43 research outputs found
Quantum Bounded Query Complexity
We combine the classical notions and techniques for bounded query classes
with those developed in quantum computing. We give strong evidence that quantum
queries to an oracle in the class NP does indeed reduce the query complexity of
decision problems. Under traditional complexity assumptions, we obtain an
exponential speedup between the quantum and the classical query complexity of
function classes.
For decision problems and function classes we obtain the following results: o
P_||^NP[2k] is included in EQP_||^NP[k] o P_||^NP[2^(k+1)-2] is included in
EQP^NP[k] o FP_||^NP[2^(k+1)-2] is included in FEQP^NP[2k] o FP_||^NP is
included in FEQP^NP[O(log n)] For sets A that are many-one complete for PSPACE
or EXP we show that FP^A is included in FEQP^A[1]. Sets A that are many-one
complete for PP have the property that FP_||^A is included in FEQP^A[1]. In
general we prove that for any set A there is a set X such that FP^A is included
in FEQP^X[1], establishing that no set is superterse in the quantum setting.Comment: 11 pages LaTeX2e, no figures, accepted for CoCo'9
Consistency of circuit lower bounds with bounded theories
Proving that there are problems in that require
boolean circuits of super-linear size is a major frontier in complexity theory.
While such lower bounds are known for larger complexity classes, existing
results only show that the corresponding problems are hard on infinitely many
input lengths. For instance, proving almost-everywhere circuit lower bounds is
open even for problems in . Giving the notorious difficulty of
proving lower bounds that hold for all large input lengths, we ask the
following question: Can we show that a large set of techniques cannot prove
that is easy infinitely often? Motivated by this and related
questions about the interaction between mathematical proofs and computations,
we investigate circuit complexity from the perspective of logic.
Among other results, we prove that for any parameter it is
consistent with theory that computational class , where is one of
the pairs: and , and , and
. In other words, these theories cannot establish
infinitely often circuit upper bounds for the corresponding problems. This is
of interest because the weaker theory already formalizes
sophisticated arguments, such as a proof of the PCP Theorem. These consistency
statements are unconditional and improve on earlier theorems of [KO17] and
[BM18] on the consistency of lower bounds with
A novel characterization of the complexity class based on counting and comparison
This is the author's accepted versionFinal version available from Elsevier via the DOI in this recordThe complexity class Θ2P, which is the class of languages recognizable by deterministic Turing machines in polynomial time with at most logarithmic many calls to an NP oracle, received extensive attention in the literature. Its complete problems can be characterized by different specific tasks, such as deciding whether the optimum solution of an NP problem is unique, or whether it is in some sense “odd” (e.g., whether its size is an odd number). In this paper, we introduce a new characterization of this class and its generalization ΘkP to the k-th level of the polynomial hierarchy. We show that problems in ΘkP are also those whose solution involves deciding, for two given sets A and B of instances of two Σk−1P-complete (or Πk−1P-complete) problems, whether the number of “yes”-instances in A is greater than those in B. Moreover, based on this new characterization, we provide a novel sufficient condition for ΘkP-hardness. We also define the general problem Comp-Validk, which is proven here Θk+1P-complete. Comp-Validk is the problem of deciding, given two sets A and B of quantified Boolean formulas with at most k alternating quantifiers, whether the number of valid formulas in A is greater than those in B. Notably, the problem Comp-Sat of deciding whether a set contains more satisfiable Boolean formulas than another set, which is a particular case of Comp-Valid1, demonstrates itself as a very intuitive Θ2P-complete problem. Nonetheless, to our knowledge, it eluded its formal definition to date. In fact, given its strict adherence to the count-and-compare semantics here introduced, Comp-Validk is among the most suitable tools to prove ΘkP-hardness of problems involving the counting and comparison of the number of “yes”-instances in two sets. We support this by showing that the Θ2P-hardness of the Max voting scheme over mCP-nets is easily obtained via the new characterization of ΘkP introduced in this paper.This work was supported by the UK EPSRC grants EP/J008346/1, EP/L012138/1, and EP/M025268/1, and by The Alan Turing Institute under the EPSRC grant EP/N510129/1. We thank Dominik Peters and the anonymous reviewers for their helpful comments on a preliminary version of the paper
Reasoning about uncertainty and explicit ignorance in generalized possibilistic logic
© 2014 The Authors and IOS Press. Generalized possibilistic logic (GPL) is a logic for reasoning about the revealed beliefs of another agent. It is a two-tier propositional logic, in which propositional formulas are encapsulated by modal operators that are interpreted in terms of uncertainty measures from possibility theory. Models of a GPL theory represent weighted epistemic states and are encoded as possibility distributions. One of the main features of GPL is that it allows us to explicitly reason about the ignorance of another agent. In this paper, we study two types of approaches for reasoning about ignorance in GPL, based on the idea of minimal specificity and on the notion of guaranteed possibility, respectively. We show how these approaches naturally lead to different flavours of the language of GPL and a number of decision problems, whose complexity ranges from the first to the third level of the polynomial hierarchy
Complexity-Theoretic Aspects of Expanding Cellular Automata
The expanding cellular automata (XCA) variant of cellular automata is
investigated and characterized from a complexity-theoretical standpoint. An XCA
is a one-dimensional cellular automaton which can dynamically create new cells
between existing ones. The respective polynomial-time complexity class is shown
to coincide with , that is, the class of decision
problems polynomial-time truth-table reducible to problems in . An
alternative characterization based on a variant of non-deterministic Turing
machines is also given. In addition, corollaries on select XCA variants are
proven: XCAs with multiple accept and reject states are shown to be
polynomial-time equivalent to the original XCA model. Finally, XCAs with
alternative acceptance conditions are considered and classified in terms of
and the Turing machine polynomial-time class
.Comment: 19 pages, 3 figure
Complexity and scalability of defeasible reasoning in many-valued weighted knowledge bases with typicality
Weighted knowledge bases for description logics with typicality under a
"concept-wise" multi-preferential semantics provide a logical interpretation of
MultiLayer Perceptrons. In this context, Answer Set Programming (ASP) has been
shown to be suitable for addressing defeasible reasoning in the finitely
many-valued case, providing a upper bound on the complexity of the
problem, nonetheless leaving unknown the exact complexity and only providing a
proof-of-concept implementation. This paper fulfils the lack by providing a
-completeness result and new ASP encodings that deal with weighted
knowledge bases with large search spaces.Comment: 14 pages 4, figure