Proof-Pattern Recognition and Lemma Discovery in ACL2

We present a novel technique for combining statistical machine learning for proof-pattern recognition with symbolic methods for lemma discovery. The resulting tool, ACL2(ml), gathers proof statistics and uses statistical pattern-recognition to pre-processes data from libraries, and then suggests auxiliary lemmas in new proofs by analogy with already seen examples. This paper presents the implementation of ACL2(ml) alongside theoretical descriptions of the proof-pattern recognition and lemma discovery methods involved in it

Hawking Radiation and Analogue Experiments: A Bayesian Analysis

We present a Bayesian analysis of the epistemology of analogue experiments with particular reference to Hawking radiation. First, we prove that such experiments can be confirmatory in Bayesian terms based upon appeal to 'universality arguments'. Second, we provide a formal model for the scaling behaviour of the confirmation measure for multiple distinct realisations of the analogue system and isolate a generic saturation feature. Finally, we demonstrate that different potential analogue realisations could provide different levels of confirmation. Our results provide a basis both to formalise the epistemic value of analogue experiments that have been conducted and to advise scientists as to the respective epistemic value of future analogue experiments.Comment: 25 pages, 5 figure

Proof by analogy in mural

One of the most important advantages of using a formal method of developing software is that one can prove that development steps are correct with respect to their specification. Conducting proofs by hand, however,can be time consuming to the extent that designers have to judge whether a proof of a particular obligation is worth conducting. Even if hand proofs are worth conducting, how do we know that they are correct? One approach to overcoming this problem is to use an automatic theorem proving system to develop and check our proofs. However, in order to enable present day theorem provers to check proofs, one has to conduct them in much more detail than hand proofs. Carrying out more detailed proofs is of course more time consuming. This paper describes the use of proof by analogy in an attempt to reduce the time spent on proofs. We develop and implement a proof follower based on analogy and present two examples to illustrate its characteristics. One example illustrates the successful use of the proof follower. The other example illustrates that the follower's failure can provide a hint that enables the user to complete a proof

Application of expert systems in project management decision aiding

The feasibility of developing an expert systems-based project management decision aid to enhance the performance of NASA project managers was assessed. The research effort included extensive literature reviews in the areas of project management, project management decision aiding, expert systems technology, and human-computer interface engineering. Literature reviews were augmented by focused interviews with NASA managers. Time estimation for project scheduling was identified as the target activity for decision augmentation, and a design was developed for an Integrated NASA System for Intelligent Time Estimation (INSITE). The proposed INSITE design was judged feasible with a low level of risk. A partial proof-of-concept experiment was performed and was successful. Specific conclusions drawn from the research and analyses are included. The INSITE concept is potentially applicable in any management sphere, commercial or government, where time estimation is required for project scheduling. As project scheduling is a nearly universal management activity, the range of possibilities is considerable. The INSITE concept also holds potential for enhancing other management tasks, especially in areas such as cost estimation, where estimation-by-analogy is already a proven method

A Graphical Language for Proof Strategies

Complex automated proof strategies are often difficult to extract, visualise, modify, and debug. Traditional tactic languages, often based on stack-based goal propagation, make it easy to write proofs that obscure the flow of goals between tactics and are fragile to minor changes in input, proof structure or changes to tactics themselves. Here, we address this by introducing a graphical language called PSGraph for writing proof strategies. Strategies are constructed visually by "wiring together" collections of tactics and evaluated by propagating goal nodes through the diagram via graph rewriting. Tactic nodes can have many output wires, and use a filtering procedure based on goal-types (predicates describing the features of a goal) to decide where best to send newly-generated sub-goals. In addition to making the flow of goal information explicit, the graphical language can fulfil the role of many tacticals using visual idioms like branching, merging, and feedback loops. We argue that this language enables development of more robust proof strategies and provide several examples, along with a prototype implementation in Isabelle

Analogical Reasoning in Mathematical Theorems

Analogical reasoning is one of the most powerful tools of mathematical thinking. For example, to prove a theorem it is necessary to see similarities with the previous theorem. This study aims to classify analogies in mathematics courses and examples. This classification is based on research results. The research was conducted use qualitative research. The research subjects are 12 lecturers who teach mathematics courses and study program managers. Analogical reasoning instruments are unstructured interview guidelines and observation sheets. Interview guides and observation sheets were made to be able to reveal mathematics analogical reasoning in the Mathematics Education Study Program course. The results of the research show that there are 3 types of analogy classifications in mathematics courses, namely definition analogy, theorem-defining analogy, and theorem analogy. First, the definition of similarity in the same or different courses. Second, the similarities between definitions and theorems in the same or different courses. Third, the theorem similarities in the same or different subjects. Our classification is related to theorems and analogical properties in several courses in the curriculum of the Mathematics Education Study Program. The analogy can be applied to certain mathematical topics related to real life. Meanwhile, to analyze other aspects of reasoning through analogy needs to be studied further

Proving Correctness and Completeness of Normal Programs - a Declarative Approach

We advocate a declarative approach to proving properties of logic programs. Total correctness can be separated into correctness, completeness and clean termination; the latter includes non-floundering. Only clean termination depends on the operational semantics, in particular on the selection rule. We show how to deal with correctness and completeness in a declarative way, treating programs only from the logical point of view. Specifications used in this approach are interpretations (or theories). We point out that specifications for correctness may differ from those for completeness, as usually there are answers which are neither considered erroneous nor required to be computed. We present proof methods for correctness and completeness for definite programs and generalize them to normal programs. For normal programs we use the 3-valued completion semantics; this is a standard semantics corresponding to negation as finite failure. The proof methods employ solely the classical 2-valued logic. We use a 2-valued characterization of the 3-valued completion semantics which may be of separate interest. The presented methods are compared with an approach based on operational semantics. We also employ the ideas of this work to generalize a known method of proving termination of normal programs.Comment: To appear in Theory and Practice of Logic Programming (TPLP). 44 page

Connectionist Inference Models

The performance of symbolic inference tasks has long been a challenge to connectionists. In this paper, we present an extended survey of this area. Existing connectionist inference systems are reviewed, with particular reference to how they perform variable binding and rule-based reasoning, and whether they involve distributed or localist representations. The benefits and disadvantages of different representations and systems are outlined, and conclusions drawn regarding the capabilities of connectionist inference systems when compared with symbolic inference systems or when used for cognitive modeling