scholarly journals Logics for Sizes with Union or Intersection

2020 ◽  
Vol 34 (03) ◽  
pp. 2870-2876
Author(s):  
Caleb Kisby ◽  
Saul Blanco ◽  
Alex Kruckman ◽  
Lawrence Moss
Keyword(s):  
Boolean Algebra ◽  
Polynomial Time ◽  
Proof System ◽  
Presburger Arithmetic

This paper presents the most basic logics for reasoning about the sizes of sets that admit either the union of terms or the intersection of terms. That is, our logics handle assertions All x y and AtLeast x y, where x and y are built up from basic terms by either unions or intersections. We present a sound, complete, and polynomial-time decidable proof system for these logics. An immediate consequence of our work is the completeness of the logic additionally permitting More x y. The logics considered here may be viewed as efficient fragments of two logics which appear in the literature: Boolean Algebra with Presburger Arithmetic and the Logic of Comparative Cardinality.

scholarly journals Augmenting the Power of (Partial) MaxSat Resolution with Extension

2020 ◽  
Vol 34 (02) ◽  
pp. 1561-1568 ◽  
Author(s):  
Javier Larrosa ◽  
Emma Rollon
Keyword(s):  
Lower Bound ◽  
Objective Function ◽  
Lower Bounds ◽  
Polynomial Time ◽  
Proof System ◽  
Extension Rule ◽  
Resolution Proof

The refutation power of SAT and MaxSAT resolution is challenged by problems like the soft and hard Pigeon Hole Problem PHP for which short refutations do not exist. In this paper we augment the MaxSAT resolution proof system with an extension rule. The new proof system MaxResE is sound and complete, and more powerful than plain MaxSAT resolution, since it can refute the soft and hard PHP in polynomial time. We show that MaxResE refutations actually subtract lower bounds from the objective function encoded by the formulas. The resulting formula is the residual after the lower bound extraction. We experimentally show that the residual of the soft PHP (once its necessary cost of 1 has been efficiently subtracted with MaxResE) is a concise, easy to solve, satisfiable problem.

scholarly journals Examining Fragments of the Quantified Propositional Calculus

2008 ◽  
Vol 73 (3) ◽  
pp. 1051-1080 ◽  
Author(s):  
Steven Perron
Keyword(s):  
Polynomial Time ◽  
Propositional Calculus ◽  
Proof System ◽  
Polynomial Size ◽  
Propositional Proof ◽  
Polynomial Time Hierarchy

AbstractWhen restricted to proving formulas, the quantified propositional proof system is closely related to the theorems of Buss's theory . Namely, has polynomial-size proofs of the translations of theorems of , and proves that is sound. However, little is known about when proving more complex formulas. In this paper, we prove a witnessing theorem for similar in style to the KPT witnessing theorem for . This witnessing theorem is then used to show that proves is sound with respect to formulas. Note that unless the polynomial-time hierarchy collapses is the weakest theory in the S2 hierarchy for which this is true. The witnessing theorem is also used to show that is p-equivalent to a quantified version of extended-Frege for prenex formulas. This is followed by a proof that Gi, p-simulates with respect to all quantified propositional formulas. We finish by proving that S2 can be axiomatized by plus axioms stating that the cut-free version of is sound. All together this shows that the connection between and does not extend to more complex formulas.

Forcing on bounded arithmetic II

1998 ◽  
Vol 63 (3) ◽  
pp. 860-868 ◽  
Author(s):  
Gaisi Takeuti ◽  
Masahiro Yasumoto
Keyword(s):  
Boolean Algebra ◽  
Polynomial Time ◽  
Bounded Arithmetic ◽  
Pigeon Hole Principle ◽  
Polynomial Size ◽  
Np Complete ◽  
Computable Functions

Forcing method on Bounded Arithmetic was first introduced by J. B. Paris and A. Wilkie in [10]. Then in [1], [2] and [3], M. Ajtai used the method to get excellent results on the pigeon hole principle and the modulo p counting principle. The forcing method on Bounded arithmetic was further developed by P. Beame, J. Krajíček and S. Riis in [4], [7], [6], [8], [5], [12], [11], [13]. It should be noted that J. Krajíček and P. Pudlák used an idea of Boolean valued in [9] and also Boolean valued notion is efficiently used for model theoretic constructions in [7], [6], [8], [5].In our previous paper [14], we developed a Boolean valued version of forcing on Bounded Arithmetic using Boolean algebra which is generated by polynomial size circuits from Boolean variables and discussed its relation with NP = co-NP problem and P = NP problem. Especially we proved the following theorem and related theorems as Theorems 2, 3 and 4 in Section 2.Theorem. If M[G] is not a model of S2, then NP ≠ co-NP and therefore P ≠ NP.However in the proof of the Theorem, we used a consequence of P = NP. More precisely we used the following as a consequence of NP = co-NP, though it is a consequence of P = NP but not a consequence of NP = co-NP.Suppose that NP = co-NP holds. Then there exists an NP-complete predicate ∃x ≤ t(a) A(x,a) with sharply bounded A(x, a) and a sharply bounded B(y, a) such that ∃x ≤ t(a) A(x,a) ↔ ∀y ≤ s(a)B(y, a). Then there exists polynomial time computable functions f and g such that the following two sequents hold.

Two-message quantum interactive proofs and the quantum separability problem

2014 ◽  
Vol 14 (5&6) ◽  
pp. 384-416
Author(s):  
Patrick Hayden ◽  
Kevin Milner ◽  
Mark M. Wilde
Keyword(s):  
Quantum State ◽  
Polynomial Time ◽  
Mixed State ◽  
Quantum Circuit ◽  
Proof System ◽  
Interactive Proof ◽  
Proof Systems ◽  
Separability Problem ◽  
Promise Problems ◽  
Interactive Proof System

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.

Deciding Boolean Algebra with Presburger Arithmetic

2006 ◽  
Vol 36 (3) ◽  
pp. 213-239 ◽  
Author(s):  
Viktor Kuncak ◽  
Huu Hai Nguyen ◽  
Martin Rinard
Keyword(s):  
Boolean Algebra ◽  
Presburger Arithmetic

scholarly journals Classical zero-knowledge arguments for quantum computations

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 266 ◽  
Author(s):  
Thomas Vidick ◽  
Tina Zhang
Keyword(s):  
Polynomial Time ◽  
Proof System ◽  
Knowledge Argument ◽  
Zero Knowledge ◽  
Quantum Computations ◽  
Quantum Polynomial ◽  
Zero Knowledge Proof

We show that every language in QMA admits a classical-verifier, quantum-prover zero-knowledge argument system which is sound against quantum polynomial-time provers and zero-knowledge for classical (and quantum) polynomial-time verifiers. The protocol builds upon two recent results: a computational zero-knowledge proof system for languages in QMA, with a quantum verifier, introduced by Broadbent et al. (FOCS 2016), and an argument system for languages in QMA, with a classical verifier, introduced by Mahadev (FOCS 2018).

scholarly journals Description Logics That Count, and What They Can and Cannot Count

10.29007/ltzn ◽  
2020 ◽  
Author(s):  
Franz Baader ◽  
Filippo De Bortoli
Keyword(s):  
Boolean Algebra ◽  
Natural Number ◽  
Predicate Logic ◽  
Description Logics ◽  
Expressive Power ◽  
Cardinality Constraints ◽  
Presburger Arithmetic ◽  
First Order ◽  
First Order Predicate Logic

Simple counting quantifiers that can be used to compare the number of role successors of an individual or the cardinality of a concept with a fixed natural number have been employed in Description Logics (DLs) for more than two decades under the respective names of number restrictions and cardinality restriction on concepts. Recently, we have considerably extended the expressivity of such quantifiers by allowing to impose set and cardinality constraints formulated in the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA) on sets of role successors and concepts, respectively. We were able to prove that this extension does not increase the complexity of reasoning.In the present paper, we investigate the expressive power of the DLs obtained this way, using appropriate bisimulation characterizations and 0--1 laws as tools for distinguishing the expressiveness of different logics. In particular, we show that, in contrast to most classical DLs, these logics are no longer expressible in first-order predicate logic (FOL), and we characterize their first-order fragments. In most of our previous work on DLs with QFBAPA-based set and cardinality constraints we have employed finiteness restrictions on interpretations to ensure that the obtained sets are finite. Here we dispense with these restrictions to make the comparison with classical DLs, where one usually considers arbitrary models rather than finite ones, easier. It turns out that doing so does not change the complexity of reasoning.

scholarly journals The computational SLR: a logic for reasoning about computational indistinguishability

2010 ◽  
Vol 20 (5) ◽  
pp. 951-975 ◽  
Author(s):  
YU ZHANG
Keyword(s):  
Polynomial Time ◽  
Formal Proof ◽  
Proof System ◽  
Pseudorandom Generator ◽  
Standard Definition ◽  
Definition Of ◽  
Probabilistic Polynomial Time

Computational indistinguishability is a notion in complexity-theoretic cryptography and is used to define many security criteria. However, in traditional cryptography, proving computational indistinguishability is usually informal and becomes error-prone when cryptographic constructions are complex. This paper presents a formal proof system based on an extension of Hofmann's SLR language, which can capture probabilistic polynomial-time computations through typing and is sufficient for expressing cryptographic constructions. In particular, we define rules that directly justify the computational indistinguishability between programs, and then prove that these rules are sound with respect to the set-theoretic semantics, and thus the standard definition of security. We also show that it is applicable in cryptography by verifying, in our proof system, Goldreich and Micali's construction of a pseudorandom generator, and the equivalence between next-bit unpredictability and pseudorandomness.

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