Beyond NP: Quantifying over Answer Sets

Quantifying into NP*

The semantic theory presented in Montague (1974) permits quantification into a number of different syntactic categories, including t, CN and IV (S, N’ and VP, respectively, in more familiar notation). In this note I point out facts concerning so-called “inverse linking constructions” which indicate that quantification must also be permitted into term phrases (NPs) if Montague’s account of inten...

Answer Sets

This chapter is an introduction to Answer Set Prolog a language for knowledge representation and reasoning based on the answer set/stable model semantics of logic programs [44, 45]. The language has roots in declarative programing [52, 65], the syntax and semantics of standard Prolog [24, 23], disjunctive databases [66, 67] and nonmonotonic logic [79, 68, 61]. Unlike “standard” Prolog it allows...

Beyond P^(NP) - NEXP

Buhrman and Torenvliet created an oracle relative to which P NP = NEXP and thus P NP = P NEXP. Their proof uses a delicate nite injury argument that leads to a nonrecursive oracle. We simplify their proof removing the injury to create a recursive oracle making P NP = NEXP. In addition, in our construction we can make P = UP = NP \ coNP. This leads to the curious situation where LOW(NP) = P but ...

High Sets for NP

Splitting NP-Complete Sets

We show that a set is m-autoreducible if and only if it is m-mitotic. This solves a long standing open question in a surprising way. As a consequence of this unconditional result and recent work by Glaßer et al., complete sets for all of the following complexity classes are m-mitotic: NP, coNP, ⊕P, PSPACE, and NEXP, as well as all levels of PH, MODPH, and the Boolean hierarchy over NP. In the c...