Predicate Learning and Selective Theory Deduction for Solving Difference Logic

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Predicate Learning and Selective Theory Deduction
for Solving Difference Logic

• Chao Wang, Aarti Gupta, Malay Ganai
• NEC Laboratories America
• Princeton, New Jersey, USA
• August 21, 2006

Presentation-only for more info. please check
Wang et al LPAR05 and Wang et al DAC06
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Difference Logic

• Logic to model systems at the word-level
• Subset of quantifier-free first order logic
• Boolean connectives predicates like (x y c)
  
• Formal verification applications
• Pipelined processors, timed systems, embedded
  software
• e.g., back-end of the UCLID Verifier
• Existing solvers
• Eager approach Strichman et al. 02, Talupur
  et al. 04, UCLID
• Lazy approach TSAT, MathSAT, DPLL(T),
  Saten, SLICE, Yices, HTP,
• Hybrid approach Seshia et al. 03, UCLID, SD-SAT

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Our contribution

• Lessons learned from previous works
• Incremental conflict detection and zero-cost
  theory backtracking Wang et al. LPAR05
• Exhaustive theory deduction
  Nieuwenhuis Oliveras CAV05
• Eager chordal transitivity constraints
  Strichman et al. FMCAD02
• Whats new?
• Incremental conflict detection PLUS selective
  theory deduction
• with little additional cost
• Dynamic predicate learning to combat exponential
  blow-up

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Outline

• Preliminaries
• Selective theory (implication) deduction
• Dynamic predicate learning
• Experiments
• Conclusions

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Preliminaries
Difference logic formula Difference
predicates Boolean skeleton Constraint graph for
assignment (A,B,C,D)
A ( x y 2 ), B ( z x -7 ) C ( y -
z 3 ), D ( w - y 10 )
A ( x y 2 ) B ( z x -7 )
C ( y - z 3 ) D ( w - y 10 )
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Theory conflict infeasible Boolean assignment

• Negative weighted cycle ?? Theory conflict
• Theory conflict ? Lemma or blocking clause ?
  Boolean conflict

A ( x y 2 ) B ( z x -7 ) C (
y - z 3 ) D ( w - y 10 )
Conflicting clause (false false false)
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Theory implication implied Boolean assignment

• If adding an edge creates a negative cycle ?
  negated edge is implied
• Theory implication ? var assignment ? Boolean
  implication (BCP)

A ( x y 2 ) B ( z x -7 ) C (
y - z 3 ) D ( w - y 10 )
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Negative cycle detection

• Called repeatedly to solve many similar
  subproblems
• For conflict detection (incremental, efficient)
• For implication deduction (often expensive)
• Incremental detection versus exhaustive deduction
• SLICE Incremental cycle detection -- O(n log
  n)
• DPLL(T) Exhaustive theory deduction -- O(n m)

SLICE LPAR05 DPLL(T) Barcelogic CAV05
Conflict detection Incremental NO
Implication deduction NO Exhaustive
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Data from LPAR05 Comparing SLICE solver
(SMT benchmarks repository,as of 08-2005)
vs. UCLID
vs. MathSAT
vs. ICS 2.0
vs. DPLL(T) Barcelogic
vs. DPLL(T)B (linear scale)
vs. TSAT
Points above the diagonals ? Wins for SLICE solver
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From the previous results

• We have learned that
• Incremental conflict detection ? more scalable
• Exhaustive theory deduction ? also helpful
• Can we combine their relative strengths?
• Our new solution
• Incremental conflict detection (SLICE)
• Zero-cost theory backtracking (SLICE)
• PLUS selective theory deduction with O(n) cost

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Outline

• Preliminaries
• Selective theory (implication) deduction
• Dynamic predicate learning
• Experiments
• Conclusions

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Constraint Propagation
Theory Constraint Propagation
Deduce() while (implications.empty())
set_var_value(implications.pop()) if
(detect_conflict()) return
CONFLICT add_new_implications() if (
ready_for_theory_propagation() ) if
(theory_detect_conflict()) return
CONFLICT theory_add_new_implications()

Boolean CP (BCP)
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Incremental conflict detection
Ramalingam 1999 Bozzano et al. 2005 Cotton
2005 Wang et al. LPAR05
Relax Edge (u,v) if ( dv gt duwu,v
) dv duwu,v piv u
0
0
x
y
w
z
-7
0

• Add an edge ? relax, relax, relax,
• Remove an edge ? do nothing (zero-cost
  backtracking in SLICE)

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Selective theory deduction
Post(x) x, z,
Pre(y) y, w,
y
w
z
x
Piy w
through relax
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Outline

• Preliminaries
• Selective theory (implication) deduction
• Dynamic predicate learning
• Experiments
• Conclusions

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Diamonds with O(2n) negative cycles
e1
e2
-1
e0
Observations With existing predicates (e1,e2,)
? exponential number of lemmas Add new
predicates (E1,E2,E3) and dummies (E1!E1)
(E2!E2) ? almost linear number
of lemmas Previous eager chordal transitivity
used by Strichmann et al. FMCAD02
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Add new predicates to reduce lemmas
z
x
y
w
Heuristics to choose GOOD predicates
(short-cuts) Nodes that show up frequently in
negative cycles Nodes that are re-convergence
points of the graph (Conceptually) adding a
dummy constraint (E3 ! E3)
Predicates E1 x y lt 5 E2 y x lt
5 Lemma ( ! E1 ! E2 )
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Experiments with SLICE

• Implemented upon SLICE
• i.e., Wang et al. LPAR05
• Controlled experiments
• Flexible theory propagation invocation
• Per predicate assignment, per BCP, or per full
  assignment
• Selective theory deduction
• No deduction, Fwd-only, or Both-directions
• Dynamic predicate learning
• With, or Without

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When to call the theory solver?
On the DTP benchmark suite
per BCP versus per predicate assignment
per BCP versus per full assignment
Points above the diagonals ? Wins for per BCP
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Comparing theory deduction schemes
On the DTP benchmark suite
Fwd-only deduction vs. no deduction total 660
seconds
Both-directions vs. no deduction total 1138
seconds
Points above the diagonals ? Wins for no deduction
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Comparing dynamic predicate learning
On the diamonds benchmark suite
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Comparing dynamic predicate learning
On the DTP benchmark suite
Dyn. pred. learning vs. No pred. learning
Points above the diagonals ? Wins for No pred.
learning
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Lessons learned

• Timing to invoke theory solver
• after every BCP finishes gives the best
  performance
• Selective implication deduction
• Little added cost, but improves the performance
  significantly
• Dynamic predicate learning
• Reduces the exponential blow-up in certain
  examples
• In the spirit of predicate abstraction

Questions ?