7,633 research outputs found
From truth to computability I
The recently initiated approach called computability logic is a formal theory
of interactive computation. See a comprehensive online source on the subject at
http://www.cis.upenn.edu/~giorgi/cl.html . The present paper contains a
soundness and completeness proof for the deductive system CL3 which axiomatizes
the most basic first-order fragment of computability logic called the
finite-depth, elementary-base fragment. Among the potential application areas
for this result are the theory of interactive computation, constructive applied
theories, knowledgebase systems, systems for resource-bound planning and
action. This paper is self-contained as it reintroduces all relevant
definitions as well as main motivations.Comment: To appear in Theoretical Computer Scienc
Constant-Soundness Interactive Proofs for Local Hamiltonians
We give a quantum multiprover interactive proof
system for the local Hamiltonian problem in which there is a constant number of
provers, questions are classical of length polynomial in the number of qubits,
and answers are of constant length. The main novelty of our protocol is that
the gap between completeness and soundness is directly proportional to the
promise gap on the (normalized) ground state energy of the Hamiltonian. This
result can be interpreted as a concrete step towards a quantum PCP theorem
giving entangled-prover interactive proof systems for QMA-complete problems.
The key ingredient is a quantum version of the classical linearity test of
Blum, Luby, and Rubinfeld, where the function is
replaced by a pair of functions \Xlin, \Zlin:\{0,1\}^n\to \text{Obs}_d(\C),
the set of -dimensional Hermitian matrices that square to identity. The test
enforces that (i) each function is exactly linear,
\Xlin(a)\Xlin(b)=\Xlin(a+b) and \Zlin(a) \Zlin(b)=\Zlin(a+b), and (ii) the
two functions are approximately complementary, \Xlin(a)\Zlin(b)\approx
(-1)^{a\cdot b} \Zlin(b)\Xlin(a).Comment: 33 page
Toward Structured Proofs for Dynamic Logics
We present Kaisar, a structured interactive proof language for differential
dynamic logic (dL), for safety-critical cyber-physical systems (CPS). The
defining feature of Kaisar is *nominal terms*, which simplify CPS proofs by
making the frequently needed historical references to past program states
first-class. To support nominals, we extend the notion of structured proof with
a first-class notion of *structured symbolic execution* of CPS models. We
implement Kaisar in the theorem prover KeYmaera X and reproduce an example on
the safe operation of a parachute and a case study on ground robot control. We
show how nominals simplify common CPS reasoning tasks when combined with other
features of structured proof. We develop an extensive metatheory for Kaisar. In
addition to soundness and completeness, we show a formal specification for
Kaisar's nominals and relate Kaisar to a nominal variant of dL
On the Power of Many One-Bit Provers
We study the class of languages, denoted by \MIP[k, 1-\epsilon, s], which
have -prover games where each prover just sends a \emph{single} bit, with
completeness and soundness error . For the case that
(i.e., for the case of interactive proofs), Goldreich, Vadhan and Wigderson
({\em Computational Complexity'02}) demonstrate that \SZK exactly
characterizes languages having 1-bit proof systems with"non-trivial" soundness
(i.e., ). We demonstrate that for the case that
, 1-bit -prover games exhibit a significantly richer structure:
+ (Folklore) When , \MIP[k, 1-\epsilon, s]
= \BPP;
+ When , \MIP[k,
1-\epsilon, s] = \SZK;
+ When , \AM \subseteq \MIP[k, 1-\epsilon,
s];
+ For and sufficiently large , \MIP[k, 1-\epsilon, s]
\subseteq \EXP;
+ For , \MIP[k, 1, 1-\epsilon, s] = \NEXP.
As such, 1-bit -prover games yield a natural "quantitative" approach to
relating complexity classes such as \BPP,\SZK,\AM, \EXP, and \NEXP.
We leave open the question of whether a more fine-grained hierarchy (between
\AM and \NEXP) can be established for the case when
Stronger Methods of Making Quantum Interactive Proofs Perfectly Complete
This paper presents stronger methods of achieving perfect completeness in
quantum interactive proofs. First, it is proved that any problem in QMA has a
two-message quantum interactive proof system of perfect completeness with
constant soundness error, where the verifier has only to send a constant number
of halves of EPR pairs. This in particular implies that the class QMA is
necessarily included by the class QIP_1(2) of problems having two-message
quantum interactive proofs of perfect completeness, which gives the first
nontrivial upper bound for QMA in terms of quantum interactive proofs. It is
also proved that any problem having an -message quantum interactive proof
system necessarily has an -message quantum interactive proof system of
perfect completeness. This improves the previous result due to Kitaev and
Watrous, where the resulting system of perfect completeness requires
messages if not using the parallelization result.Comment: 41 pages; v2: soundness parameters improved, correction of a minor
error in Lemma 23, and removal of the sentences claiming that our techniques
are quantumly nonrelativizin
Generalized Quantum Arthur-Merlin Games
This paper investigates the role of interaction and coins in public-coin
quantum interactive proof systems (also called quantum Arthur-Merlin games).
While prior works focused on classical public coins even in the quantum
setting, the present work introduces a generalized version of quantum
Arthur-Merlin games where the public coins can be quantum as well: the verifier
can send not only random bits, but also halves of EPR pairs. First, it is
proved that the class of two-turn quantum Arthur-Merlin games with quantum
public coins, denoted qq-QAM in this paper, does not change by adding a
constant number of turns of classical interactions prior to the communications
of the qq-QAM proof systems. This can be viewed as a quantum analogue of the
celebrated collapse theorem for AM due to Babai. To prove this collapse
theorem, this paper provides a natural complete problem for qq-QAM: deciding
whether the output of a given quantum circuit is close to a totally mixed
state. This complete problem is on the very line of the previous studies
investigating the hardness of checking the properties related to quantum
circuits, and is of independent interest. It is further proved that the class
qq-QAM_1 of two-turn quantum-public-coin quantum Arthur-Merlin proof systems
with perfect completeness gives new bounds for standard well-studied classes of
two-turn interactive proof systems. Finally, the collapse theorem above is
extended to comprehensively classify the role of interaction and public coins
in quantum Arthur-Merlin games: it is proved that, for any constant m>1, the
class of problems having an m-turn quantum Arthur-Merlin proof system is either
equal to PSPACE or equal to the class of problems having a two-turn quantum
Arthur-Merlin game of a specific type, which provides a complete set of quantum
analogues of Babai's collapse theorem.Comment: 31 pages + cover page, the proof of Lemma 27 (Lemma 24 in v1) is
corrected, and a new completeness result is adde
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