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- W2914656101 abstract "Given a fixed computable binary operation f , we study the complexity of the following generation problem: the input consists of strings a 1 , … , a n , b . The question is whether b is in the closure of { a 1 , … , a n } under operation f . For several subclasses of operations we prove tight upper and lower bounds for the generation problems. For example, we prove exponential-time upper and lower bounds for generation problems of length-monotonic polynomial-time computable operations. Other bounds involve classes like NP and PSPACE . Here, the class of bivariate polynomials with positive coefficients turns out to be the most interesting class of operations. We show that many of the corresponding generation problems belong to NP . However, we do not know this for all of them, e.g., for x 2 + 2 y this is an open question. We prove NP -completeness for polynomials x a y b c where a , b , c ⩾ 1 . Also, we show NP -hardness for polynomials like x 2 + 2 y . As a by-product we obtain NP -completeness of the extended sum-of-subset problem SOS c = { ( w 1 , … , w n , z ) : ∃ I ⊆ { 1 , … , n } ( ∑ i ∈ I w i c = z ) } for any c ⩾ 1 ." @default.
- W2914656101 created "2019-02-21" @default.
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- W2914656101 date "2005-11-01" @default.
- W2914656101 modified "2023-09-24" @default.
- W2914656101 title "Generation problems" @default.
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- W2914656101 doi "https://doi.org/10.1016/j.tcs.2005.07.011" @default.
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