SMELS: Sat Modulo Equality with Lazy Superposition

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SMELS Sat Modulo Equality with Lazy Superposition

• Christopher Lynch Clarkson
• Duc-Khanh Tran - MPI

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Interest

• Verification problems often reduce to formulas
  containing
• mostly ground equations and
• quantified equations representing properties or
  theories

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Goal

• Efficient inference system for deciding
  satisfiability of sets of equational clauses,
  mostly ground

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Assumptions

• DPLL(cc) most efficient way of solving ground
  equational clauses
• Superposition most efficient way of solving
  nonground equational clauses
• Develop complete implementable combination of the
  two methods
• DPLL(cc(Sup))

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Contents of Talk

• DPLL(cc)
• Superposition
• SMELS DPLL(cc) with Lazy Superposition
• Completeness
• Implementation plans

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DPLL(cc)

• DPLL Given set of clauses S, tries to build
  model of S by adding literals one by one
• DPLL(cc) Given set of equational clauses, tries
  to build model by adding literals one by one, and
  checking consistency in background theory (Cong.
  Closure)

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Responsibility of cc

• Receives set M of (dis)equations
• Notifies DPLL procedure if M inconsistent
• Returns J µ M, justification of inconsistency
• Clause J (or alternative) can be added as lemma

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Using cc for implication

• Given M find L where M ² L
• And find small J µ M where J ² L
• DPLL adds J Ç L (or alternative) as lemma

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Example

• f(a)b Ç d!e
• ac Ç i!j
• de Ç g!h
• ij
• DPLL generates ij, ac, gh, de, f(a)b
• gh is justification for f(c)b (not only one)
• Then g!h Ç f(c)b added as lemma

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Definition of Justification

• Let S be set of clauses, M (partial) model
• Model is set of (dis)equations
• Let L 2 M
• j is a function where
• j(L) µ M and
• S j(L) ² L

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Summary so far

• DPLL sends partial model M to cc
• cc determines consistency of M
• If M ² L, there 9 just. j(L)
• It is sound to add j(L) Ç L
• Note We can always have j(L) L
• Self-justification

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Superposition

• Ç us v Ç st
• --------------------------------
• ( Ç Ç ut v)¾
• ¾ mgu(s,s) and s not variable
• s ! t, us ! v, st max, us v max
• Also for us ! v

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Orderings are crucial

• Without orderings, no hope of termination
• Example
• gt(x,0) Ç gt(s(x),0)
• gt(c,0)
• With orderings it immediately halts

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SMELS

• Let S be set of clauses, g(S) ground clauses in
  S, v(S) nonground clauses in S
• DPLL receives g(S) and passes M to cc
• cc passes reduced implied (dis)equations T to Sup
• Sup performs inferences between T and v(S),
  justified ground clauses sent to DPLL

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Superposition in DPLL(cc(Sup)

• There are two kinds of Superposition
• Superposition among nonground clauses
• Superposition among nonground clause and implied
  (dis)equation from cc (Justified Sup)
• No Superposition between ground clauses

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Nonground Superposition

• We modify Superposition so that inferences
  involve maximal literals of nonground part of
  clause (as opposed to max of entire clause)
• Equational Factoring and Equation Resolution also
  involve maximal nonground literal

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Example of Nonground Sup

• Premises
• f(g(a))b Ç g(x)x Ç f(g(x))x
• f(f(a))c Ç g(a)c Ç g(y)y
• Conclusion
• f(g(a))b Ç f(f(a))c Ç g(a)c Ç g(x)x Ç f(x)x

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Justified Superposition

• Between nonground clause and literal L from cc,
  After Superposition, we add negation of
  justification
• Equivalently, a Superposition inference between
  nonground clause and j(L) Ç L

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Examples of Justified Sup

• Suppose j(f(a)b) de, f(b)e
• Let g(f(c))c Ç f(x)x Ç f(x)g(x) 2 v(S)
• Then Justified Superposition gives d!e Ç f(b)!e
  Ç g(f(c))c Ç f(a)a Ç bg(a)
• This is ground, so passed back to DPLL
  

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Example of DPLL(cc(Sup))

• p(a,b) p1
• p(c,d) p2
• p(e,f) p3
• p1 p2 Ç p1 p3
• a ! c
• a ! e
• p(x1,y1) ! p(x2,y2) Ç x1 x2

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DPLL

• Input g(S) p(a,b)p1, p(c,d)p2, p(e,f) p3,
  p1p2 Ç p1p3, a ! c, a ! e
• Output M p(a,b)p1, p(c,d)p2, p(e,f) p3,
  p1p2, a!c, a!e
• j(p1p2) p1p2
• For all other L 2 M, j(L)

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cc

• Input M p(a,b)p1, p(c,d)p2, p(e,f) p3,
  p1p2, a!c, a!e
• Output T p(a,b)p2, p(c,d)p2, p(e,f) p3,
  p1p2, a!c, a!e
• j(p(a,b)p2) p1p2

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Sup

• Input T p(a,b)p2, p(c,d)p2, p(e,f) p3,
  p1p2, a!c, a!e
• v(S) p(x1,y1) ! p(x2,y2) Ç x1 x2
• Justified Superposition gives p1!p2 Ç
  p2!p(x2,y2) Ç ax2, p2!p(x2,y2) Ç cx2,
  p3!p(x2,y2) Ç ex2
• Also p1!p2 Ç ac

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DPLL

• Input g(S) p(a,b)p1, p(c,d)p2, p(e,f) p3,
  p1p2 Ç p1p3, a ! c, a ! e, p1!p2 Ç ac
• Output M p(a,b)p1, p(c,d)p2, p(e,f) p3,
  p1p3, a!c, a!e
• j(p1p3)

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cc

• Input M p(a,b)p1, p(c,d)p2, p(e,f) p3,
  p1p3, a!c, a!e
• Output T p(a,b)p3, p(c,d)p2, p(e,f) p3,
  p1p3, a!c, a!e
• j(p(a,b)p3)

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Sup

• Input T p(a,b)p3, p(c,d)p2, p(e,f) p3,
  p1p3, a!c, a!e
• v(S) p(x1,y1) ! p(x2,y2) Ç x1 x2
• Justified Superposition gives ae

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DPLL

• Input g(S) p(a,b)p1, p(c,d)p2, p(e,f) p3,
  p1p2 Ç p1p3, a ! c, a ! e, p1!p2 Ç ac, ae
• Output UNSAT

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Example 2

• Repeat example, suppose that original set did not
  contain a!e
• Then everything is the same up until the last
  DPLL step

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DPLL

• Input g(S) p(a,b)p1, p(c,d)p2, p(e,f) p3,
  p1p2 Ç p1p3, a ! c, p1!p2 Ç ac, ae
• Output M p(a,b)p1, p(c,d)p2, p(e,f) p3,
  p1p3, a!c, ae

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cc

• Input M p(a,b)p1, p(c,d)p2, p(e,f) p3,
  p1p3, a!c, ae
• Output T p(e,b)p3, p(c,d)p2, p(e,f) p3,
  p1p3, c!e, ae
• All justifications empty

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Sup

• Input T p(e,b)p3, p(c,d)p2, p(e,f) p3,
  p1p3, c!e, ae
• v(S) p(x1,y1) ! p(x2,y2) Ç x1 x2
• Justified Superposition gives nothing new
• Therefore T is a model modulo v(S)

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Schematic Saturation

• Example theory v(S) is decidable
• We could use Schematic Saturation to prove the
  decidability
• We could also use Schematic Saturation to compile
  nonground theory and efficiently perform
  Justified Superposition

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Instantiation

• Resolution self-justification Instantiation
• j(p(a)) p(a)
• Nonground clause q(x) Ç p(x)
• Justified Resolution gives q(a) Ç p(a)
• As far as we know, first combination of
  instantiation with ordered resolution

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Completeness

• Suppose S is saturated by SMELS
• Let M be model of g(S)
• Then M is v(S) model of g(S)

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Completeness Proof

• Modifed version of BG model generation
• May have implications for selection rules and
  goal-directed Superposition
• Justifications are key

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Completeness implies

• S is SAT implies
• Ground model M (modulo v(S)) is generated in
  finite time, or
• M (modulo v(S)) is generated in infinite time
• S is UNSAT implies
• Unsatisfiable ground g(S) is found

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Comparisons

• BE Uses Eager Superposition (works for some
  theories)
• SPASST FOL theorem prover is driver, which
  calls SMT, not complete
• InstGen Instantiates clauses but no orderings
• Simplify Instantiates terms but is not complete

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Conclusions

• SMELS DPLL(cc(Sup))
• DPLL sends partial model to cc
• cc passes reduced implications to Sup
• Sup handles nonground part using powerful
  orderings

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Future Work

• Implement using compilation of Justification
  Superposition using Schematic Saturation
• Combine with other theories like Linear Arithmetic