Boolean Satisfiability with Transitivity Constraints

1
Boolean Satisfiability with Transitivity
Constraints
Randal E. Bryant Miroslav N. Velev
Carnegie Mellon University
http//www.cs.cmu.edu/bryant
2
Outline

• Application Domain
• Verify correctness of a pipelined processor
• Based on Burch-Dill correspondence checking
• Burch Dill, CAV 94
• Verification Task
• Decide validity of formula in logic of equality
  with uninterpreted functions
• Translate into equational logic
• Propositional logic with equations of form vi
  vj
• Bryant, German Velev, CAV 99
• Goel, Sahid, Zhou, Aziz, Singhal, CAV 98
• New Contribution
• Efficient handling of transitivity constraints

3
Decision Problem

• Logic of Equality with Uninterpreted Functions
  (EUF)
• Truth Values
• Dashed Lines
• Model control signals
• Domain Values
• Solid lines
• Model data words
• Task
• Determine whether formula is universally valid
• True for all interpretations of variables and
  function symbols

4
Eliminating Function Applications

• Verification Task
• Prove x f(f(x)) ? x f(f(f(x))) ? x
  f(x)
• Instance of x y ? x f(y) ? x f(x)
• Ackermanns Method
• Replace f(x) ? f1 f( f(x)) ? f2 f(f( f(x))) ?
  f3
• Gives x f2 ? x f3 ? x f1
• Functional Consistency Constraints
• x f1 ? f1 f2
• f1 f2 ? f2 f3
• x f2 ? f1 f3

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Eliminating Funct. Apps. (cont.)

• Equational Formula
• Complement of substituted formula consistency
  constraints
• Clauses Origin
• x f2 ? x f3 ? x ? f1 ?x f2 ? x
  f3 ? x f1
• ? (x ? f1 ? f1 f2) x f1 ? f1 f2
• ? (f1 ? f2 ? f2 f3) f1 f2 ? f2 f3
• ? (x ? f2 ? f1 f3) x f2 ? f1 f3
• Verification Task
• Prove that equational formula is not satisfiable

x f2 ? x f3 ? x ? f1 ? (x ? f1 ?
f1 f2) ? (f1 ? f2 ? f2 f3) ? (x ?
f2 ? f1 f3)
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Solving Equational Formulas

• Historically
• E.g., Nelson Oppen 80
• Create special purpose search engine
• Davis-Putnam search
• Data structure to maintain equivalence classes
• Question
• Can we translate problem into pure propositional
  logic?
• Would enable use of BDDs or SAT checkers

7
Replacing Equations by Variables

• Relational Variables
• Goel, Sahid, Zhou, Aziz, Singhal, CAV 98
• Replace vi vj by propositional variable ei,j
• Propositional Formula Fsat
• Relabeling x ? v1 f1 ? v2 f2 ? v3 f3 ? v4
• Clauses Origin
• e13 ? e14 ? ?e12 x f2 ? x f3 ? x ?
  f1
• ? (?e12 ? e23) ? (x ? f1 ? f1 f2)
• ? (?e23 ? e34) ? (f1 ? f2 ? f2 f3)
• ? (?e13 ? e24) ? (x ? f2 ? f1 f3)

e13 ? e14 ? ?e12 ? (?e12 ? e23) ?
(?e23 ? e34) ? (?e13 ? e24)
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Need for Transitivity Constraints

• Propositional Formula Fsat
• e13 ? e14 ? ?e12
• ? (?e12 ? e23)
• ? (?e23 ? e34)
• ? (?e13 ? e24)
• Solution
• e13 true e14 true e12 false e23
  true e34 true e24 true
• Transitivity Violation in Solution
• e13 true e23 true e12 false
• Corresponds to x f2 and f2 f1 but x ? f1

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Handling Transitivity Constraints Goel, et al.,
CAV 98

• Complexity
• Finding solution to Fsat that satisfies
  transitivity constraints is NP-Hard
• Even when Fsat represented as OBDD
• Their method
• Enumerate implicants of Fsat from OBDD
  representation
• Discard any implicant that contains transitivity
  violation
• Eventually find solution or run out of implicants
• Our Experiments
• Works well for small benchmarks
• Far too many implicants for larger benchmarks

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Handling Transitivity Constraints Our Method

• Idea
• Generate propositional formula Ftrans expressing
  transitivity constraints
• Satisfy formula Fsat ? Ftrans
• Using OBDDs or SAT checker
• Sources of Efficiency
• Equational structure very sparse
• Far fewer than n(n-1)/2 relational variables
• Only need to enforce limited set of transitivity
  constraints
• With OBDDs, can reduce set of relational
  variables
• Only those in true support of Fsat

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Benchmark Circuits

• Single Issue Pipeline 1xDLX-C
• Analogous to DLX model in Hennessy Patterson
• Verified in 94 by Burch Dill
• Dual Issue Pipeline 1 2xDLX-CA
• Second pipeline can only handle R-R and R-I
  instructions
• Burch (DAC 96) required 28 manual case splits, 3
  commutative diagrams, and 1800s.
• Dual Issue Pipeline 2 2xDLX-CC
• Second pipeline can also handle all instructions

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Verifying Original Benchmarks

• None Require Transitivity Constraints
• Fsat is unsatisfiable in every case
• Circuits dont make use of transitivity in
  forwarding or stall decisions
• Performance
• Circuit OBDD Secs. FGRASP Secs.
• 1xDLX-C 0.2 3
• 2xDLX-CA 11. 176
• 2xDLX-CC 29. 5,035

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Transitivity Benchmarks

• Modified, but Correct Circuits
• Modify forwarding logic
• ESrc1MDest ?
• ESrc1MDest ? (ESrc1ESrc2 ? ESrc2MDest)
• Equivalent under transitivity
• Circuit names 1xDLX-Ct, 2xDLX-CAt, 2xDLX-CCt
• Buggy Circuits
• 100 buggy versions of 2xDLX-CC
• Each contains single modification of control
  logic
• Must ensure that counterexample satisfies
  transitivity constraints

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1xDLX-C Equation Structure

• Vertices
• For each vi
• 13 different register identifiers
• Edges
• For each equation
• Control stalling and forwarding logic
• 27 relational variables
• Out of 78 possible

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2DLX-CCt Equation Structure

• Equations
• Between 25 different register identifiers
• 143 relational variables
• Out of 300 possible

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Graph Interpretation of Transitivity

• Transitivity Violation
• Cycle in graph
• Exactly one edge has ei,j false

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Exploiting Chords

• Chord
• Edge connecting two non-adjacent vertices in cycle

• Property
• Sufficient to enforce transitivity constraints
  for all chord-free cycles
• If transitivity holds for all chord-free cycles,
  then holds for arbitrary cycles

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Enumerating Chord-Free Cycles

• Strategy
• Enumerate chord-free cycles in graph
• Each cycle of length k yields k transitivity
  constraints

• Problem
• Potentially exponential number of chord-free
  cycles

1
2
k

2kk chord-free cycles

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Adding Chords

• Strategy
• Add edges to graph to reduce number of chord-free
  cycles

1
2
k

2kk chord-free cycles

• Trade-Off
• Reduces formula size
• Increases number of relational variables

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Chordal Graph

• Definition
• Every cycle of length gt 3 has a chord
• Goal
• Add minimum number of edges to make graph chordal
• Relation to Sparse Gaussian Elimination
• Choose pivot ordering that minimizes fill-in
• NP-hard
• Simple heuristics effective

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Adding Chordal Edges to 1xDLX-C

• Original
• 27 relational variables
• 286 cycles
• 858 clauses

• Augmented
• 33 relational variables
• 40 cycles
• 120 clauses

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Adding Chordal Edges to 2xDLX-CCt

• Original
• 143 relational variables
• 2,136 cycles
• 8,364 clauses

• Augmented
• 193 relational variables
• 858 cycles
• 2,574 clauses

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SAT Checker on Good Circuits

• Strategy
• Run on clauses encoding Fsat and Ftrans
• FGRASP Performance (Secs.)
• Circuit Fsat Fsat ? Ftrans
• 1xDLX-C 3 4
• 1xDLX-Ct --- 9
• 2xDLX-CA 176 1,275
• 2xDLX-CAt --- 896
• 2xDLX-CC 5,035 9,932
• 2xDLX-CCt --- 15,003
• Observation
• Much more challenging with transitivity
  constraints imposed

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SAT Checker on Buggy Circuits

• Performance Penalty with Transitivity Constraints
• Geometric average slowdown 2.3X

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Using OBDDs

• Possible Strategy
• Build OBDDs for Fsat and Ftrans
• Compute Fsat ? Ftrans
• Find satisfying solution

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Limitation of OBDDs

• OBDD for Ftrans can be of exponential size
• Regardless of variable ordering
• Formal result
• Relational variables forming k X k mesh
• OBDD representation has ?(2k/4) nodes
• Experimental Results
• Unable to build OBDD of Ftrans for large
  benchmarks

6 X 6 mesh
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Better Use of OBDDs

• Strategy
• Build OBDD for Fsat
• Determine relational variables in true support
• Easy with OBDD
• Generate Ftrans for these variables
• Compute conjunction and find satisfying solution
• Performance
• When Fsat unsatisfiable, no further steps
  required
• For other benchmarks, yields tractable Ftrans

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2DLX-CCt Reduced Constraints

• Relational variables
• 46 original
• 6 chordal
• OBDD Representation
• 7,168 nodes

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Reduced Constraints Average-Case Buggy Circuit

• Relational Variables
• 17 original
• 3 chordal
• OBDD Representation
• 70 nodes

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Reduced Constraints Worst-Case Buggy Circuit

• Relational variables
• 52 original
• 16 chordal
• OBDD Representation
• 93,937 nodes

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OBDDs on Good Circuits

• CUDD Performance (Secs.)
• Circuit Time
• 1xDLX-C 0.2
• 1xDLX-Ct 2
• 2xDLX-CA 11
• 2xDLX-CAt 109
• 2xDLX-CC 29
• 2xDLX-CCt 441
• Observation
• Significantly more effort with transitivity
  constraints
• Better performance than FGRASP

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OBDDs on Buggy Circuits

• Performance Penalty with Transitivity Constraints
• Geometric average slowdown 1.01X

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Conclusion

• Equational Formulas can be Solved by
  Propositional Methods
• Exploit sparse structure of equations
• Reduces number of variables
• Reduces formula size
• With OBDDs, can identify essential relational
  variables
• In true support of Fsat
• Can use either SAT checker or OBDDs
• OBDDs do best for unsatisfiable formulas

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Extension

• Formulas with Ordering Constraints
• Constraints of form vi ? vj
• Symbolic Solution
• Introduce variables ai,j and aj,i for each
  constraint vi rel vj
• ai,j true when vi ? vj
• Solution defines partial ordering
• Application
• Scheduling problems