False Conjectures Research Articles

The $p$-adic Duffin-Schaeffer conjecture

We prove Haynes' version of the Duffin-Schaeffer conjecture for the $p$-adic numbers. In addition, we prove several results about an associated related but false conjecture, related to $p$-adic approximation in the spirit of Jarník and Lutz.

Feedback linearization and backstepping: an equivalence in control design of strict-feedback form

Abstract This paper investigates an equivalence between feedback linearization and backstepping control. Implications from equivalence are that stability and performance properties of one method are the same for another method. Thus, a property known to exist only for one method could be used to prove property also holds for another. Also, a suspected advantage of one method over the other could be proven to be a false conjecture. Control laws in both approaches are achieved by coordinate transformations and non-linear feedbacks. Further, resulting non-linear feedback control law achieved by feedback linearization method matches exactly with non-linear controller achieved by the backstepping control design. This equivalence is a general analytical match within the specific class of non-linear dynamic systems under investigation. Demonstrations are considered and validated via flight control of longitudinal dynamics of a high performance aircraft simulation model. Algorithms are tested and evaluated with analytical models and non-linear closed-loop simulation.

Tests and proofs for custom data generators

Abstract We address automated testing and interactive proving of properties involving complex data structures with constraints, like the ones studied in enumerative combinatorics, e.g., permutations and maps. In this paper we show testing techniques to check properties of custom data generators for these structures. We focus on random property-based testing and bounded exhaustive testing, to find counterexamples for false conjectures in the Coq proof assistant. For random testing we rely on the existing Coq plugin QuickChick and its toolbox to write random generators. For bounded exhaustive testing, we use logic programming to generate all the data up to a given size. We also propose an extension of QuickChick with bounded exhaustive testing based on generators developed inside Coq, but also on correct-by-construction generators developed with Why3. These tools are applied to an original Coq formalization of the combinatorial structures of permutations and rooted maps, together with some operations on them and properties about them. Recursive generators are defined for each combinatorial family. They are used for debugging properties which are finally proved in Coq. This large case study is also a contribution in enumerative combinatorics.

Pre-service Elementary Teachers’ Conceptions of Counterexamples

This study aimed at investigating two main issues related to counterexample construction: the appropriateness of counterexamples and the types of arguments that are often used when refuting a false conjecture. Twelve pre-service elementary teachers who demonstrated a wide range of reasoning skills participated in this study. The data revealed various phenomena among pre-service teachers’ conceptions of refutations. While all participants who demonstrated deductive reasoning were aware of the fact that one counterexample was sufficient to refute a false statement, the majority of the participants who had not yet developed deductive reasoning failed to recognize that one counterexample would be enough and/or they tend to believe that providing more than one counterexample would support the argument further to refute the statement. Furthermore, the participants from the deductive proof scheme attempted to construct a general refutation or to provide a justification for their constructed counterexamples while the participants from different proof schemes only provided specific counterexamples without further explanation or justification. This study also documented various misconceptions that PSTs held regarding to the underlying concepts in which they were being asked to refute—the concepts of area and perimeter or quadrilaterals—and how their misconceptions affected their constructed counterexamples or decisions as to whether the presented statements were true or false.

Descubrimiento de conocimiento matemático mediante la reformulación de conjeturas falsas en un ambiente de pruebas y refutaciones

Este trabajo expone los resultados obtenidos en un estudio llevado a cabo con futuros profesores de educación secundaria. Se presenta cómo descubrieron conocimiento matemático a partir de una conjetura falsa, la cual reformularon en un proceso de discusión basado en algunos elementos del método cuasiempírico de pruebas y refutaciones. También se describen las etapas del proceso junto con los elementos que fueron observados, y cómo el uso de un software de geometría dinámica facilitó a los estudiantes la exploración, verificación, análisis y aclaración de conceptos matemáticos relaciona¬dos con las conjeturas reformuladas.

Counterexamples as Starting Points for Reasoning and Sense Making

Asked to “fix” a false conjecture, students combine their reasoning and observations about absolute value inequalities, signed numbers, and distance to write true mathematical statements.

THE QUANTUM sl(n, ℂ) REPRESENTATION THEORY AND ITS APPLICATIONS

In this paper, we study the quantum sl(<TEX>$n$</TEX>) representation category using the web space. Specially, we extend sl(<TEX>$n$</TEX>) web space for <TEX>$n{\geq}4$</TEX> as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial <TEX>$P_n(q)$</TEX> specialized to a one variable polynomial can be computed by a linear expansion with respect to a presentation of the quantum representation category of sl(<TEX>$n$</TEX>). Moreover, we correct the false conjecture [30] given by Chbili, which addresses the relation between some link polynomials of a periodic link and its factor link such as Alexander polynomial (<TEX>$n=0$</TEX>) and Jones polynomial (<TEX>$n=2$</TEX>) and prove the corrected conjecture not only for HOMFLY polynomial but also for the colored HOMFLY polynomial specialized to a one variable polynomial.

Power and Meaninglessness

Many medical practitioners have an aversion to statistics. Some may even prefer to believe that statistics are a game played by academics that is only tangentially related to their own practice. Therefore, they do not pursue expertise in statistics. But avoiding statistics constitutes a disservice to their own intellect, and renders practitioners vulnerable to illusions and misinformation. Some practitioners might have the impression that statistics are used (or misused) to prove a conjecture. This is not correct. Truth cannot be proved. However, falsehoods can be proved; and the role of statistics is to refute false conjectures. Statistics serve practitioners not by proving truth but by protecting them from errors and false assertions. Particularly difficult concepts in statistics are those pertaining to type I and type II errors in controlled trials, and the power of a trial. These concepts pose difficulties because they rely on dealing with simultaneity and double negatives. One way of explaining these concepts starts with pointing out that any controlled study is theoretically vulnerable to errors because of rogue samples. Differences arise, or fail to arise, not because of the biological effects of an intervention but because of the samples used to test the intervention. There is a risk that a difference, between an index intervention and a control intervention, might be detected when such a difference should not occur. When it does occur, it is known as type I error. It arises when, by chance alone, the subjects who undergo the index …

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.

Reply to Farina and Enoka: The Reconstruct-and-Test Approach Is the Most Appropriate Validation for Surface EMG Signal Decomposition to Date

reply: We thank Drs. Farina and Enoka for recognizing our surface electromyographic (sEMG) signal decomposition technology as being “impressive” and for appreciating that “the approach is far superior to even the most optimistic expectations in the field.” We will use this opportunity to assuage doubts they and others might have, to correct their misunderstanding, and to expand on the advantages of our accuracy verification approach. They begin by casting doubt on our algorithm's ability to resolve N overlapping action potentials in complex signal segments by relying on the notion that “global optimization of overlapping action potentials is a…NP-hard problem that cannot be solved by polynomial complexity algorithms.” To those not versed in complexity theory, the jargon of NP-hard (nondeterministic polynomial time-hard) can create the mistaken impression that such problems are impossible to solve in any practical way unless N is small (say N 95.2% on average) is an empirical validation that for relatively large values of N, a great preponderance of segments in real sEMG signals can be resolved correctly. They proceed to question the accuracy assessments we reported for our algorithm. They make four claims, each of which shows a major misunderstanding of our reconstruct-and-test approach.

Reducing Symmetries to Generate Easier SAT Instances

Finding countermodels is an effective way of disproving false conjectures. In first-order predicate logic, model finding is an undecidable problem. But if a finite model exists, it can be found by exhaustive search. The finite model generation problem in the first-order logic can also be translated to the satisfiability problem in the propositional logic. But a direct translation may not be very efficient. This paper discusses how to take the symmetries into account so as to make the resulting problem easier. A static method for adding constraints is presented, which can be thought of as an approximation of the least number heuristic (LNH). Also described is a dynamic method, which asks a model searcher like SEM to generate a set of partial models, and then gives each partial model to a propositional prover. The two methods are analyzed, and compared with each other.

Simplify: a theorem prover for program checking

This article provides a detailed description of the automatic theorem prover Simplify, which is the proof engine of the Extended Static Checkers ESC/Java and ESC/Modula-3. Simplify uses the Nelson--Oppen method to combine decision procedures for several important theories, and also employs a matcher to reason about quantifiers. Instead of conventional matching in a term DAG, Simplify matches up to equivalence in an E-graph, which detects many relevant pattern instances that would be missed by the conventional approach. The article describes two techniques, error context reporting and error localization, for helping the user to determine the reason that a false conjecture is false. The article includes detailed performance figures on conjectures derived from realistic program-checking problems.

Verifying Haskell programs by combining testing, model checking and interactive theorem proving

We propose a program verification method that combines random testing, model checking and interactive theorem proving. Testing and model checking are used for debugging programs and specifications before a costly interactive proof attempt. During proof development, testing and model checking quickly eliminate false conjectures and generate counterexamples which help to correct them. With an interactive theorem prover we also ensure the correctness of the reduction of a top level problem to subproblems that can be tested or proved. We demonstrate the method using our random testing tool and binary decision diagrams-based (BDDs) tautology checker, which are added to the Agda/Alfa interactive proof assistant for dependent type theory. In particular we apply our techniques to the verification of Haskell programs. The first example verifies the BDD checker itself by testing its components. The second uses the tautology checker to verify bitonic sort together with a proof that the reduction of the problem to the checked form is correct.

Controllability Revisited: A Generalization for the Modular Approach

In the Supervisory Control Theory, originally presented by Ramadge and Wonham, Ramadge and Wonham (1989), an important property is controllability. The definition of controllability as originally presented assumed equal event sets, alphabets, in the specification and the plant. When this assumption does not hold, it has been argued that the differences in the alphabets can easily be removed. However, no matter how we choose to handle the differences, pursuing verification with the original definition will lead to false conjectures in some cases. A few examples on this is presented in this paper. In order to overcome the problem, a generalized controllability definition, allowing non-equal alphabets, is presented. It is of great importance to be able to verify controllability of subsystems with different alphabets when exploiting the modularity of discrete event systems for computationally efficient verification and synthesis.

Automated Theorem Proving by Test Set Induction

Test set induction is a goal-directed proof technique which combines the full power of explicit induction and proof by consistency. It works by computing an appropriate explicit induction scheme calleda test set, to trigger the induction proof, and then applies a refutation principle using proof by consistency techniques. We present a general scheme for test set induction together with a simple soundness proof. Our method is based on new notions of test sets,induction variables, andprovable inconsistency, which allow us to refute false conjectures even in the case where the functions are not completely defined. We show how test sets can be computed when the constructors are not free, and give an algorithm for computing induction variables. Finally, we present a procedure for proof by test set induction which is refutationally complete for a larger class of specifications than has been shown in previous work. The method has been implemented in the proverSPIKE. Based on computer experiments dealing with mutual induction,SPIKEappears to be more practical and efficient than explicit induction based systems.

True Theorems From False Conjectures

In 1933 R. K. Morley noted that there are just four proper fractions with denominators less than 100 where illegitimate cancellation of a digit produces the correct answer.1