Positive Boolean Functions as Multiheaded Clauses*

1
Positive Boolean Functions asMultiheaded Clauses

• Jacob Howe
• City University, London, UK
• jacob_at_soi.city.ac.uk
• Andy King
• University of Kent, Canterbury, UK
• a.m.king_at_kent.ac.uk

International Conference on Logic Programming,
Cyprus, 2001 Keywords Abstract interpretation,
(constraint) logic programs, Boolean functions,
groundness analysis Funded by EPSRC grant
GR/MO8769
2
Applications Context

• Dependency analysis takes, as input, a program
  and produces, as output, dependencies such as
  x ? y ? z which means that variable x satisfies
  some property if both y and z satisfy the
  property.
• Dependencies arise in CLP in rigidity analysis
  Giacobazzi95, finiteness analysis Bigot92,
  BTA Glynn01, groundness analysis.
• From a logic programming point of view,
  groundness analysis is the most practical
  strength test of Boolean function manipulation.

3
Positive and Definite Boolean functions

• Let f denote the set of models for a
  propositional formulae over X. Then x?y
  x, y, x,y where X x,y
• A function f is positive iff X ? f PosX
  denotes the set of positive Boolean functions
  over X
• A function f is definite iff M ? M' ? f for
  all M, M' ? f DefX denotes the set of
  positive functions over X that are definite

4
Examples of Positive and Definite functions

• Let X x, y, z and consider the following
  table, which states, for some Boolean functions,
  whether they are in PosX or DefX

f Def Pos Ø x y z x,y x,z y,z x,y,z
false
x?y ? ? ? ?
x?y ? ? ? ? ? ? ?
x?y ? ? ? ? ? ? ? ?
x?(y?z) ? ? ? ? ? ? ? ?
true ? ? ? ? ? ? ? ? ? ?
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Implementation Context

• Traditionally, Boolean function manipulation has
  been performed using BDDs Bagnara98,Fecht97.
• There has been a stream of work on
  representations amenable to Prolog implementation
  Codish95, in particular for the subclass of
  definite positive functions, Def,
  GarciadelaBanda96,Genaim01.
• Most of these implementations, included those
  based on BDDs, require widening for large
  benchmarks.

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Implementation Methodology

• A Def-based groundness analyser HoweKing03 that
  does not require widening, was constructed by
  ensuring that frequent operations are the most
  lightweight.
• This suggests that a Pos-based groundness
  analysis should represent functions as
  multiheaded clauses (CNF).

file sim_v5-2 peval aircraft essln chat_80 aqua_c
meet 2192 2198 7063 8406 15483 112455
join 536 632 2742 1668 4663 35007
join (diff) 2 185 26 177 693 5173
equiv 536 632 2742 1668 4663 35007
project 788 805 3230 2035 5523 38163
rename 2052 2149 8963 5738 14540 103795
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Subtlety of Representation

• In fact Armstrong98 investigated Reduced
  Conjunctive Normal Form (RCNF), concluding that
  it performed reasonably well, but RCNF was
  ultimately rejected because BDDs performed 40
  faster.
• The subtlety of domain representation is
  illustrated by considering CNF and RNCF.
• RCNF is reduced so that no clause subsumes
  another CNF can include redundant clauses.
• Meet in O(n2) in RNCF O(1) in CNF. Message 1

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Subtlety of Stratification

• Neither RCNF nor CNF is canonical, so that
  equivalence cannot by checked syntactically.
• Armstrong98 compute the dual Blake canonical
  form (DBCF) of a function, and then test for
  syntactic identity.
• DBCF may be exponentially larger than RNCF and it
  must be completely computed, so cannot filter.
• Entailment of formulae can be detected using a
  hierarchy of incomplete algorithms (filters) of
  increasing complexity. Last complete. Message 2

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Pos as Multiheaded Clauses

• A multiheaded clause y1 ? ... ? yn ? x1 ? .. ? xm
  collapses to true if xi and yj overlap.
• A Boolean function is positive if and only if
  every clause in its CNF representation contains
  at least one positive literal.
• If m1, the multiheaded clause is a propositional
  Horn clause, which suggests that entailment
  checking might be specialised (filtered).
• Multiheaded clauses can be widened in linear time
  by discarding clauses with more than, say k,
  heads.

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Join for Multiheaded Clauses

• If f1 c1?? cn and f2 d1?? dm then
• f1 ? f2 ?i1n(?j1m(ci ? dj))
• Suppose ci y1 ? ... ? yk ? x1 ? ... ? xl and di
  u1 ? ... ? up ? v1 ? ... ? vq
• Then ci \/ di y1 ? ... ? yk ? u1 ? ... ? up ?
  x1 ? ... ? xl ? v1 ? ... ? vq so join is
  quadratic.

11
Entailment for Multiheaded Clauses

• A sound but incomplete test (entailslite) checks
  for
• ? i1l (Bi ? Hi) (B ? H) where B b1 ? ...
  ? bn and H h1 ? ? hm
• It binds each hi to false, each bj to true, and
  then propagates deterministic bindings in an
  attempt to find a solution to ?i1l (Bi ? Hi). It
  contains forward chaining for Horn clauses as a
  special case.
• It terminates either when a contradiction is
  found in ?i1l (Bi ? Hi) (then entailment
  follows) or when no more bindings can be
  propagated.
• Running-time is O(n) in variable occurrences.

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Entailment for Multiheaded Clauses (cont)

• The sound and complete algorithm entailsheavy
  applies case splitting if entailslite does not
  detect entailment.
• The number of cases is potentially exponential in
  the number of variables left unbound by
  entailslite.
• However, propagation occurs after each binding,
  and deep case splitting is rarely required.

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Projection for Multiheaded Clauses

• Projection is calculated using a Fourier-Motzkin
  style algorithm.
• Projection of a single variable, z, out of a pair
  of clauses (syllogising) is performed by
• z . ((y1 ? ... ? yp ? z ? x1 ? ... ? xq) ?
• (z ? yp1 ? ... ? yn ? xq1 ?
  ... ? xm))
• (y1 ? ... ? yn ? x1 ? ... ? xm)
• Syllogising gives a O(n2) blow-up and thus it is
  followed by a compaction step, based on
  entailslite, so the overall running time is
  O(n4).

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Experimental Results
file read MHC RNCF BDD GI
sim 0.22 1.09 24.78 0.25 0.62
rubik 0.21 0.22 25.32 0.20 0.16
parser 0.36 0.29 1.75 0.26 0.30
sim_v5-2 0.23 0.07 0.33 0.16 0.10
peval 0.18 0.64 4.63 0.16 1.30
aircraft 0.54 0.15 0.70 0.41 0.12
essln 0.48 0.19 20.72 0.37 0.30
chat_80 1.43 0.88 4.28 0.84 0.64
aqua_c 3.55 7.68 67.04 gt 120 6.59
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Experimental Results (cont)

• Multiheaded clauses perform consistently better
  than RCNF. This is not surprising given the cost
  of meet and equivalence checking.
• MHC compares favourably with BDDs, especially
  considering that the BDD operations exploit
  memoisation and are coded in C.
• In terms of runtime, MHC and BDDs give similar
  results, although BDDs perform well on sim.pl,
  whereas MHC perform well on sim_v5-2.pl.

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Instrumentation Results

• Instrumentation has revealed that the total
  number of times entailslite is invoked in
  checking F f almost never exceeds var(F).
• Therefore in practice entailsheavy exhibits cubic
  behaviour in the size of the input formulae.
• Further instrumentation has shown that the
  maximum number of heads observed in a clause is
  four.
• So only a few bindings typically need to be made
  to check entailment.

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Conclusions

• Illustrates the subtlety of choosing a
  representation and its associated operations,
  even for a well known domain (Pos).
• Success due to O(1) meet and stratified
  entailment.
• Analysers do not fire widening (for reasonable
  programs).
• Can code analysers in Prolog.