11,821 research outputs found
Characterizing the Existence of Optimal Proof Systems and Complete Sets for Promise Classes.
In this paper we investigate the following two questions: Q1: Do there exist optimal proof systems for a given language L? Q2: Do there exist complete problems for a given promise class C? For concrete languages L (such as TAUT or SAT) and concrete promise classes C (such as NP∩coNP, UP, BPP, disjoint NP-pairs etc.), these ques-tions have been intensively studied during the last years, and a number of characterizations have been obtained. Here we provide new character-izations for Q1 and Q2 that apply to almost all promise classes C and languages L, thus creating a unifying framework for the study of these practically relevant questions. While questions Q1 and Q2 are left open by our results, we show that they receive affirmative answers when a small amount on advice is avail-able in the underlying machine model. This continues a recent line of research on proof systems with advice started by Cook and Kraj́ıček [6]
Different Approaches to Proof Systems
The classical approach to proof complexity perceives proof systems as deterministic, uniform, surjective, polynomial-time computable functions that map strings to (propositional) tautologies. This approach has been intensively studied since the late 70’s and a lot of progress has been made. During the last years research was started investigating alternative notions of proof systems. There are interesting results stemming from dropping the uniformity requirement, allowing oracle access, using quantum computations, or employing probabilism. These lead to different notions of proof systems for which we survey recent results in this paper
Two-message quantum interactive proofs and the quantum separability problem
Suppose that a polynomial-time mixed-state quantum circuit, described as a
sequence of local unitary interactions followed by a partial trace, generates a
quantum state shared between two parties. One might then wonder, does this
quantum circuit produce a state that is separable or entangled? Here, we give
evidence that it is computationally hard to decide the answer to this question,
even if one has access to the power of quantum computation. We begin by
exhibiting a two-message quantum interactive proof system that can decide the
answer to a promise version of the question. We then prove that the promise
problem is hard for the class of promise problems with "quantum statistical
zero knowledge" (QSZK) proof systems by demonstrating a polynomial-time Karp
reduction from the QSZK-complete promise problem "quantum state
distinguishability" to our quantum separability problem. By exploiting Knill's
efficient encoding of a matrix description of a state into a description of a
circuit to generate the state, we can show that our promise problem is NP-hard
with respect to Cook reductions. Thus, the quantum separability problem (as
phrased above) constitutes the first nontrivial promise problem decidable by a
two-message quantum interactive proof system while being hard for both NP and
QSZK. We also consider a variant of the problem, in which a given
polynomial-time mixed-state quantum circuit accepts a quantum state as input,
and the question is to decide if there is an input to this circuit which makes
its output separable across some bipartite cut. We prove that this problem is a
complete promise problem for the class QIP of problems decidable by quantum
interactive proof systems. Finally, we show that a two-message quantum
interactive proof system can also decide a multipartite generalization of the
quantum separability problem.Comment: 34 pages, 6 figures; v2: technical improvements and new result for
the multipartite quantum separability problem; v3: minor changes to address
referee comments, accepted for presentation at the 2013 IEEE Conference on
Computational Complexity; v4: changed problem names; v5: updated references
and added a paragraph to the conclusion to connect with prior work on
separability testin
Logical closure properties of propositional proof systems - (Extended abstract)
In this paper we define and investigate basic logical closure properties of propositional proof systems such as closure of arbitrary proof systems under modus ponens or substitutions. As our main result we obtain a purely logical characterization of the degrees of schematic extensions of EF in terms of a simple combination of these properties. This result underlines the empirical evidence that EF and its extensions admit a robust definition which rests on only a few central concepts from propositional logic
Infinite subgame perfect equilibrium in the Hausdorff difference hierarchy
Subgame perfect equilibria are specific Nash equilibria in perfect
information games in extensive form. They are important because they relate to
the rationality of the players. They always exist in infinite games with
continuous real-valued payoffs, but may fail to exist even in simple games with
slightly discontinuous payoffs. This article considers only games whose outcome
functions are measurable in the Hausdorff difference hierarchy of the open sets
(\textit{i.e.} when in the Baire space), and it characterizes the
families of linear preferences such that every game using these preferences has
a subgame perfect equilibrium: the preferences without infinite ascending
chains (of course), and such that for all players and and outcomes
we have . Moreover at
each node of the game, the equilibrium constructed for the proof is
Pareto-optimal among all the outcomes occurring in the subgame. Additional
results for non-linear preferences are presented.Comment: The alternative definition of the difference hierarchy has changed
slightl
Quantum Proofs
Quantum information and computation provide a fascinating twist on the notion
of proofs in computational complexity theory. For instance, one may consider a
quantum computational analogue of the complexity class \class{NP}, known as
QMA, in which a quantum state plays the role of a proof (also called a
certificate or witness), and is checked by a polynomial-time quantum
computation. For some problems, the fact that a quantum proof state could be a
superposition over exponentially many classical states appears to offer
computational advantages over classical proof strings. In the interactive proof
system setting, one may consider a verifier and one or more provers that
exchange and process quantum information rather than classical information
during an interaction for a given input string, giving rise to quantum
complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum
analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit
some properties from their classical counterparts, they also possess distinct
and uniquely quantum features that lead to an interesting landscape of
complexity classes based on variants of this model.
In this survey we provide an overview of many of the known results concerning
quantum proofs, computational models based on this concept, and properties of
the complexity classes they define. In particular, we discuss non-interactive
proofs and the complexity class QMA, single-prover quantum interactive proof
systems and the complexity class QIP, statistical zero-knowledge quantum
interactive proof systems and the complexity class \class{QSZK}, and
multiprover interactive proof systems and the complexity classes QMIP, QMIP*,
and MIP*.Comment: Survey published by NOW publisher
Information-Sharing and Privacy in Social Networks
We present a new model for reasoning about the way information is shared
among friends in a social network, and the resulting ways in which it spreads.
Our model formalizes the intuition that revealing personal information in
social settings involves a trade-off between the benefits of sharing
information with friends, and the risks that additional gossiping will
propagate it to people with whom one is not on friendly terms. We study the
behavior of rational agents in such a situation, and we characterize the
existence and computability of stable information-sharing networks, in which
agents do not have an incentive to change the partners with whom they share
information. We analyze the implications of these stable networks for social
welfare, and the resulting fragmentation of the social network
On The Power of Tree Projections: Structural Tractability of Enumerating CSP Solutions
The problem of deciding whether CSP instances admit solutions has been deeply
studied in the literature, and several structural tractability results have
been derived so far. However, constraint satisfaction comes in practice as a
computation problem where the focus is either on finding one solution, or on
enumerating all solutions, possibly projected to some given set of output
variables. The paper investigates the structural tractability of the problem of
enumerating (possibly projected) solutions, where tractability means here
computable with polynomial delay (WPD), since in general exponentially many
solutions may be computed. A general framework based on the notion of tree
projection of hypergraphs is considered, which generalizes all known
decomposition methods. Tractability results have been obtained both for classes
of structures where output variables are part of their specification, and for
classes of structures where computability WPD must be ensured for any possible
set of output variables. These results are shown to be tight, by exhibiting
dichotomies for classes of structures having bounded arity and where the tree
decomposition method is considered
- …