An Interpolating Theorem Prover

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An Interpolating Theorem Prover

• K.L. McMillan
• Cadence Berkley Labs

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Agenda

• Concepts
• Inerpolants from Proofs
• Linear Inequalities (LI)
• Equality and Uninterpreted Functions (EUF)
• Combining LI and EUF
• An Interpolating Prover
• Generating Proofs
• Interpolants for Structured Formulas
• Applications

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Agenda

• Concepts
• Inerpolants from Proofs
• Linear Inequalities (LI)
• Equality and Uninterpreted Functions (EUF)
• Combining LI and EUF
• An Interpolating Prover
• Generating Proofs
• Interpolants for Structured Formulas
• Applications

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Concepts

• term - linear combination c0c1v1cnvn
• v1vn distinct individual variables
• c0cn rational constants, c1cn?0
• x,y terms
• x is 1a, y is b-2a gt 2xy is term 2b
• atomic predicate
• 0 x (x is term)
• propositional variable

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Concepts Cont.

• literal - atomic predicate or its negation
• clause - (l1 v v ln) l1..n literals
• ? - set of literals
• lt?gt - clause from literal of ?
• ltgt - empty clause False
• sequent - ? ? ?, ? set of clauses
• conjunctions of ? entails disjunction of ?.
• lower case letters - formulas
• upper case letters - sets of formulas
• Example ?,?? ??,A ? ? U ? ? U A

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Concepts cont.

• - - 0-1 (False)
• is interpolant
  (deduction of )

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Agenda

• Concepts
• Inerpolants from Proofs
• Linear Inequalities (LI)
• Equality and Uninterpreted Functions (EUF)
• Combining LI and EUF
• An Interpolating Prover
• Generating Proofs
• Interpolants for Structured Formulas
• Applications

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Proof Rules for LI
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Proof Example

• Yaels example

This is a refutation proof
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Motivation for interpolant definition

• A - 0 w-x, 0 x-y
• B - 0 y-z
• F AB 0 w-y
• Contribution from A FA 0 w-y
• A FA
• FA,B F
• Coefficient of w is the same in A and FA
• When F 0 -1 then FA B

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Inequality Interpolation

• Definition 1 (A,B) - 0 x x, ?, ?
• A, B clause sets
• x, x terms
• ??, ? formulas such that
• A, ? 0 x ?
• B ? and B, ? 0 x - x
• ?, ? B x, ?, ? A (x-x) B
• For the current system, the formulas ? and ? are
  always T.

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Hypotheses

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Hypotheses
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Soundness

• Comb
• Comb
• Condition 3 is trivial

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Example

• We want to derive an interpolant for (A,B) where
• A - (0y-x),(0z-y)
• B - (0x-z-1)
• In example

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Solution

• A - (0y-x),(0z-y)
• B - (0x-z-1)
• Step 1,2
• Step 3
• Step 4
• Step 5

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Interpolation syntax for clauses

• Definition 2 (A,B) - lt ? gt ?
• A, B clause sets
• ? literal set
• ? formula
• A f v lt ?\Bgt
• B, f lt ??Bgt
• ? B and ? A
• If ? is empty, ? is an interpolant for (A,B).

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Hypotheses
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Resolution Rules
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Resolution(A) Soundness

• Condition 1,2

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Resolution(B) Soundness

• Condition 1,2

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Contradiction Rule
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Soundness

• Condition 1
• Definition 1
• DeMorgan

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Soundness

• Condition 2
• Definition 1 (condition 2)
• Previous DeMorgan
• Condition 3
• Third condition of definition 1 guaranties that.
  Because coefficient of every must be 0.

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Example

• We want to derive an interpolant for (A,B) where
• Step 1
• Step 2

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Example (Cont.)

• Step 3
• Step 4
• Result

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Agenda

• Concepts
• Inerpolants from Proofs
• Linear Inequalities (LI)
• Equality and Uninterpreted Functions (EUF)
• Combining LI and EUF
• An Interpolating Prover
• Generating Proofs
• Interpolants for Structured Formulas
• Applications

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Proof rules for EUF

• terms are x1xn fn(x1xn)

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Proof rules for EUF

• CONTRA and RES rules the same as in previous
  system.

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Motivation for interpolant definition

• ??gt(xt1)(t1t2)(tny)
• All equalities? (A,B)
• At least one global term in ?
• ? - leftmost global term in ? (A,B)
• ? - right most global term in ? (A,B)
• A x? and y? (everything from the left and
  right are from A)
• There are (tktm) only from A can be summarized
  by a single (tktm) such that ?tk and tm ?
  by location.
• tk, tmare common between A,B

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Motivation for interpolant definition

• ? - will present conjunction of such subchains
• A ?
• B, ? ??
• ? consists only from common variables from (A,B)
• ? is interpolant for xy
• If ? not contains global terms ? degenerate case
  ? ?x and ?y ? ? T

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Equality Interpulation

• Definition 3 (A,B) - xy x, y, ?, ?
• A,B clause sets
• x, y, x, y terms
• ?, ? formulas
• A, ? xx yy ?
• B ? and
• x y and y x (the degenerate case), or
• x,y B and B, ? xy
• ?,? B and ?,? B, and if x B then x x
  else x A (similarity for y,y)

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More Concepts

• (x,y) or if x B then x else y
• (x,y) or if y B then y else x
• if then else T
• if then T else xy
• x(y/z) if then y else x
• syntactic equality, equality pass
  contains global variable

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Hypotheses

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Hypotheses
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Transitivity Rule
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Transitivity Rule - Motivation

• Solid lines equalities from A
• Dotted lines equalities from B,?
• Not degenerate case
• x z ? x z
• If y is local then y,y A else y y

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Transitivity Sound Prove

• Condition 1

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Transitivity Sound Prove (cont.)

• Condition 2 Suppose B, ?, ?, yy

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Transitivity Sound Prove (cont.)

• Condition 3 Trivial

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Transitivity degenerate

• Now yz is solution for xz
• B,? yz

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Transitivity Rule (degenerate) - Sound

• Condition 1 Suppose A, ?, ?
• Same for zz(y/y)

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Transitivity Rule (degenerate) - Sound

• Condition 2 Suppose B, ?, ?

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Transitivity Rule (degenerate) - Sound

• Condition 3

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Cong-Rule
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Cong-Rule Soundness

• Condition 1

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Cong-Rule Soundness

• Condition 2

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Cong-Rule Soundness

• Condition 3

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EqNeq Rules

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Example

• We want to derive an interpolant for f(x)f(y)
• A xy
• B yz
• Step 1,2 Two hypotheses
• Step 3
• Step 4

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Agenda

• Concepts
• Inerpolants from Proofs
• Linear Inequalities (LI)
• Equality and Uninterpreted Functions (EUF)
• Combining LI and EUF
• An Interpolating Prover
• Generating Proofs
• Interpolants for Structured Formulas
• Applications

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Combining LI and EUF - Rules

• Pass from equality to inequality
• From inequality to equality

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Interpolating Rules

• From equality to inequality

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LeqEq - Soundness

• Condition 1
• Condition 2
• Condition 3

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Interpolating Rules

• From inequality to euality

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EqLeq - Soundness

• Condition 1 Trivial
• Condition 2
• Condition 3

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Soundness and Completeness

• Definition 4 ? is interpolant for (A,B)
• A ?
• B, ? False
• ? A and ? B
• Theorem 1 (Soundness) If a clause interpolation
  of the form (A,B)- ltgt f is derivable, then f
  is an interpolant for (A,B).
• Theorem 2 (Completeness) For any derivable
  sequent A,B - ?, there is a derivable
  interpolation of the form (A,B) - ? X.

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Agenda

• Concepts
• Inerpolants from Proofs
• Linear Inequalities (LI)
• Equality and Uninterpreted Functions (EUF)
• Combining LI and EUF
• An Interpolating Prover
• Generating Proofs
• Interpolants for Structured Formulas
• Applications

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Generating Proofs

• Use combination of DPLL based SAT solver
  (propositional reasoning) Nelson-Oppen style
  ground decision procedure (theory reasoning)
  using lazy approach.

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Interpolants for structured formulas

• Problem A, B have arbitrary nesting of Boolean
  operators and not CNF structure.
• We will transfer general (A,B) into (Ac,Bc) where
  Ac, Bc are in clause form.
• Tseitin encoding is used for convert to CNF
  structure.
• Theorem 3 An interpolant for (Ac,Bc) is also an
  interpolant for (A,B).

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Applications

• Using Interpolation for Predicate Refinement.
• Model Checking with Interpolation

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Contribution

• Development of combined proof system for LI and
  EUF.
• Interpolant extraction from combination of two
  theories LIEUF based on proof system rools.

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• Thank you