Narrow Proofs May Be Maximally Long

We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n^Omega(w). This shows that the simple counting argument that any formula refutable in width w must have a proof in size n^O(w) is essentially tight. Moreover, our lower bound generalizes to polynomial calculus resolution (PCR) and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. Our results do not extend all the way to Lasserre, however, where the formulas we study have proofs of constant rank and size polynomial in both n and w

Narrow proofs may be maximally long

We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n(Omega(w)). This shows that the simple counting argument that any formula refutable in width w must have a proof in size n(O(w)) is essentially tight. Moreover, our lower bound generalizes to polynomial calculus resolution and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. The lower bound does not extend all the way to Lasserre, however, since we show that there the formulas we study have proofs of constant rank and size polynomial in both n and w.Peer ReviewedPostprint (author's final draft

A Decision Procedure for Herbrand Formulas without Skolemization

This paper describes a decision procedure for disjunctions of conjunctions of anti-prenex normal forms of pure first-order logic (FOLDNFs) that do not contain V within the scope of quantifiers. The disjuncts of these FOLDNFs are equivalent to prenex normal forms whose quantifier-free parts are conjunctions of atomic and negated atomic formulae (= Herbrand formulae). In contrast to the usual algorithms for Herbrand formulae, neither skolemization nor unification algorithms with function symbols are applied. Instead, a procedure is described that rests on nothing but equivalence transformations within pure first-order logic (FOL). This procedure involves the application of a calculus for negative normal forms (the NNF-calculus) with A -||- A & A (= &I) as the sole rule that increases the complexity of given FOLDNFs. The described algorithm illustrates how, in the case of Herbrand formulae, decision problems can be solved through a systematic search for proofs that reduce the number of applications of the rule &I to a minimum in the NNF-calculus. In the case of Herbrand formulae, it is even possible to entirely abstain from applying &I. Finally, it is shown how the described procedure can be used within an optimized general search for proofs of contradiction and what kind of questions arise for a &I-minimal proof strategy in the case of a general search for proofs of contradiction

A Partitioning Algorithm for Detecting Eventuality Coincidence in Temporal Double recurrence

A logical theory of regular double or multiple recurrence of eventualities, which are regular patterns of occurrences that are repeated, in time, has been developed within the context of temporal reasoning that enabled reasoning about the problem of coincidence. i.e. if two complex eventualities, or eventuality sequences consisting respectively of component eventualities x0, x1,....,xr and y0, y1, ..,ys both recur over an interval k and all eventualities are of fixed durations, is there a subinterval of k over which the occurrence xp and yq for p between 1 and r and q between 1 and s coincide. We present the ideas behind a new algorithm for detecting the coincidence of eventualities xp and yq within a cycle of the double recurrence of x and y. The algorithm is based on the novel concept of gcd partitions that requires the partitioning of each of the incidences of both x and y into eventuality sequences each of which components have a duration that is equal to the greatest common divisor of the durations of x and y. The worst case running time of the partitioning algorithm is linear in the maximum of the duration of x and that of y, while the worst case running time of an algorithm exploring a complete cycle is quadratic in the durations of x and y. Hence the partitioning algorithm works faster than the cyclical exploration in the worst case

Experimental Demonstration of Quantum Fully Homomorphic Encryption with Application in a Two-Party Secure Protocol

A fully homomorphic encryption system hides data from unauthorized parties while still allowing them to perform computations on the encrypted data. Aside from the straightforward benefit of allowing users to delegate computations to a more powerful server without revealing their inputs, a fully homomorphic cryptosystem can be used as a building block in the construction of a number of cryptographic functionalities. Designing such a scheme remained an open problem until 2009, decades after the idea was first conceived, and the past few years have seen the generalization of this functionality to the world of quantum machines. Quantum schemes prior to the one implemented here were able to replicate some features in particular use cases often associated with homomorphic encryption but lacked other crucial properties, for example, relying on continual interaction to perform a computation or leaking information about the encrypted data. We present the first experimental realization of a quantum fully homomorphic encryption scheme. To demonstrate the versatility of a a quantum fully homomorphic encryption scheme, we further present a toy two-party secure computation task enabled by our scheme

Testing the accuracy of reflection-based supermassive black hole spin measurements in AGN

X-ray reflection is a very powerful method to assess the spin of supermassive black holes (SMBHs) in active galactic nuclei (AGN), yet this technique is not universally accepted. Indeed, complex reprocessing (absorption, scattering) of the intrinsic spectra along the line of sight can mimic the relativistic effects on which the spin measure is based. In this work, we test the reliability of SMBH spin measurements that can currently be achieved through the simulations of high-quality XMM-Newton and NuSTAR spectra. Each member of our group simulated ten spectra with multiple components that are typically seen in AGN, such as warm and (partial-covering) neutral absorbers, relativistic and distant reflection, and thermal emission. The resulting spectra were blindly analysed by the other two members. Out of the 60 fits, 42 turn out to be physically accurate when compared to the input model. The SMBH spin is retrieved with success in 31 cases, some of which (9) are even found among formally inaccurate fits (although with looser constraints). We show that, at the high signal-to-noise ratio assumed in our simulations, neither the complexity of the multi-layer, partial-covering absorber nor the input value of the spin are the major drivers of our results. The height of the X-ray source (in a lamp-post geometry) instead plays a crucial role in recovering the spin. In particular, a success rate of 16 out of 16 is found among the accurate fits for a dimensionless spin parameter larger than 0.8 and a lamp-post height lower than five gravitational radii.Comment: 20 pages, 9 figures, 4 tables. Accepted for publication in A&

Experimental implementation of a Kochen-Specker set of quantum tests

The conflict between classical and quantum physics can be identified through a series of yes-no tests on quantum systems, without it being necessary that these systems be in special quantum states. Kochen-Specker (KS) sets of yes-no tests have this property and provide a quantum-versus-classical advantage that is free of the initialization problem that affects some quantum computers. Here, we report the first experimental implementation of a complete KS set that consists of 18 yes-no tests on four-dimensional quantum systems and show how to use the KS set to obtain a state-independent quantum advantage. We first demonstrate the unique power of this KS set for solving a task while avoiding the problem of state initialization. Such a demonstration is done by showing that, for 28 different quantum states encoded in the orbital-angular-momentum and polarization degrees of freedom of single photons, the KS set provides an impossible-to-beat solution. In a second experiment, we generate maximally contextual quantum correlations by performing compatible sequential measurements of the polarization and path of single photons. In this case, state independence is demonstrated for 15 different initial states. Maximum contextuality and state independence follow from the fact that the sequences of measurements project any initial quantum state onto one of the KS set's eigenstates. Our results show that KS sets can be used for quantum-information processing and quantum computation and pave the way for future developments.Comment: REVTeX, 15 pages, 4 figure

Higher order Maass forms

We determine the size of spaces of higher order Maass forms of even weight for cofinite discrete subgroups of PSL(2,R) with cusps. If exponential growth at the cusps is allowed, the spaces of Maass forms of a given order are as large as algebraic constrictions allow. We show the analogous statement for the spaces of holomorphic forms. For functions on the universal covering group of PSL(2,R) we introduce the concept of generalized weight. For the resulting spaces of higher order Maass forms with even generalized weight we show that the size is maximal

A Probabilistic Defense of Proper De Jure Objections to Theism

A common view among nontheists combines the de jure objection that theism is epistemically unacceptable with agnosticism about the de facto objection that theism is false. Following Plantinga, we can call this a “proper” de jure objection—a de jure objection that does not depend on any de facto objection. In his Warranted Christian Belief, Plantinga has produced a general argument against all proper de jure objections. Here I first show that this argument is logically fallacious (it makes subtle probabilistic fallacies disguised by scope ambiguities), and proceed to lay the groundwork for the construction of actual proper de jure objections