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The exhaustivity checks in the pattern matcher build a propositional
formula that must be converted into conjunctive normal form (CNF) in
order to be amenable to the following DPLL decision procedure.
However, the simple conversion into CNF via negation normal form and
Shannon expansion that was used has exponential worst-case complexity
and thus even simple problems could become untractable.
A better approach is to use a transformation into an _equisatisfiable_
CNF-formula (by generating auxiliary variables) that runs with linear
complexity. The commonly known Tseitin transformation uses bi-
implication. I have choosen for an enhancement: the Plaisted
transformation which uses implication only, thus the resulting CNF
formula has (on average) only half of the clauses of a Tseitin
transformation.
The Plaisted transformation uses the polarities of sub-expressions to
figure out which part of the bi-implication can be omitted. However,
if all sub-expressions have positive polarity (e.g., after
transformation into negation normal form) then the conversion is
rather simple and the pseudo-normalization via NNF increases chances
only one side of the bi-implication is needed.
I implemented only optimizations that had a substantial
effect on formula size:
- formula simplification, extraction of multi argument operands
- if a formula is already in CNF then the Tseitin/Plaisted
transformation is omitted
- Plaisted via NNF
- omitted: (sharing of sub-formulas is also not implemented)
- omitted: (clause subsumption)
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