On W[1]-Hardness as Evidence for Intractability

The central conjecture of parameterized complexity states that FPT !=W[1], and is generally regarded as the parameterized counterpart to P !=NP. We revisit the issue of the plausibility of FPT !=W[1], focusing on two aspects: the difficulty of proving the conjecture (assuming it holds), and how the relation between the two classes might differ from the one between P and NP. Regarding the first aspect, we give new evidence that separating FPT from W[1] would be considerably harder than doing the same for P and NP. Our main result regarding the relation between FPT and W[1] states that the closure of W[1] under relativization with FPT-oracles is precisely the class W[P], implying that either FPT is not low for W[1], or the W-Hierarchy collapses. This theorem also has consequences for the A-Hierarchy (a parameterized version of the Polynomial Hierarchy), namely that unless W[P] is a subset of some level A[t], there are structural differences between the A-Hierarchy and the Polynomial Hierarchy. We also prove that under the unlikely assumption that W[P] collapses to W[1] in a specific way, the collapse of any two consecutive levels of the A-Hierarchy implies the collapse of the entire hierarchy to a finite level; this extends a result of Chen, Flum, and Grohe (2005). Finally, we give weak (oracle-based) evidence that the inclusion W[t]subseteqA[t] is strict for t>1, and that the W-Hierarchy is proper. The latter result answers a question of Downey and Fellows (1993)

Diagonalizations over polynomial time computable sets

AbstractA formal notion of diagonalization is developed which allows to enforce properties that are related to the class of polynomial time computable sets (the class of polynomial time computable functions respectively), like, e.g., p-immunity. It is shown that there are sets—called p-generic— which have all properties enforceable by such diagonalizations. We study the behaviour and the complexity of p-generic sets. In particular, we show that the existence of p-generic sets in NP is oracle dependent, even if we assume P ≠ NP

Immunity and Pseudorandomness of Context-Free Languages

We discuss the computational complexity of context-free languages, concentrating on two well-known structural properties---immunity and pseudorandomness. An infinite language is REG-immune (resp., CFL-immune) if it contains no infinite subset that is a regular (resp., context-free) language. We prove that (i) there is a context-free REG-immune language outside REG/n and (ii) there is a REG-bi-immune language that can be computed deterministically using logarithmic space. We also show that (iii) there is a CFL-simple set, where a CFL-simple language is an infinite context-free language whose complement is CFL-immune. Similar to the REG-immunity, a REG-primeimmune language has no polynomially dense subsets that are also regular. We further prove that (iv) there is a context-free language that is REG/n-bi-primeimmune. Concerning pseudorandomness of context-free languages, we show that (v) CFL contains REG/n-pseudorandom languages. Finally, we prove that (vi) against REG/n, there exists an almost 1-1 pseudorandom generator computable in nondeterministic pushdown automata equipped with a write-only output tape and (vii) against REG, there is no almost 1-1 weakly pseudorandom generator computable deterministically in linear time by a single-tape Turing machine.Comment: A4, 23 pages, 10 pt. A complete revision of the initial version that was posted in February 200

The complexity of parameters for probabilistic and quantum computation

In this dissertation we study some effects of allowing computational models that use parameters whose own computational complexity has a strong effect on the computational complexity of the languages computable from the model. We show that in the probabilistic and quantum models there are parameter sets that allow one to obtain noncomputable outcomes;In Chapter 3 we define BP[beta]P the BPP class based on a coin with bias [beta]. We then show that if [beta] is BPP-computable then it is the case that BP[beta]P = BPP. We also show that each language L in P/CLog is in BP[beta]P for some [beta]. Hence there are some [beta] from which we can compute noncomputable languages. We also examine the robustness of the class BPP with respect to small variations from fairness in the coin;In Chapter 4 we consider measures that are based on polynomial-time computable sequences of biased coins in which the biases are bounded away from both zero and one (strongly positive P-sequences). We show that such a sequence [vector][beta] generates a measure [mu][vector][beta] equivalent to the uniform measure in the sense that if C is a class of languages closed under positive, polynomial-time, truth-table reductions with queries of linear length then C has [mu][vector][beta]-measure zero if and only if it has measure zero relative to the uniform measure [mu]. The classes P, NP, BPP, P/Poly, PH, and PSPACE are among those to which this result applies. Thus the measures of these much-studied classes are robust with respect to changes of this type in the underlying probability measure;In Chapter 5 we introduce the quantum computation model and the quantum complexity class BQP. We claim that the computational complexity of the amplitudes is a critical factor in determining the languages computable using the quantum model. Using results from chapter 3 we show that the quantum model can also compute noncomputable languages from some amplitude sets. Finally, we determine a restriction on the amplitude set to limit the model to the range of languages implicit in others\u27 typical meaning of the class BQP

Lower Bounds and Derandomization

A major open problem in complexity theory is to determine whether randomized complexity classes such as BPP, AM, and MA have any nontrivial derandomization. This thesis investigates the derandomization of two randomized versions of the polynomial hierarchy

A predicative variant of a realizability tripos for the Minimalist Foundation.

open2noHere we present a predicative variant of a realizability tripos validating the intensional level of the Minimalist Foundation extended with Formal Church thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel