Proof Tools Research Articles

Modifications of sequence selection principles

We study modifications of the sequence selection principles for real functions obtained by allowing other than continuous functions and/or by replacing the pointwise convergence by quasi-normal or discrete convergence. We show that for important families of real functions we obtain already known sequence selection principles. Replacing the pointwise convergence we obtain plenty of sequence selection principles and we try to find equivalences or implications among them. Finally we attempt to show that these modifications are useful tools for proofs of known results or their strengthening.

Convexity and the difference property

We show that the class of all delta-convex selfmappings of R (differences of two convex functions) enjoys the difference property in the sense of N.G. de Bruijn. The Q-differentiability technique has been applied as a proof tool.

Graphs of Hecke operators

Let $X$ be a curve over $\F_q$ with function field $F$. In this paper, we define a graph for each Hecke operator with fixed ramification. A priori, these graphs can be seen as a convenient language to organize formulas for the action of Hecke operators on automorphic forms. However, they will prove to be a powerful tool for explicit calculations and proofs of finite dimensionality results. We develop a structure theory for certain graphs $G_x$ of unramified Hecke operators, which is of a similar vein to Serre's theory of quotients of Bruhat Tits trees. To be precise, $G_x$ is locally a quotient of a Bruhat Tits tree and has finitely many components. An interpretation of $G_x$ in terms of rank 2 bundles on $X$ and methods from reduction theory show that $G_x$ is the union of finitely many cusps, which are infinite subgraphs of a simple nature, and a nucleus, which is a finite subgraph that depends heavily on the arithmetics of $F$. We describe how one recovers unramified automorphic forms as functions on the graphs $G_x$. In the exemplary cases of the cuspidal and the toroidal condition, we show how a linear condition on functions on $G_x$ leads to a finite dimensionality result. In particular, we re-obtain the finite-dimensionality of the space of unramified cusp forms and the space of unramified toroidal automorphic forms. In an Appendix, we calculate a variety of examples of graphs over rational function fields.

Stability regions for linear fractional differential systems and their discretizations

This paper concerns with basic stability properties of linear autonomous fractional differential and difference systems involving derivative operators of the Riemann–Liouville type. We derive stability regions for special discretizations of the studied fractional differential systems including a precise description of their asymptotics. Our analysis particularly shows that discretizations based on backward differences can retain the key qualitative properties of underlying fractional differential systems. In addition, we introduce the backward discrete Laplace transform and employ some of its properties as the main proof tool.

The speed with which an orbit approaches a limit point

If is a continuous map of a compact metric space , and if is a sequence of positive reals converging to 0, we investigate the properties of the set . We show that is a dense subset of for every when x is a recurrent point, even though can be disjoint with the orbit of x for some . Under the assumption that f has an invariant non-atomic Borel probability measure , we prove results to the effect that (i) there is a uniform upper limit to the speed with which the orbit of each x can approach y for -almost every , (ii) if is ergodic with full support and if is the set of points having dense orbits, then for -almost every and for every there is a uniform upper limit to the speed with which the orbit of x can approach y. Next, using as a useful tool in proofs, we establish the following. If f is totally transitive and X is infinite, then there is a dense subset which is a countable union of Cantor sets such that and for any two distinct and any two distinct . If f is a transitive map enjoying a certain type of continuity in the backward direction, then f has a residual set of points with dense backward orbits.

Epsilon substitution for first- and second-order predicate logic

The epsilon substitution method was proposed by D. Hilbert as a tool for consistency proofs. A version for first order predicate logic had been described and proved to terminate in the monograph “Grundlagen der Mathematik”. As far as the author knows, there have been no attempts to extend this approach to the second order case. We discuss possible directions for and obstacles to such extensions.

The HOL Light Theory of Euclidean Space

We describe the library of theorems about N-dimensional Euclidean space that has been formalized in the HOL Light prover. This formalization was started in 2005 and has been extensively developed since then, partly in direct support of the Flyspeck project, partly out of a general desire to develop a well-rounded and comprehensive theory of basic analytical, geometrical and topological machinery. The library includes various `big name' theorems (Brouwer's fixed point theorem, the Stone-Weierstrass theorem, the Tietze extension theorem), numerous non-trivial results that are useful in applications (second mean value theorem for integrals, power series for real and complex transcendental functions) and a host of supporting definitions and lemmas. It also includes some specialized automated proof tools. The library has as planned been applied to the Flyspeck project and has become the basis of a significant development of results in complex analysis, among others.

Formal Specifications of Trusted OLSR Protocol of Ad hoc Network in Z

Mobile Ad hoc Network is a network of mobile nodes operating in an infrastructure-less network. These nodes not have the defense rendered by firewalls in infrastructure-based networks. Trust oriented system aids to improvise this situation. So, the incorporation of trust in routing decisions yields a more secure and reliable framework for such type of networks. As if any suggested model does not achieve as projected, it is reduced quality of service. The validation before deployment of any model leads to a more stable model. A good way to confirm a model is to use formal specification and verification techniques. In the present study, with the aim to include trust component in conventional OLSR protocol of ad hoc network and also to rule out invalid actions, formal specifications of the various procedures of trusted OLSR Protocol are given using ―Z‖ specification language. Z is a state-oriented formal specification language based on set theory and predicate logic. Z/EVES, a proof tool based on EVES and ZF set theory that supports the Z notation is used for the formal specifications.

Review/PerspectiveTranslational Medicine: What is it and what could it be?

Translational medicine is not a traditional and established discipline in the biosciences, certainly not in medicine. It has become a very popular phrase which is being more and more widely used in variable contexts both by the pharmaceutical industry, funding agencies and academia. The idea of translational medicine is as obvious as persuasive if introduced into grant applications or industry tools of operation. It is driven by the general observation that there is a deep and rapidly widening trench between medical in vitro and animal research on one hand and clinical innovation on the other. Whereas the knowledge in medicine in general terms is exploding, this does certainly not translate into an explosion of novelties in patient treatments, exceptions granted. In particular, the <i>lamento</i> is coming increasingly from the pharmaceutical industry which complains about surging R&D costs at yet steadily decreasing marketing approval rates of new medicines. Academia and its sponsors join into this choir not understanding why all the knowledge obtained in the test tubes and experimental animals (e. g. knock-out mice) does help so little in changing realities of daily medical practice. Thus, translation has become a widespread phrase to indicate that research and development aims at bedside innovation or facilitates the transition from ‘bench to bedside’. For this, it utilizes biomarkers which should be predictive at an early stage of development (e. g. animal experiments) for later stages, especially at the human level. Human exploratory (non-regulatory) trials are important tools for early proof of principle. In general, however, a clear set of methods and technology is still lacking, and no solid science supports the wish to translate. If biomedical research should survive increasing competition for resources by other areas of research demands from e. g. environmental or energy concerns, it has to get the act of translation right in that a solid armamentarium of paradigms, tools, methods needs to be created. This includes translation planning, translatability assessment, and, thus, portfolio considerations to balance translational risks in both public and private funding situations.

Formal Verification of Numerical Programs: From C Annotated Programs to Mechanical Proofs

Numerical programs may require a high level of guarantee. This can be achieved by applying formal methods, such as machine-checked proofs. But these tools handle mathematical theorems while we are interested in C code, in which numerical computations are performed using floating-point arithmetic, whereas proof tools typically handle exact real arithmetic. To achieve this high level of confidence on C programs, we use a chain of tools: Frama-C, its Jessie plugin, Why and provers among Coq, Gappa, Alt-Ergo, CVC3 and Z3. This approach requires the C program to be annotated: each function must be precisely specified, and we prove the correctness of the program by proving both that it meets its specifications and that no runtime error may occur. The purpose of this paper is to illustrate, on various examples, the features of this approach.

On the coalescence time of reversible random walks

Consider a system of coalescing random walks where each individual performs a random walk over a finite graph $\mathbf {G}$ or (more generally) evolves according to some reversible Markov chain generator $Q$. Let $C$ be the first time at which all walkers have coalesced into a single cluster. $C$ is closely related to the consensus time of the voter model for this $\mathbf {G}$ or $Q$. We prove that the expected value of $C$ is at most a constant multiple of the largest hitting time of an element in the state space. This solves a problem posed by Aldous and Fill and gives sharp bounds in many examples, including all vertex-transitive graphs. We also obtain results on the expected time until only $k\geq 2$ clusters remain. Our proof tools include a new exponential inequality for the meeting time of a reversible Markov chain and a deterministic trajectory, which we believe to be of independent interest.

Scheme-based theorem discovery and concept invention

We describe an approach to automatically invent/explore new mathematical theories, with the goal of producing results comparable to those produced by humans, as represented, for example, in the libraries of the Isabelle proof assistant. Our approach is based on ‘schemes’, which are formulae in higher-order logic. We show that it is possible to automate the instantiation process of schemes to generate conjectures and definitions. We also show how the new definitions and the lemmata discovered during the exploration of a theory can be used, not only to help with the proof obligations during the exploration, but also to reduce redundancies inherent in most theory-formation systems. We exploit associative-commutative (AC) operators using ordered rewriting to avoid AC variations of the same instantiation. We implemented our ideas in an automated tool, called IsaScheme, which employs Knuth–Bendix completion and recent automatic inductive proof tools. We have evaluated our system in a theory of natural numbers and a theory of lists.

Refined typing to localize the impact of forced strictness on free theorems

Free theorems establish interesting properties of parametrically polymorphic functions, solely from their types, and serve as a nice proof tool. For pure and lazy functional programming languages, they can be used with very few preconditions. Unfortunately, in the presence of selective strictness, as provided in languages like Haskell, their original strength is reduced. In this paper we present an approach for overcoming this weakness in specific situations. Employing a refined type system which tracks the use of enforced strict evaluation, we rule out unnecessary restrictions that otherwise emerge. Additionally, we provide (and implement) an algorithm determining all refined types for a given term.

Concentration of Measure for Quantum States with a Fixed Expectation Value

Given some observable H of a finite-dimensional quantum system, we investigate the typical properties of random quantum state vectors that have a fixed expectation value with respect to H. Under some some conditions on the spectrum, we prove that this manifold of quantum states shows a concentration of measure phenomenon: any continuous function on this set is almost everywhere close to its mean. We also give a method to estimate the corresponding expectation values analytically, and we prove a formula for the typical reduced density matrix in the case that H is a sum of local observables. We discuss the implications of our results as new proof tools in quantum information theory and to study phenomena in quantum statistical mechanics. As a by-product, we derive a method to sample the resulting distribution numerically, which generalizes the well-known Gaussian method to draw random states from the sphere.

Cartesian effect categories are Freyd-categories

Most often, in a categorical semantics for a programming language, the substitution of terms is expressed by composition and finite products. However this does not deal with the order of evaluation of arguments, which may have major consequences when there are side-effects. In this paper Cartesian effect categories are introduced for solving this issue, and they are compared with strong monads, Freyd-categories and Haskell’s Arrows. It is proved that a Cartesian effect category is a Freyd-category where the premonoidal structure is provided by a kind of binary product, called the sequential product. The universal property of the sequential product provides Cartesian effect categories with a powerful tool for constructions and proofs. To our knowledge, both effect categories and sequential products are new notions.

Deductive verification of cryptographic software

We apply state-of-the art deductive verification tools to check security-relevant properties of cryptographic software, including safety, absence of error propagation, and correctness with respect to reference implementations. We also develop techniques to help us in our task, focusing on methods oriented towards increased levels of automation, in scenarios where there are clear obvious limits to such automation. These techniques allow us to integrate automatic proof tools with an interactive proof assistant, where the latter is used off-line to prove once-and-for-all fundamental lemmas about properties of programs. The techniques developed have independent interest for practical deductive verification in general.

Accessible privacy and security: a universally usable human-interaction proof tool

Despite growing interest in designing usable systems for managing privacy and security, recent efforts have generally failed to address the needs of users with disabilities. As security and privacy tools often rely upon subtle visual cues or other potentially inaccessible indicators, users with perceptual limitations might find such tools particularly challenging. To understand the needs of an important group of users with disabilities, a focus group was conducted with blind users to determine their perceptions of security-related challenges. Human-interaction proof (HIP) tools, commonly known as CAPTCHAs, are used by web pages to defeat robots and were identified in the focus group as a major concern. Therefore, a usability test was conducted to see how well blind users were able to use audio equivalents of these graphical tools. Finally, an accessible HIP tool was developed which combines audio and matching images, supporting both visual and audio output. Encouraging results from a small usability evaluation of the prototype with five sighted users and five blind users show that this new form of HIP is preferred by both blind and visual users to previous forms of text-based HIPs. Future directions for research are also discussed.

Secretory Leukocyte Protease Inhibitor Antagonizes Paclitaxel in Ovarian Cancer Cells

Ovarian cancer recurrence with the development of paclitaxel resistance is an obstacle to long-term survival. We showed that secretory leukocyte protease inhibitor (SLPI) is a survival factor for ovarian cancer. We hypothesize that SLPI may antagonize paclitaxel injury. Differential SLPI induction in response to paclitaxel and in response to stable forced expression of SLPI was shown in A2780-1A9 cells and their paclitaxel-resistant sublines, PTX10 and PTX22, and confirmed with HEY-A8 cells. SLPI-mediated survival was reduced by the MAP/extracellular signal-regulated kinase (ERK) kinase inhibitor, U0126, and a humanized neutralizing monoclonal anti-SLPI antibody, CR012. OVCAR3 xenographs tested the role of CR012 in vivo. SLPI expression was lower in A2780-1A9 ovarian cancer cells than in PTX10 and PTX22, and SLPI was induced by paclitaxel exposure. Stable SLPI expression yielded a proliferation advantage (P = 0.01); expression of and response to SLPI in OVCAR3 cells were abrogated by exposure to CR012. SLPI reduced the paclitaxel susceptibility of 1A9 and HEY-A8 cells (P <or= 0.05), and SLPI expression did not increase the resistance of PTX10 and PTX22 cells. Both paclitaxel and SLPI overexpression induced ERK activation. Inhibition of MAP/ERK kinase with U0126 increased paclitaxel injury and overcame SLPI-mediated cell protection. It did not reinstate PTX10 sensitivity to paclitaxel, which was associated with AKT activation. Significant inhibition of OVCAR3 xenograft growth was observed with CR012 and paclitaxel, over single agents (P <or= 0.001). A two-pronged approach confirmed that SLPI overcomes paclitaxel in part through activation of ERK1/2. These results credential SLPI as a molecular target for ovarian cancer and suggest CR012 as a tool for proof of concept.

Modelling and Verification of Pipelined Micro-Architectures: Functional Approach

Formal verification of a pipelined micro-architectural (PMA) implementation against an instruction-set-architecture (ISA) specification is a common approach, which still requires considerable efforts because it is impossible to find a meaningful point where the implementation state and the specification state can be compared easily. An alternative solution to such problem is to verify a PMA implementation against a sequential multi-cycle implementation. Because both models are formalized in terms of clock cycles, all synchronous intermediate states represent interesting points where the comparison could be achieved easily. Furthermore, by decomposing the state, the overall proof decomposes systematically into a set of verification conditions more simple to reason about and to verify. A major advantage of this elegant choice is the ability to carry out the proof by induction within the same specification language rather than by symbolic simulation through a proof tool which remains very tedious. Also, because both models relate to the micro-architectural (MA) level, there is no need for a data abstraction function, only a time abstraction function is needed to map between the times used by the two models. The potential features of the proposed proof methodology are demonstrated over the pipelined MIPS RISC processor within Haskell framework.

CSP-Prover — a Proof Tool for the Verification of Scalable Concurrent Systems

CSP-Prover — a Proof Tool for the Verification of Scalable Concurrent Systems