scholarly journals QMA variants with polynomially many provers

2013 ◽  
Vol 13 (1&2) ◽  
pp. 135-157
Author(s):  
Sevag Gharibian ◽  
Jamie Sikora ◽  
Sarvagya Upadhyay
Keyword(s):  
Bounded Function ◽  
Expressive Power ◽  
Proof System ◽  
Alternate Proof ◽  
Cone Programming ◽  
Parallel Repetition ◽  
Proof Systems ◽  
Classical Proof

We study three variants of multi-prover quantum Merlin-Arthur proof systems. We first show that the class of problems that can be efficiently verified using polynomially many quantum proofs, each of logarithmic-size, is exactly \class{MQA} (also known as QCMA), the class of problems which can be efficiently verified via a classical proof and a quantum verifier. We then study the class $\class{BellQMA}(\poly)$, characterized by a verifier who first applies unentangled, nonadaptive measurements to each of the polynomially many proofs, followed by an arbitrary but efficient quantum verification circuit on the resulting measurement outcomes. We show that if the number of outcomes per nonadaptive measurement is a polynomially-bounded function, then the expressive power of the proof system is exactly \class{QMA}. Finally, we study a class equivalent to \class{QMA}($m$), denoted $\class{SepQMA}(m)$, where the verifier's measurement operator corresponding to outcome {\it accept} is a fully separable operator across the $m$ quantum proofs. Using cone programming duality, we give an alternate proof of a result of Harrow and Montanaro [FOCS, pp. 633--642 (2010)] that shows a perfect parallel repetition theorem for $\class{SepQMA}(m)$ for any $m$.

scholarly journals A Tableau Proof System with Names for Modal Mu-calculus

10.29007/lwqm ◽  
2018 ◽  
Author(s):  
Colin Stirling
Keyword(s):  
Modal Logic ◽  
Expressive Power ◽  
Proof System ◽  
Temporal Logics ◽  
General Issue ◽  
Proof Systems

Howard Barringer was a pioneer in the study of temporal logics withfixpoints. Their addition adds considerable expressive power.One general issue is how to defineproof systems for such logics. Here we examine proof systems formodal logic with fixpoints. We present a tableau proof systemfor checking validity of formulaswhich uses names to keep track of unfoldings of fixpoint variables.

scholarly journals Computational and Proof Complexity of Partial String Avoidability

2021 ◽  
Vol 13 (1) ◽  
pp. 1-25
Author(s):  
Dmitry Itsykson ◽  
Alexander Okhotin ◽  
Vsevolod Oparin
Keyword(s):  
Lower Bounds ◽  
Circuit Complexity ◽  
Conjunctive Normal Form ◽  
Proof Complexity ◽  
Proof Systems ◽  
Finite Set ◽  
Propositional Proof Systems ◽  
Classical Proof ◽  
Exponential Size ◽  
Infinite String

The partial string avoidability problem is stated as follows: given a finite set of strings with possible “holes” (wildcard symbols), determine whether there exists a two-sided infinite string containing no substrings from this set, assuming that a hole matches every symbol. The problem is known to be NP-hard and in PSPACE, and this article establishes its PSPACE-completeness. Next, string avoidability over the binary alphabet is interpreted as a version of conjunctive normal form satisfiability problem, where each clause has infinitely many shifted variants. Non-satisfiability of these formulas can be proved using variants of classical propositional proof systems, augmented with derivation rules for shifting proof lines (such as clauses, inequalities, polynomials, etc.). First, it is proved that there is a particular formula that has a short refutation in Resolution with a shift rule but requires classical proofs of exponential size. At the same time, it is shown that exponential lower bounds for classical proof systems can be translated for their shifted versions. Finally, it is shown that superpolynomial lower bounds on the size of shifted proofs would separate NP from PSPACE; a connection to lower bounds on circuit complexity is also established.

scholarly journals Constructing weak simulations from linear implications for processes with private names

2019 ◽  
Vol 29 (8) ◽  
pp. 1275-1308 ◽  
Author(s):  
Ross Horne ◽  
Alwen Tiu
Keyword(s):  
Expressive Power ◽  
Dual Pair ◽  
Proof System ◽  
Branching Time ◽  
First Order ◽  
Order Extension

AbstractThis paper clarifies that linear implication defines a branching-time preorder, preserved in all contexts, when used to compare embeddings of process in non-commutative logic. The logic considered is a first-order extension of the proof system BV featuring a de Morgan dual pair of nominal quantifiers, called BV1. An embedding of π-calculus processes as formulae in BV1 is defined, and the soundness of linear implication in BV1 with respect to a notion of weak simulation in the π -calculus is established. A novel contribution of this work is that we generalise the notion of a ‘left proof’ to a class of formulae sufficiently large to compare embeddings of processes, from which simulating execution steps are extracted. We illustrate the expressive power of BV1 by demonstrating that results extend to the internal π -calculus, where privacy of inputs is guaranteed. We also remark that linear implication is strictly finer than any interleaving preorder.

REMARKS ON A QUERY-BASED VARIANT OF THE PARALLEL REPETITION THEOREM

2001 ◽  
Vol 12 (04) ◽  
pp. 517-531
Author(s):  
OLEG VERBITSKY
Keyword(s):  
Strong Correlation ◽  
Error Probability ◽  
Cryptographic Protocols ◽  
Parallel Execution ◽  
Proof System ◽  
Interactive Proof ◽  
Parallel Repetition ◽  
Interactive Proof System

The Parallel Repetition Theorem says that n-fold parallel execution of a two-prover one-round interactive proof system reduces the error probability exponentially in n. The bound on the error probability of the parallelized system depends on the error probability and the answer size of the single proof system. It is still unknown whether the theorem holds true with a bound depending only on the query size. This kind of a bound may be preferable whenever the query size is considerably smaller than the answer size, what really happens in some cryptographic protocols. Such a bound is only known in the case that queries to the provers are independent. The present paper extends this result to some cases of strong correlation between queries. In particular, a query-based variant of the Parallel Repetition Theorem is proven when the graph of dependence between queries to the provers is a tree and, in a bit weaker form, when this graph is a cycle.

scholarly journals A duality between proof systems for cyclic term graphs

2007 ◽  
Vol 17 (3) ◽  
pp. 439-484 ◽  
Author(s):  
CLEMENS GRABMAYER
Keyword(s):  
Close Connection ◽  
Proof System ◽  
Consistency Checking ◽  
Duality Result ◽  
First Order ◽  
Proof Systems ◽  
Similar Proof ◽  
Equational Specifications ◽  
Cyclic Term

This paper presents a proof-theoretic observation about two kinds of proof systems for bisimilarity between cyclic term graphs.First we consider proof systems for demonstrating that μ term specifications of cyclic term graphs have the same tree unwinding. We establish a close connection between adaptations for μ terms over a general first-order signature of the coinductive axiomatisation of recursive type equivalence by Brandt and Henglein (Brandt and Henglein 1998) and of a proof system by Ariola and Klop (Ariola and Klop 1995) for consistency checking. We show that there exists a simple duality by mirroring between derivations in the former system and formalised consistency checks, which are called ‘consistency unfoldings', in the latter. This result sheds additional light on the axiomatisation of Brandt and Henglein: it provides an alternative soundness proof for the adaptation considered here.We then outline an analogous duality result that holds for a pair of similar proof systems for proving that equational specifications of cyclic term graphs are bisimilar.

scholarly journals Universal Hash Proofs and a Paradigm for Adaptive Chosen Ciphertext Secure Public-Key Encryption

2001 ◽  
Vol 8 (37) ◽  
Author(s):  
Ronald Cramer ◽  
Victor Shoup
Keyword(s):  
Special Kind ◽  
Public Key ◽  
Proof System ◽  
Reasonable Assumption ◽  
Encryption Scheme ◽  
Public Key Encryption ◽  
Proof Systems ◽  
Encryption Schemes ◽  
Hash Proof System ◽  
Chosen Ciphertext Attack

We present several new and fairly practical public-key encryption schemes and prove them secure against adaptive chosen ciphertext attack. One scheme is based on Paillier's Decision Composite Residuosity (DCR) assumption, while another is based in the classical Quadratic Residuosity (QR) assumption. The analysis is in the standard cryptographic model, i.e., the security of our schemes does not rely on the Random Oracle model.<br /> <br />We also introduce the notion of a universal hash proof system. Essentially, this is a special kind of non-interactive zero-knowledge proof system for an NP language. We do not show that universal hash proof systems exist for all NP languages, but we do show how to construct very efficient universal hash proof systems for a general class of group-theoretic language membership problems.<br /> <br />Given an efficient universal hash proof system for a language with certain natural cryptographic indistinguishability properties, we show how to construct an efficient public-key encryption schemes secure against adaptive chosen ciphertext attack in the standard model. Our construction only uses the universal hash proof system as a primitive: no other primitives are required, although even more efficient encryption schemes can be obtained by using hash functions with appropriate collision-resistance properties. We show how to construct efficient universal hash proof systems for languages related to the DCR and QR assumptions. From these we get corresponding public-key encryption schemes that are secure under these assumptions. We also show that the Cramer-Shoup encryption scheme (which up until now was the only practical encryption scheme that could be proved secure against adaptive chosen ciphertext attack under a reasonable assumption, namely, the Decision Diffie-Hellman assumption) is also a special case of our general theory.

Relational proof system for relevant logics

1992 ◽  
Vol 57 (4) ◽  
pp. 1425-1440 ◽  
Author(s):  
Ewa Orlowska
Keyword(s):  
Natural Deduction ◽  
Proof System ◽  
Algebraic Semantics ◽  
Proof Systems ◽  
Relevant Logics ◽  
Deduction Rules

AbstractA method is presented for constructing natural deduction-style systems for propositional relevant logics. The method consists in first translating formulas of relevant logics into ternary relations, and then defining deduction rules for a corresponding logic of ternary relations. Proof systems of that form are given for various relevant logics. A class of algebras of ternary relations is introduced that provides a relation-algebraic semantics for relevant logics.

The relative efficiency of propositional proof systems

1979 ◽  
Vol 44 (1) ◽  
pp. 36-50 ◽  
Author(s):  
Stephen A. Cook ◽  
Robert A. Reckhow
Keyword(s):  
Relative Efficiency ◽  
Natural Deduction ◽  
Type Systems ◽  
Linear Increase ◽  
Proof System ◽  
Proof Systems ◽  
System A ◽  
Frege Systems ◽  
Propositional Proof Systems ◽  
Hilbert Type

We are interested in studying the length of the shortest proof of a propositional tautology in various proof systems as a function of the length of the tautology. The smallest upper bound known for this function is exponential, no matter what the proof system. A question we would like to answer (but have not been able to) is whether this function has a polynomial bound for some proof system. (This question is motivated below.) Our results here are relative results.In §§2 and 3 we indicate that all standard Hilbert type systems (or Frege systems, as we call them) and natural deduction systems are equivalent, up to application of a polynomial, as far as minimum proof length goes. In §4 we introduce extended Frege systems, which allow introduction of abbreviations for formulas. Since these abbreviations can be iterated, they eliminate the need for a possible exponential growth in formula length in a proof, as is illustrated by an example (the pigeonhole principle). In fact, Theorem 4.6 (which is a variation of a theorem of Statman) states that with a penalty of at most a linear increase in the number of lines of a proof in an extended Frege system, no line in the proof need be more than a constant times the length of the formula proved.

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