯β₯2 )β¨Β¬(π₯2+π¦2β€1)
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MCSat
π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
π₯β₯2 π₯β₯1 π¦ β₯1Β¬(π₯2+π¦2β€1)
Β¬(π₯2+π¦2β€1)β¨π₯β€1Β¬ (π₯β₯2 )β¨Β¬(π₯2+π¦2β€1)
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MCSat β Finite BasisEvery theory that admits quantifier elimination has a finite basis (given a fixed assignment order)
πΉ [π₯1 ,β¦,π₯π , π¦1 ,β¦, π¦π]
βπ₯1 ,β¦, π₯π :πΉ [π₯1 ,β¦,π₯π , π¦ ]
πΆ1[π¦1 ,β¦, π¦π]β§β¦β§πΆπ[π¦1 ,β¦, π¦π ]
Β¬πΉ [π₯1 ,β¦,π₯π , π¦1 ,β¦, π¦π ]β¨πΆπ[π¦1 ,β¦, π¦π]
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MCSat β Finite Basis
πΉ 1[π₯1]
πΉ 2[π₯1 ,π₯2]
πΉ π[π₯1 ,π₯2,β¦, π₯πβ1 ,π₯π]
πΉ πβ1[π₯1 ,π₯2,β¦, π₯πβ1]β¦
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MCSat β Finite Basis
πΉ 1[π₯1]
πΉ 2[π₯1 ,π₯2]
πΉ π[π₯1 ,π₯2,β¦, π₯πβ1 ,π₯π]
πΉ πβ1[π₯1 ,π₯2,β¦, π₯πβ1]β¦
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MCSat β Finite Basis
πΉ 1[π₯1]
πΉ 2[π₯1 ,π₯2]
πΉ π[π₯1 ,π₯2,β¦, π₯πβ1 ,π₯π]
πΉ πβ1[π₯1 ,π₯2,β¦, π₯πβ1]β¦
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MCSat β Finite Basis
πΉ 1[π₯1]
πΉ 2[π₯1 ,π₯2]
πΉ π[π₯1 ,π₯2,β¦, π₯πβ1 ,π₯π]
πΉ πβ1[π₯1 ,π₯2,β¦, π₯πβ1]β¦
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MCSat β Finite BasisEvery βfiniteβ theory has a finite basis
πΉ [π₯1 ,β¦,π₯π , π¦1 ,β¦, π¦π]
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MCSat β Finite BasisTheory 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.
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MCSat: Termination
Propagations
Decisions
Model Assignments
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MCSat
β»
Propagations
Decisions
Model Assignments
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MCSat
β»
Propagations
Decisions
Model Assignments
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MCSat
ΒΏπΉππππ‘ππ΅ππ ππ β¨ΒΏ
β¦Maximal Elements
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π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
π₯β₯2 π₯β₯1 π¦ β₯1π₯2+ π¦2β€1 π₯β€1
Β¬(π₯2+π¦2β€1)β¨π₯β€1ConflictΒ¬ (π₯β₯2 )β¨Β¬(π₯β€1)
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π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
π₯β₯2 π₯β₯1 π¦ β₯1π₯2+ π¦2β€1 π₯β€1
Β¬(π₯2+π¦2β€1)β¨π₯β€1ConflictΒ¬ (π₯β₯2 )β¨Β¬(π₯β€1)
π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
π₯β₯2 π₯β₯1 π¦ β₯1Β¬(π₯2+π¦2β€1)
Β¬(π₯2+π¦2β€1)β¨π₯β€1Β¬ (π₯β₯2 )β¨Β¬(π₯2+π¦2β€1)
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π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
π₯β₯2 π₯β₯1 π¦ β₯1π₯2+ π¦2β€1 π₯β€1
Β¬(π₯2+π¦2β€1)β¨π₯β€1ConflictΒ¬ (π₯β₯2 )β¨Β¬(π₯β€1)
π₯β₯2 , (Β¬π₯β₯1β¨ π¦ β₯1 ) ,(π₯2+π¦2β€1β¨π₯π¦>1)
π₯β₯2 π₯β₯1 π¦ β₯1Β¬(π₯2+π¦2β€1)
Β¬(π₯2+π¦2β€1)β¨π₯β€1Β¬ (π₯β₯2 )β¨Β¬(π₯2+π¦2β€1)
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π₯<1β¨π ,Β¬πβ¨π₯=2
π₯β1
MCSat
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π₯<1β¨π ,Β¬πβ¨π₯=2
π₯β1
MCSat
π
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π₯<1β¨π ,Β¬πβ¨π₯=2
π₯β1
MCSat
π
Conflict (evaluates to false)
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π₯<1β¨π ,Β¬πβ¨π₯=2
π₯β1
MCSat
π
New clause
π₯<1β¨π₯=2
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π₯<1β¨π ,Β¬πβ¨π₯=2
π₯β1
MCSat
π
New clause
π₯<1β¨π₯=2
π₯<1
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π₯<1β¨π ,Β¬πβ¨π₯=2
π₯β1
MCSat
π
New clause
π₯<1β¨π₯=2
π₯<1
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MCSat: Architecture
Arithmetic
Boolean Lists
Arrays
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MCSat: development