Decision Procedures for Equality Logic 3

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Decision Procedures for Equality Logic 3
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Decision Procedures for Equality Logic

• We will first investigate methods that solve
  Equality Logic. Uninterpreted functions are
  eliminated with one of the reduction schemes.
• Our starting point the E-Graph GE(?E)
• Recall GE(?E) represents an abstraction of
  ?EIt represents ALL equality formulas with the
  same set of equality predicates as ?E

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From Equality to Propositional LogicBryant
Velev 2000 the Sparse method

• ?E x1 x2 x2 x3 x1 ? x3
• ?enc e1 e2 e3
• Encode all edges with Boolean variables
• (note for now, ignore polarity)?
• This is an abstraction
• Transitivity of equality is lost!
• Must add transitivity constraints!

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From Equality to Propositional Logic

• ?E x1 x2 x2 x3 x1 ? x3
• ?enc e1 e2 e3
• For each cycle add a transitivity constraint
• ?trans (e1 e2 ! e3) (e1 e3 ! e2)
  
• (e3 e2 ! e1)?
• Check ?enc ?trans

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From Equality to Propositional Logic

• There can be an exponential number of cycles, so
  lets try to make it better.
• Thm It is sufficient to constrain simple cycles
  only

e2
e3
e4
e1
e5
e6
Only two simple cycles here.
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From Equality to Propositional Logic

• Still, there is an exponential number of simple
  cycles.
• Def A chord of a cycle is an edge connecting two
  non-adjacent nodes of the cycle.
• Thm Bryant Velev It is sufficient to
  constrain chord-free simple cycles only.

x
y
The edge xv is a chord of the cycle xyvz.
z
v
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From Equality to Propositional Logic

• Still, there can be an exponential number of
  chord-free simple cycles
• Solution make the graph chordal by adding
  edges.

.
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From Equality to Propositional Logic

• Def. A graph is chordal iff every cycle of size 4
  or more has a chord.
• Def. A vertex v of an arbitrary graph G is called
  simplicial if all vertices adjacent to v are
  pairwise adjacent.
• How to recognize chordal graphs?Repeatedly find
  a simplicial vertex and eliminate it from the
  graph until no vertices remain (and the graph is
  chordal) or no simplicial vertices remain (and
  the graph is non-chordal).

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From Equality to Propositional Logic

• How to make a graph chordal ? Make
  non-simplicial vertices simplicial by adding
  edges.

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From Equality to Propositional Logic

• Once the graph is chordal, we can constrain only
  the triangles.
• Note that this procedure adds not more than a
  polynomial of edges, and results in a
  polynomial no. of constraints.

T
T
T
T
Contradiction!
T
F
T
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Improvement

• So far we did not consider the polarity of the
  edges.
• Claim in the following graph ?trans e3 e2 !
  e1 is sufficient
• This is only true because of monotonicity of NNF

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Definitions

• Def. A contradictory cycle C is constrained under
  T? if T does not allow this assignment

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Theorem
red

• If ?trans constrains all simple contradictory
  cycles and
• for every assignment ?, ? ² ?trans ! ? ² ?trans
• then ?E is satisfiable iff ?enc ?trans is
  satisfiable

red
red
From the Sparse method
The equality formula