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# Small Formulas for Large Programs: On-line Constraint ...

Small Formulas for Large Programs: On-line Constraint ...

## Leaves of a Formula ●

Leaves of a Formula ● We consider quantifier­free formulas using the boolean connectives AND, OR, and NOT over any decidable theory . ● We assume formulas are in NNF. ● A formula that does not contain conjunction or disjunction is an atomic formula. ● Each syntactic occurrence of an atomic formula is a leaf. ● Example: :f(x) = 1 _ (:f(x) = 1 ^ x + y · 1) 3 distinct leaves

Redundant Leaves ● A leaf L is non­constraining in formula F if replacing L with true in F yields an equivalent formula. ● L is non­relaxing in F if replacing L with false is equivalent to F. ● L is redundant if it is non­constraining or non­relaxing. x = y | {z } L 0 ^ (f(x) = 1 | {z } L 1 Non­relaxing because formula is equivalent when it is replaced by false. _ (f(y) = 1 | {z } L 2 ^ x + y · 1)) | {z } L 3 Both non­constraining and non­relaxing.

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