Finding Counterexamples Research Articles

MAPPINGS AND DECOMPOSITIONS OF (μ, σ)-CONTINUITY ON μ-NEARLY COMPACT SPACES

In generalised topological space (X,μ), a μ-space is said to be μ-nearly compact if every μ-regular open cover of X has a finite subcover. In literature, many generalised continuities in a topological space are constructed using many generalised open sets. Relationships among them are studied by proving their implications and finding counterexamples for their independent relationships. Many researchers called this type of study a decomposition of continuities. In this paper, we further investigate some decompositions of (μ,σ)-continuity and applying them to μ-nearly compact spaces. Moreover, we also analyse the effect of mappings on these spaces and obtain some results. The main result is that the (μ,σ)-δ-continuous image of μ-nearly compact space is σ-nearly compact.

Testing noninterference, quickly

AbstractInformation-flow control mechanisms are difficult both to design and to prove correct. To reduce the time wasted on doomed proof attempts due to broken definitions, we advocate modern random-testing techniques for finding counterexamples during the design process. We show how to use QuickCheck, a property-based random-testing tool, to guide the design of increasingly complex information-flow abstract machines, leading up to a sophisticated register machine with a novel and highly permissive flow-sensitive dynamic enforcement mechanism that is sound in the presence of first-class public labels. We find that both sophisticated strategies for generating well-distributed random programs and readily falsifiable formulations of noninterference properties are critically important for efficient testing. We propose several approaches and evaluate their effectiveness on a collection of injected bugs of varying subtlety. We also present an effective technique for shrinking large counterexamples to minimal, easily comprehensible ones. Taken together, our best methods enable us to quickly and automatically generate simple counterexamples for more than 45 bugs. Moreover, we show how testing guides the discovery of the sophisticated invariants needed for the noninterference proof of our most complex machine.

Automating grammar comparison

We consider from a practical perspective the problem of checking equivalence of context-free grammars. We present techniques for proving equivalence, as well as techniques for finding counter-examples that establish non-equivalence. Among the key building blocks of our approach is a novel algorithm for efficiently enumerating and sampling words and parse trees from arbitrary context-free grammars; the algorithm supports polynomial time random access to words belonging to the grammar. Furthermore, we propose an algorithm for proving equivalence of context-free grammars that is complete for LL grammars, yet can be invoked on any context-free grammar, including ambiguous grammars. Our techniques successfully find discrepancies between different syntax specifications of several real-world languages, and are capable of detecting fine-grained incremental modifications performed on grammars. Our evaluation shows that our tool improves significantly on the existing available state of the art tools. In addition, we used these algorithms to develop an online tutoring system for grammars that we then used in an undergraduate course on computer language processing. On questions involving grammar constructions, our system was able to automatically evaluate the correctness of 95% of the solutions submitted by students: it disproved 74% of cases and proved 21% of them.

Testing noninterference, quickly

Information-flow control mechanisms are difficult to design and labor intensive to prove correct. To reduce the time wasted on proof attempts doomed to fail due to broken definitions, we advocate modern random testing techniques for finding counterexamples during the design process. We show how to use QuickCheck, a property-based random-testing tool, to guide the design of a simple information-flow abstract machine. We find that both sophisticated strategies for generating well-distributed random programs and readily falsifiable formulations of noninterference properties are critically important. We propose several approaches and evaluate their effectiveness on a collection of injected bugs of varying subtlety. We also present an effective technique for shrinking large counterexamples to minimal, easily comprehensible ones. Taken together, our best methods enable us to quickly and automatically generate simple counterexamples for all these bugs.

A Sharp Error Estimate for Numerical Fourier Fransform of Band-Limited Functions Based on Windowed Samples

W. Dickmeis and R. J. Nessel published their first version of a quantitative extension of the classical uniform boundedness principle in [J. Approx. Theory 31 (1981), 161–174]. It is a general approach to finding counterexamples that prove sharpness of error estimates. So far applications of this principle include error bounds for approximation processes, cubature rules, ordinary and partial differential equations, and reconstruction from samples. Here we discuss the error of discrete approximations of the Fourier transform based on windowed samples for band-limited functions. The results can be applied to the Hann- and Blackmann–Harris-window but also to window-functions that enable higher orders of convergence. We describe a class of such windows.

Integrating Testing and Interactive Theorem Proving

Using an interactive theorem prover to reason about programs involves a sequence of interactions where the user challenges the theorem prover with conjectures. Invariably, many of the conjectures posed are in fact false, and users often spend considerable effort examining the theorem prover's output before realizing this. We present a synergistic integration of testing with theorem proving, implemented in the ACL2 Sedan (ACL2s), for automatically generating concrete counterexamples. Our method uses the full power of the theorem prover and associated libraries to simplify conjectures; this simplification can transform conjectures for which finding counterexamples is hard into conjectures where finding counterexamples is trivial. In fact, our approach even leads to better theorem proving, e.g. if testing shows that a generalization step leads to a false conjecture, we force the theorem prover to backtrack, allowing it to pursue more fruitful options that may yield a proof. The focus of the paper is on the engineering of a synergistic integration of testing with interactive theorem proving; this includes extending ACL2 with new functionality that we expect to be of general interest. We also discuss our experience in using ACL2s to teach freshman students how to reason about their programs.

On using data abstractions for model checking refinements

In this paper we investigate how standard model checkers can be applied to checking refinement relationships between Z specifications. The major obstacle to such a use are the (potentially) infinite data domains in specifications. Consequently, we examine the application of data abstraction techniques for reducing the infinite to a finite state space. Since data abstractions do, however, decrease the amount of information in a specification, refinement can—in general—not be proven on the abstractions anymore, it can only be disproved. The model checker can thus be used to generate counter examples to a refinement relationship. Here, we show how abstract specifications can be systematically constructed (from a given data abstraction) and how a standard model checker (FDR) can be applied to find counter examples in case when refinement is absent. We especially discuss the applicability of the construction method: it constructs abstract specifications which are either upward or downward simulations of the original specifications, and depending on the operations in the specification and the data abstraction chosen, such a construction might succeed or fail. The construction abstracts both the input/output as well as the state.

Software abstractions: logic, language and analysis

In Software Abstractions Daniel Jackson introduces an approach to software design that draws on traditional formal but exploits automated tools to find flaws as early as possible. This approach--which Jackson calls lightweight formal methods or agile modeling--takes from formal specification idea of a precise and expressive notation based on a tiny core of simple and robust concepts but replaces conventional analysis based on theorem proving with a fully automated analysis that gives designers immediate feedback. Jackson has developed Alloy, a language that captures essence of software abstractions simply and succinctly, using a minimal toolkit of mathematical notions. This revised edition updates text, examples, and appendixes to be fully compatible with latest version of Alloy (Alloy 4). The designer can use automated analysis not only to correct errors but also to make models that are more precise and elegant. This approach, Jackson says, can rescue designers from the tarpit of implementation technologies and return them to thinking deeply about underlying concepts. Software Abstractions introduces key elements: a logic, which provides building blocks of language; a language, which adds a small amount of syntax to logic for structuring descriptions; and an analysis, a form of constraint solving that offers both simulation (generating sample states and executions) and checking (finding counterexamples to claimed properties).

Weakly abelian lattice-ordered groups

Every nilpotent lattice-ordered group is weakly Abelian; i.e., satisfies the identity $x^{-1}(y\vee 1)x\vee (y\vee 1)^2=(y\vee 1)^2$. In 1984, V. M. Kopytov asked if every weakly Abelian lattice-ordered group belongs to the variety generated by all nilpotent lattice-ordered groups [The Black Swamp Problem Book, Question 40]. In the past 15 years, all attempts have centred on finding counterexamples. We show that two constructions of weakly Abelian lattice-ordered groups fail to be counterexamples. They include all preiously considered potential counterexamples and also many weakly Abelian ordered free groups on finitely many generators. If every weakly Abelian ordered free group on finitely many generators belongs to the variety generated by all nilpotent lattice-ordered groups, then every weakly Abelian lattice-ordered group belongs to this variety. This paper therefore redresses the balance and suggests that Kopytov’s problem is even more intriguing.

Quantum aspects of triangular billiards

The energy spectrum of a particle enclosed in a two-dimensional triangular box can in general not be calculated analytically. Therefore, in order to establish the degree of intergrability, using some well known criteria, this spectrum must be calculated numerically. For more than 800 triangles these calculations were performed and then used to check a number of conjectures found in the literature. For almost all of them we find counter examples. Our own suggestion is that in general acute triangles are more integrable than obtuse ones. However, also this rule has its exceptions, which cannot be explained away by some triangles being more rational than others.

Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands

The study of regularity properties for minimizers of problems of the form min , u, u’) dt: u E IV’-‘(Q; R”), u(a) = a, u(b) = B (1.1) has been undertaken for a long time in Calculus of Variations, and a number of results are by now available on this subject. When f(t, s, z) is a smooth coercive function, problem ( 1.1) was considered by Tonelli (see [ 18, 19]), who proved that every minimizer u(t) of the problem (1.1) is locally Lipschitz on a relatively open set of full measure in [a, b]. Under some additional assumptions on f, it is also possible to prove global regularity theorems (see for instance Cesari [3, Chap 2, Sect. 61). In recent years, much attention has been devoted to problems of the form (l.l), on the one hand in weakening the hypotheses of Tonelli’s regularity results, and on the other in finding counterexamples of nonregular minimizers for coercive functionals. With respect to this last subject, we recall that it has been shown (see Ball and Mizel [l] and Clarke and Vinter [6]) that there exist polynomial functions f(t, s, z) such that, for suitable C > 0 and p,

Definition of noise temperatures θ in three dimensions and the conjecture θ ≥ T in nonequilibrium systems

A system of electrons in a strong electric field obeying a linear transport equation is considered. Various possibilities to define a second-rank noise-temperature tensor θ αβ by means of the velocity fluctuation spectrum S αβ and the velocity-field response function μ αβ are proposed. The noise-temperature conjecture θ ≥ T can obviously be generalized in three different forms to three dimensions: 1) weak form θ ∥ F (parallel to field) ≥ T, 2) intermediate form Sp θ ≥ 3 T, 3) strong form θ sym - T 1 positive definite. They are supposed to hold for all electric fields F independently of band structure and collision operator (scattering mechanism). In the constant relaxation-time model of the collision operator forms 1) and 2) are shown to hold at least for small fields (warm electron), whereas for case 3) it is easy to find counter examples which are isotropic and therefore independent of the θ-tensor definition chosen. Noise temperatures are given for two examples with nontrivial collision operators: the LKDPYW model with deformation potential scattering and n-type GaAs using the data of Monte-Carlo calculations.