Step In Proof Research Articles

The ability of senior high school students in comprehending mathematical proofs

Mathematical proving ability is an ability which has function to improve mathematical high order thinking and reasoning skill. This ability consists of two sub-abilities, which are mathematical proof comprehension and mathematical proof construction. Both of the sub-abilities are related each other. To construct mathematical proofs, students need to have good mathematical proof comprehension. Due to the importance of mathematical proof comprehension, a qualitative descriptive study was conducted to describe senior high school students’ ability in mathematical proof comprehension. A test was given to 10 students of 11th grade in one of senior high schools at Bandung, West Java, Indonesia. The students’ works were analyzed for each proof comprehension indicator. Overall, the research found four results, the students: (i) were good in giving a reason for each step of proofs; (ii) got difficulties in validating a step of proof; (iii) were good in applying the steps of a proof to prove similar statement; and (iv) failed in determining and proving a notion based on pattern from the given statements. This study concludes that the senior high school students’ ability in comprehending mathematical proofs has not been satisfied. In several parts, it needs to be improved during mathematics learning process.

Generalisation of proof simulation procedures for Frege systems by M.L. Bonet and S.R. Buss

ABSTRACTIn this paper, we present a generalisation of proof simulation procedures for Frege systems by Bonet and Buss to some logics for which the deduction theorem does not hold. In particular, we study the case of finite-valued Łukasiewicz logics. To this end, we provide proof systems and which augment Avron's Frege system HŁuk with nested and general versions of the disjunction elimination rule, respectively. For these systems, we provide upper bounds on speed-ups w.r.t. both the number of steps in proofs and the length of proofs. We also consider Tamminga's natural deduction and Avron's hypersequent calculus GŁuk for 3-valued Łukasiewicz logic and generalise our results considering the disjunction elimination rule to all finite-valued Łukasiewicz logics.

Cohesive fracture with irreversibility: Quasistatic evolution for a model subject to fatigue

In this paper we prove the existence of quasistatic evolutions for a cohesive fracture on a prescribed crack surface, in small-strain antiplane elasticity. The main feature of the model is that the density of the energy dissipated in the fracture process depends on the total variation of the amplitude of the jump. Thus, any change in the crack opening entails a loss of energy, until the crack is complete. In particular this implies a fatigue phenomenon, i.e. a complete fracture may be produced by oscillation of small jumps. The first step of the existence proof is the construction of approximate evolutions obtained by solving discrete-time incremental minimum problems. The main difficulty in the passage to the continuous-time limit is that we lack of controls on the variations of the jump of the approximate evolutions. Therefore we resort to a weak formulation where the variation of the jump is replaced by a Young measure. Eventually, after proving the existence in this weak formulation, we improve the result by showing that the Young measure is concentrated on a function and coincides with the variation of the jump of the displacement.

Mechanical synthesis of sorting algorithms for binary trees by logic and combinatorial techniques

We develop logic and combinatorial methods for automating the generation of sorting algorithms for binary trees, starting from input-output specifications and producing conditional rewrite rules. The main approach consists in proving (constructively) the existence of an appropriate output from every input. The proof may fail if some necessary sub-algorithms are lacking. Then, their specifications are suggested and their synthesis is performed by the same principles.Our main goal is to avoid the possibly prohibitive cost of pure resolution proofs by using a natural-style proving in which domain-specific strategies and inference steps lead to a significant increase of efficiency. In addition to classical techniques for natural-style proving, we introduce novel ones (priority of certain types of assumptions, transformation of elementary goals into conditions, special criteria for decomposition of the goal and of the assumptions), as well as methods based on the properties of domain-specific relations and functions. In particular, we use combinatorial techniques in order to generate possible witnesses, which in certain cases lead to the discovery of new induction principles. From the proof, the algorithm is extracted by transforming inductive proof steps into recursions, and case-based proof steps into conditionals.The approach is demonstrated in parallel using the Theorema system, by developing the theory, implementing the prover, and performing the proofs of the necessary properties and synthesis conjectures. It is also validated in the Coq system, which allows to compare the facilities of the two systems from the point of view of our application.

Axiomatizations of team logics

In a modular approach, we lift Hilbert-style proof systems for propositional, modal and first-order logic to generalized systems for their respective team-based extensions. We obtain sound and complete axiomatizations for the dependence-free fragment FO(∼) of Väänänen's first-order team logic TL, for propositional team logic PTL, quantified propositional team logic QPTL, modal team logic MTL, and for the corresponding logics of dependence, independence, inclusion and exclusion.As a crucial step in the completeness proof, we show that the above logics admit, in a particular sense, a semantics-preserving elimination of modalities and quantifiers from formulas.

Differential Privacy With Bias-Control Limited Sources

In the design of differentially private mechanisms, it is usually assumed that a uniformly random source is available. However, in many situations it seems unrealistic, and one must deal with various imperfect random sources. Dodis et al. (CRYPTO'12) presented that differential privacy can be achieved with Santha-Vazirani (SV) sources, introduced in 1986, via adding a stronger property called SV-consistent sampling and left open the question if differential privacy is possible with more realistic, i.e., less structured, sources. A new source, called bias-control limited (BCL) source, introduced by Dodis (ICALP'01), is more realistic. It can be considered as a generalization of the SV and sequential bit-fixing sources. Unfortunately, the natural extension of SV-consistent sampling to the BCL source is hopeless to achieve differential privacy, mainly because SV-consistent sampling requires “consecutive” strings, while some strings cannot be generated from “non-trivial” BCL sources. Moreover, there exists some inconsistency in the bit lengths of random sources between the mechanism construction and the proofs for differentially privacy presented by Dodis et al. (CRYPTO'12). More appropriate mechanisms and more rigourous proofs are needed. Motivated by these questions, we introduce a new appealing property, called compact BCL-consistent sampling, the degeneration of which is different from SV-consistent sampling shown by Dodis et al. (CRYPTO'12). We prove that if the mechanism based on the BCL source satisfies this property, then it is differentially private for certain parameters. Even if the BCL source is degenerated into the SV source, our proof is much more intuitive and simpler than that of Dodis et al. Further, we construct an explicit finite-precision mechanism using an improved construction of infinite mechanism and a new truncation technique. We also propose its concrete results for differential privacy and utility, one core proof step of which is trickier and much shorter than the counterpart by Dodis et al. (CRYPTO'12). While the results of Dodis and Yao (CRYPTO'15) imply that if there exist differentially private mechanisms for imperfect randomness, then the parameters should have some constraints, we show an explicit construction of such mechanisms, whose parameters match the prior constraints.

Mostly Automated Formal Verification of Loop Dependencies with Applications to Distributed Stencil Algorithms

The class of stencil programs involves repeatedly updating elements of arrays according to fixed patterns, referred to as stencils. Stencil problems are ubiquitous in scientific computing and are used as an ingredient to solve more involved problems. Their high regularity allows massive parallelization. Two important challenges in designing such algorithms are cache efficiency and minimizing the number of communication steps between nodes. In this paper, we introduce a mathematical framework for a crucial aspect of formal verification of both sequential and distributed stencil algorithms, and we describe its Coq implementation. We present a domain-specific embedded programming language with support for automating the most tedious steps of proofs that nested loops respect dependencies, applicable to sequential and distributed examples. Finally, we evaluate the robustness of our library by proving the dependency-correctness of some real-world stencil algorithms, including a state-of-the-art cache-oblivious sequential algorithm, as well as two optimized distributed kernels.

On a theorem of Campana and P\u{a}un

Let $X$ be a smooth projective variety over the complex numbers, and $\Delta \subseteq X$ a reduced divisor with normal crossings. We present a slightly simplified proof for the following theorem of Campana and P\u{a}un: If some tensor power of the bundle $\Omega_X^1(\log \Delta)$ contains a subsheaf with big determinant, then $(X, \Delta)$ is of log general type. This result is a key step in the recent proof of Viehweg's hyperbolicity conjecture.

Complete Fabrication of a Traversable 3 µm Thick NbN Film Superconducting Coil with Cu plated layer of 42m in Length in a Spiral Three-Storied Trench Engraved in a Si Wafer of 76.2 mm in Diameter Formed by MEMS Technology for a Compact SMES with High Energy Storage Volume Density

Based on the concept of a novel approach to make a compact SMES unit composed of a stack of Si wafers using MEMS process proposed previously, a complete fabrication of a traversable 3 µam thick NbN film superconducting coil lined with Cu plated layer of 42m in length in a spiral three-storied trench engraved in and extended over a whole Si-wafer of 76.2 mm in diameter was attained for the first time. With decrease in temperature, the DC resistivity showed a metallic decrease indicating the current pass was in the Cu plated layer and then made a sudden fall to residual contact resistance indicating the shift of current pass from the Cu plated layer to the NbN film at the critical temperature Tc of 15.5K by superconducting transition. The temperature dependence of I-V curve showed the increase in the critical current with decrease in the temperature and the highest critical current measured was 220 mA at 4K which is five times as large as that obtained in the test fabrication as the experimental proof of concept presented in the previous report. This completion of a one wafer superconducting NbN coil is an indispensable step for the next proof of concept of fabrication of series-connected two wafer coils via superconductive joint which will read to series connected 600 wafer coils finally, and for replacement of NbN by high Tc superconductor such as YBa2Cu3O7-x for operation under the cold energy of liquid hydrogen or liquid nitrogen.

The optimal constants in Khintchine’s inequality for the case $2 < p < 3$

A mean step in Haagerup's proof for the optimal constants in Khintchine's inequality is to show integral inequalities of type $\int(g^s-f^s)\mathrm{d}\mu\geq 0$. F.L. Nazarov and A.N. Podkorytov made Haagerup's proof much more clearer for the case 0<p<2 by using a lemma on distribution functions. In this article we want to treat the case 2<p<3 with their technique.

Compact SMES with a superconducting film in a spiral groove on a Si wafer formed by MEMS technology with possible high-energy storage volume density comparable to that of rechargeable batteries

The concept of a novel approach to make a compact SMES unit composed of a stack of Si wafers using a well-established MEMS process was proposed. The concept was backed up by pilot estimations for energy storage capacity and mechanical strength to endure electromagnetic stress. The estimated volume density of the storable energy is comparable to that of rechargeable batteries and the mechanical strength of Si wafer endures the electromagnetic stress imposed on it. These estimations support the feasibility of this novel concept, although there needs to be more detailed design of the system for its practical realization. Furthermore, there are a lot of challenges to overcome. The first step of the experimental proof of this new concept was successfully performed through several repeated test fabrications. In one of these test fabrications, the theoretically estimated upper limit value of the energy storage corresponding to a pilot design of a spiral superconducting NbN coil in the spiral trench formed on a Si wafer 10.15 cm in diameter was attained.

Finding Proofs in Tarskian Geometry

We report on a project to use a theorem prover to find proofs of the theorems in Tarskian geometry. These theorems start with fundamental properties of betweenness, proceed through the derivations of several famous theorems due to Gupta and end with the derivation from Tarski’s axioms of Hilbert’s 1899 axioms for geometry. They include the four challenge problems left unsolved by Quaife, who two decades ago found some OTTER proofs in Tarskian geometry (solving challenges issued in Wos’s 1998 book). There are 212 theorems in this collection. We were able to find OTTER proofs of all these theorems. We developed a methodology for the automated preparation and checking of the input files for those theorems, to ensure that no human error has corrupted the formal development of an entire theory as embodied in two hundred input files and proofs. We distinguish between proofs that were found completely mechanically (without reference to the steps of a book proof) and proofs that were constructed by some technique that involved a human knowing the steps of a book proof. Proofs of length 40–100, roughly speaking, are difficult exercises for a human, and proofs of 100–250 steps belong in a Ph.D. thesis or publication. 29 of the proofs in our collection are longer than 40 steps, and ten are longer than 90 steps. We were able to derive completely mechanically all but 26 of the 183 theorems that have “short” proofs (40 or fewer deduction steps). We found proofs of the rest, as well as the 29 “hard” theorems, using a method that requires consulting the book proof at the outset. Our “subformula strategy” enabled us to prove four of the 29 hard theorems completely mechanically. These are Ph.D. level proofs, of length up to 108.

A Complete Uniform Substitution Calculus for Differential Dynamic Logic

This article introduces a relatively complete proof calculus for differential dynamic logic (dL) that is entirely based on uniform substitution, a proof rule that substitutes a formula for a predicate symbol everywhere. Uniform substitutions make it possible to use axioms instead of axiom schemata, thereby substantially simplifying implementations. Instead of subtle schema variables and soundness-critical side conditions on the occurrence patterns of logical variables to restrict infinitely many axiom schema instances to sound ones, the resulting calculus adopts only a finite number of ordinary dLformulas as axioms, which uniform substitutions instantiate soundly. The static semantics of differential dynamic logic and the soundness-critical restrictions it imposes on proof steps is captured exclusively in uniform substitutions and variable renamings as opposed to being spread in delicate ways across the prover implementation. In addition to sound uniform substitutions, this article introduces differential forms for differential dynamic logic that make it possible to internalize differential invariants, differential substitutions, and derivatives as first-class axioms to reason about differential equations axiomatically. The resulting axiomatization of differential dynamic logic is proved to be sound and relatively complete.

Can quantum transition state theory be defined as an exact t = 0+ limit?

The definition of the classical transition state theory (TST) as a t → 0+ limit of the flux-side time correlation function relies on the assumption that simultaneous measurement of population and flux is a well defined physical process. However, the noncommutativity of the two measurements in quantum mechanics makes the extension of such a concept to the quantum regime impossible. For this reason, quantum TST (QTST) has been generally accepted as any kind of quantum rate theory reproducing the TST in the classical limit, and there has been a broad consensus that no unique QTST retaining all the properties of TST can be defined. Contrary to this widely held view, Hele and Althorpe (HA) [J. Chem. Phys. 138, 084108 (2013)] recently suggested that a true QTST can be defined as the exact t → 0+ limit of a certain kind of quantum flux-side time correlation function and that it is equivalent to the ring polymer molecular dynamics (RPMD) TST. This work seeks to question and clarify certain assumptions underlying these suggestions and their implications. First, the time correlation function used by HA as a starting expression is not related to the kinetic rate constant by virtue of linear response theory, which is the first important step in relating a t = 0+ limit to a physically measurable rate. Second, a theoretical analysis calls into question a key step in HA's proof which appears not to rely on an exact quantum mechanical identity. The correction of this makes the true t = 0+ limit of HA's QTST different from the RPMD-TST rate expression, but rather equal to the well-known path integral quantum transition state theory rate expression for the case of centroid dividing surface. An alternative quantum rate expression is then formulated starting from the linear response theory and by applying a recently developed formalism of real time dynamics of imaginary time path integrals [S. Jang, A. V. Sinitskiy, and G. A. Voth, J. Chem. Phys. 140, 154103 (2014)]. It is shown that the t → 0+ limit of the new rate expression vanishes in the exact quantum limit.

Simplicity and Finiteness of Discrete Spectrum of the Benjamin--Ono Scattering Operator

A spectral analysis is done on the $L$ operator of the Lax pair for the Benjamin--Ono equation. Simplicity and finiteness of the discrete spectrum are established as are needed for the Fokas and Ablowitz inverse scattering transform scheme. A crucial step in the simplicity proof is the discovery of a new identity connecting the $L^2$ norm of the eigenvector to its inner product with the scattering potential. The proof for finiteness is an extension of the ideas involved in the Birman--Schwinger bound for Schrödinger operators.

Speedith: A Reasoner for Spider Diagrams

In this paper, we introduce Speedith which is an interactive diagrammatic theorem prover for the well-known language of spider diagrams. Speedith provides a way to input spider diagrams, transform them via the diagrammatic inference rules, and prove diagrammatic theorems. Speedith's inference rules are sound and complete, extending previous research by including all the classical logic connectives. In addition to being a stand-alone proof system, Speedith is also designed as a program that plugs into existing general purpose theorem provers. This allows for other systems to access diagrammatic reasoning via Speedith, as well as a formal verification of diagrammatic proof steps within standard sentential proof assistants. We describe the general structure of Speedith, the diagrammatic language, the automatic mechanism that draws the diagrams when inference rules are applied on them, and how formal diagrammatic proofs are constructed.

Incoherence-Optimal Matrix Completion

This paper considers the matrix completion problem. We show that it is not necessary to assume joint incoherence, which is a standard but unintuitive and restrictive condition that is imposed by previous studies. This leads to a sample complexity bound that is order-wise optimal with respect to the incoherence parameter (as well as to the rank $r$ and the matrix dimension $n$ up to a log factor). As a consequence, we improve the sample complexity of recovering a semidefinite matrix from $O(nr^{2}\log^{2}n)$ to $O(nr\log^{2}n)$, and the highest allowable rank from $\Theta(\sqrt{n}/\log n)$ to $\Theta(n/\log^{2}n)$. The key step in proof is to obtain new bounds on the $\ell_{\infty,2}$-norm, defined as the maximum of the row and column norms of a matrix. To illustrate the applicability of our techniques, we discuss extensions to SVD projection, structured matrix completion and semi-supervised clustering, for which we provide order-wise improvements over existing results. Finally, we turn to the closely-related problem of low-rank-plus-sparse matrix decomposition. We show that the joint incoherence condition is unavoidable here for polynomial-time algorithms conditioned on the Planted Clique conjecture. This means it is intractable in general to separate a rank-$\omega(\sqrt{n})$ positive semidefinite matrix and a sparse matrix. Interestingly, our results show that the standard and joint incoherence conditions are associated respectively with the information (statistical) and computational aspects of the matrix decomposition problem.

A multi-focused proof system isomorphic to expansion proofs

The sequent calculus is often criticized for requiring proofs to contain large amounts of low-level syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, sequent proofs can separate closely related steps---such as instantiating a block of quantifiers---by irrelevant noise. Moreover, the sequential nature of sequent proofs forces proof steps that are syntactically non-interfering and permutable to nevertheless be written in some arbitrary order. The sequent calculus thus lacks a notion of canonicity: proofs that should be considered essentially the same may not have a common syntactic form. To fix this problem, many researchers have proposed replacing the sequent calculus with proof structures that are more parallel or geometric. Proof-nets, matings, and atomic flows are examples of such revolutionary formalisms. We propose, instead, an evolutionary approach to recover canonicity within the sequent calculus, which we illustrate for classical first-order logic. The essential element of our approach is the use of a multi-focused sequent calculus as the means for abstracting away low-level details from classical cut-free sequent proofs. We show that, among the multi-focused proofs, the maximally multi-focused proofs that collect together all possible parallel foci are canonical. Moreover, if we start with a certain focused sequent proof system, such proofs are isomorphic to expansion proofs---a well known, minimalistic, and parallel generalization of Herbrand disjunctions---for classical first-order logic. This technique appears to be a systematic way to recover the "essence of proof" from within sequent calculus proofs.

RNAi-derived field resistance to Cassava brown streak disease persists across the vegetative cropping cycle

A confined field trial was established to determine durability of RNAi-mediated resistance to Cassava brown streak disease (CBSD). Stem cuttings were obtained from field-grown cassava plants of cv 60444 transgenic for construct p718, consisting of an 894 bp inverted repeat sequence from the Ugandan Cassava brown streak virus (UCBSV) coat protein. Plants were established from three transgenic lines previously shown to provide complete resistance to UCBSV and differing levels of protection to the non-homologous virus species Cassava brown streak virus (CBSV), and grown for 11 months. CBSD symptoms were observed on shoots and storage roots of all non-transgenic cv 60444 control plants and transgenic lines p718–002 and p718–005, but not on p718–001. RT-PCR diagnostic showed tissues of plant lines p718–002 and p718–005 to be infected with CBSV, but free of UCBSV. All leaves and roots of p718–001 plants were to carry no detectable levels of either pathogen. Plants of cv 60444 in this field trial showed severe cassava mosaic disease symptoms, indicating that presence of replicating geminiviruses did not cause significant suppression of RNAi-mediated resistance to CBSD. Resistance to CBSD across a vegetative cropping cycle confirms earlier field data, and provides an important step in proof of concept for application of RNAi technology to control of CBSD under conditions encountered in farmers’ fields.

The inositol monophosphatase inhibitor L-690,330 affects pilocarpine-behavior and the forced swim test

Lithium has been a standard pharmacological treatment for bipolar disorder over the last 60years; however, the molecular targets through which lithium exerts its therapeutic effects are still not defined. Attenuation of the phosphatidylinositol signal transduction pathway as a consequence of inhibition of inositol monophosphatase (IMPase) has been proposed as one of the possible mechanisms for lithium-induced mood stabilization. The objective was to study the behavioral effect of the specific competitive IMPase inhibitor L-690,330 in mice in the lithium-sensitive pilocarpine-induced seizures paradigm and the forced swim test (FST). The inhibitor was administered intracerebroventricularly in liposomes. L-690,330 increased the sensitivity to subconvulsive doses of pilocarpine and decreased immobility time in the FST. It is possible that the behavioral effects of lithium in the pilocarpine-induced seizures and in the FST are mediated through the inhibition of IMPase, but reversal of the inhibitor's effect with intracerebroventricular inositol would be an important further step in proof.