Decision Procedures for Equality Logic 1

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Decision Procedures for Equality Logic 1
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Equality logic

• Def. An equality logic formula is defined by the
  following grammarformula formula formula
  formula atomatom term termterm
  identifier constantwhere the
  identifiers are variables over a single infinite
  domain like the reals or integers. Constants are
  elements from the same domain as identifiers.

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Complexity

• Equality logic and propositional logic are both
  NP-complete.
• Thus they model the same decision problems.
• Why to study both?- convenience of modeling-
  efficiency
• Extensions different domains, Boolean variables

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Basic assumptions and notations

• Input formulas are in NNF
• Input formulas are checked for satisfiability
• Equality formula ?E

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Removing constantsA simplification

• Thm. Given an equality logic formula ?E, there is
  an algorithm that generates an equisatisfiable
  formula ?E' without constants, in polynomial
  time.
• Algorithm Input An equality logic formula ?E
  with constants c1,...,cnOutput An equality
  logic formula ?E' such that ?E and ?E' are
  equisatisfiable and ?E' has no constants.

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Removing constantsA simplification

• 1. ?E' ?E2. In ?E', replace each constant
  ci, 1in, with a new variable Cci .3.
  For each pair of constants ci, cj such that
  1iltjn, add the constraint Cci ?Ccj to
  ?E' .
• In the following we assume that the input
  equality formulas do not have constants.

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Uninterpreted functions

• Def. An equality logic formula with uninterpreted
  functions and uninterpreted predicates is defined
  by the following grammarformula formula
  formula formula (formula)
  atomatom term term
  predicate-symbol(list of terms)term
  identifier function-symbol(list of terms)?
• Example F(x)F(G(x))
• Congruence xG(x) ) F(x) F(G(x))

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Usage of uninterpreted functions

• Replacing functions by uninterpreted functions in
  a given formula is a common technique to make
  reasoning easier.
• Makes the formula weaker ?UF ) ???
• Ignore the semantics of the function, but
• Functional congruence Instances of the same
  function return the same value for equal
  arguments.

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From uninterpreted functions to equality logic

• Two possible reductions- Ackermann's
  reduction- Bryant's reduction
• Ackermann's reductionGiven an input formula
  ?UF, add explicit constraints for functional
  congruence and transform the formula to an
  equality logic formula ?? of the form
  ??????FCE ) flatEwhere FCE is a
  conjunction of functional-consistency
  constraints, and flatE is a flattening of ?UF.

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• Algoritm Ackermann's reductionInput ?UF with m
  instances of an uninterpreted function
  F Output ???such that ???is valid iff ?UF is
  valid1. Assign indices to the UF-instances.2.
  flatE T(?UF) where T replaces each Fi by a fresh
  fi3. FCE i1..m-1 ji1..m
  (T(arg(Fi)) T(arg(Fj))) ) fifj4. Return
  ??? FCE ) flatE

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Ackermann's reductionExample

• (x1 ? x2) _ (F(x1) F(x2)) _ (F(x1) ? F(x3))?
• flatE (x1 ? x2) _ (f1 f2) _ (f1 ? f3))?
• FCE (x1x2 ) f1f2) (x1x3 )
  f1f3) (x2x3 ) f2f3)
• ??? FCE ) flatE

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Ackermann's reduction validity vs. satisfiability

• ??? FCE ) flatE?is valid iff ?UF is valid
• Validity check of ?UF check ?? for validity or
  ???for unsatisfiability
• What if we want to check satisfiability of ?UF?
• Ackermann's reduction in the above form maintains
  validity, not satisfiability!
• Solution check satisfiability of ??? FCE
  flatE

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Bryant's reduction

• Case expression Fi case x1xi f1
  x2xi
  f2
  ...
  true fiwhere xi is the argument
  arg(Fi) of Fi for all i
• Semantics _j1,...,i (Fi fj (xjxi)
  k1,...,j-1 (xk ? xi))?

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Bryant's reductionAlgorithm

• Input An EUF formula ?UF with m instances of an
  uninterpreted function F
• Output An EF formula ?? such that ?? is valid
  iff ?UF is valid
• 1. Assign indices to the uninterpreted-function
  instances from subexpressions outwards.2.
  Return ?E T (?UF) where T replaces each
  Fi(arg(Fi)) by case T (arg(F1))
  T (arg(Fi)) f1
  ... T
  (arg(Fi-1)) T (arg(Fi)) fi-1
  true
  fi

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Bryant's reduction Example

• int power3_con (int in) int i, out
  out in for (i 0 i lt 2 i)
  out out in return out
• int power3_con_new (int in) return ((in
  in) in)

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Bryant's reduction Example

• int power3_con (int in) int i, out
  out in for (i 0 i lt 2 i)
  out out in return out
• int power3_con_new (int in) return (in
  in) in
• ?? out0 in out1 out0 in out2 out1
  in
• ?2 out0_new (in in) in
• ??? ????) out2 out0_new

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• ? (out0 in out1 out0 in out2
  out1 in ??out0_new (in in) in)) out2
  out0_new
• ?UF (out0 in out1 G(out0,in) out2
  G(out1,in) out0_new G(G(in,in),in))) out2
  out0_new

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• ?UF (out0 in out1 G(out0,in) out2
  G(out1,in) ?out0_new G(G(in,in),in)) ) out2
  out0_new
• ?E (out0 in out1 G1 out2 G2
  ?????out0_new G4 ) out2 out0_newwith G1
  g1 G2 case out0out1 inin
  g1 true
  g2 G3 case
  out0in inin g1
  case out1in inin g2
  true g3

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• ?UF (out0 in out1 G(out0,in) out2
  G(out1,in) ?out0_new G(G(in,in),in)) ) out2
  out0_new
• ?E (out0 in out1 G1 out2 G2
  ?????out0_new G4 ) out2 out0_newand
  with G4 case out0G3 inin g1
  case out1 G3 inin
  g2 case in G3
  inin g3
  true g4

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EUF where are we now?

• We introduced equality logic and uninterpreted
  functions
• We showed how to eliminate constants
• We used Ackermann's and Bryant's algorithms to
  reduce the validity question in equality logic
  with uninterpreted functions to validity
  questions in equality logic
• Next Decision procedures for equality logic and
  uninterpreted functions