Truth criterion quantitative constraints

Theorem 5 (Modal Truth Criterion for Quantitative Constraints). A constraint x.. k is necessarily satisfied in a situation s iff there is a situation t equal to or necessarily previous to s in which x.. k is satisfied and for every step C possibly before s that possibly violates x.. k, there is a step W necessarily before C that ensures x.. k in s. 

The statement and proof of the truth criterion (see Lakshmanan, 1989) for quantitative constraints is along the lines of Chapman s truth criterion for domain independent nonmonotonic planning. From this criterion, the plan modification operations that would ensure satisfaction of constraints are... 

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