Integrating Automated and Interactive Theorem Proving

Artificial Intelligence for Theorem Proving (AITP) has given rise to a plethora of benchmarks and methodologies, particularly in Interactive Theorem Proving (ITP). Research in the area is fragmented, with a diverse set of approaches being spread across several ITP systems. This presents a significant challenge to the comparison of methods, which are often complex and difficult to replicate. Addressing this, we present BAIT, a framework for the fair and streamlined comparison of learning approaches in ITP. We demonstrate BAIT’s capabilities with an in-depth comparison, across several ITP benchmarks, of state-of-the-art architectures applied to the problem of formula embedding. We find that Structure Aware Transformers perform particularly well, improving on techniques associated with the original problem sets. BAIT also allows us to assess the end-to-end proving performance of systems built on interactive environments. This unified perspective reveals a novel end-to-end system that improves on prior work. We also provide a qualitative analysis, illustrating that improved performance is associated with more semantically-aware embeddings. By streamlining the implementation and comparison of Machine Learning algorithms in the ITP context, we anticipate BAIT will be a springboard for future research.

The formalization of mathematics in practice relies heavily on proof assistants and automatic theorem provers, therefore we studied what are the state of the art proof assitants and their limitations to understand what are the main challenges in making formalized mathematics common practice among mathematicians. We found out that curretly the two major dificulties in formalizing mathematics with proof assistants are due to steep learning curves in how to use these tools and due to a wide gap between the notation employed in these proof assistants and the currently used mathematical notation. We also developed a C++ library to develop proof assistants with great notational flexibility.

This volume contains the proceedings of the Workshop on Partiality and Recursion in Interactive Theorem Provers (PAR 2010) which took place on July 15 in Edinburgh, UK. This workshop was held as a satellite workshop of the International Conference on Interactive Theorem Proving (ITP 2010), itself part of the Federated Logic Conference 2010 (FLoC 2010). This workshop is a venue for researchers working on new approaches to cope with partial functions and terminating general (co)recursion in theorem provers.

Unpacking the Relationship between Prosecutors and Democracy in the United States

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Agile software engineering increasingly seeks to incorporate design modification and continuous refactoring in order to maintain code quality even in highly dynamic environments. However, there does not currently appear to be an industry-wide consensus on how to do this and research in this area expresses conflicting opinions. This paper presents an empirical study based upon an industry survey aimed at understanding the different ways that refactoring is thought of by the different people carrying out different roles in agile processes and how these different people weigh the importance of refactoring versus other kinds of tasks in the process. The study found good support for the importance of refactoring, but most respondents agreed that deferred refactoring impacts the agility of their process. Thus there was no universally agreed-upon strategy for planning refactoring. The survey findings also indicated that different roles have different perspectives on the different kinds of tasks in an agile process although all seem to want to increase the priority given to refactoring during planning for the iterations in agile development. Analysis of the survey raised many interesting questions suggesting the need for a considerable amount of future research.

Most computer checked proofs are tied to the particular technology of a prover’s software. While sharing results between proof assistants is a recognized and desirable goal, the current organization of theorem proving tools makes such sharing an exception instead of the rule. In this talk, I argue that we need to turn the current architecture of proof assistants and formal proofs inside-out. That is, instead of having a few mature theorem provers include within them their formally checked theorems and proofs, I propose that proof assistants should sit on the edge of a web of formal proofs and that proof assistant should be exporting their proofs so that they can exist independently of any theorem prover. While it is necessary to maintain the dependencies between definitions, theories, and theorems, no explicit library structure should be imposed on this web of formal proofs. Thus a theorem and its proofs should not necessarily be located at a particular URL or within a particular prover’s library. While the world of symbolic logic and proof theory certainly allows for proofs to be seen as global and permanent objects, there is a lot of research and engineering work that is needed to make this possible. I describe some of the required research and development that must be done to achieve this goal.

We introduce a heuristic for automatically checking the validity of first-order formulas of the form forallalpham existbetan middot Psi(alpham,betan) that are encountered in inductive proofs of hardware correctness. The heuristic introduced in this paper is used to automatically check the validity of k-step induction formulas needed to verify hardware designs. The heuristic works on word-level designs that can have data and address buses of arbitrary widths. Our refinement heuristic relies on the idea of predicate instantiation introduced in (H. Abu-Haimed et al., 2003). The heuristic proves quantified formulas by the use of a validity checker, CVC (A. Stump et al., 2002), and a first-order theorem prover, Otter (W.W. McCune, 1994). Our heuristic can be used as a stand-alone technique to verify word-level designs or as a component in an interactive theorem prover. We show the effectiveness of this heuristic for hardware verification by verifying a number of hardware designs completely automatically. The large size of the quantified formulas encountered in these examples shows the effectiveness of our heuristic as a component of a theorem prover

And-Inverter Graphs (AIGs) are a popular way to represent Boolean functions (like circuits). AIG simplification algorithms can dramatically reduce an AIG, and play an important role in modern hardware verification tools like equivalence checkers. In practice, these tricky algorithms are implemented with optimized C or C++ routines with no guarantee of correctness. Meanwhile, many interactive theorem provers can now employ SAT or SMT solvers to automatically solve finite goals, but no theorem prover makes use of these advanced, AIG-based approaches. We have developed two ways to represent AIGs within the ACL2 theorem prover. One representation, Hons-AIGs, is especially convenient to use and reason about. The other, Aignet, is the opposite; it is styled after modern AIG packages and allows for efficient algorithms. We have implemented functions for converting between these representations, random vector simulation, conversion to CNF, etc., and developed reasoning strategies for verifying these algorithms. Aside from these contributions towards verifying AIG algorithms, this work has an immediate, practical benefit for ACL2 users who are using GL to bit-blast finite ACL2 theorems: they can now optionally trust an off-the-shelf SAT solver to carry out the proof, instead of using the built-in BDD package. Looking to the future, it is a first step toward implementing verified AIG simplification algorithms that might further improve GL performance.

Predicting disease progress with imprecise lab test results

This chapter asks what changes should be made, in the here and now, to improve the moral balance sheet of organizations. It introduces the notion of ‘bottom-up’ requirements on the basic structure of a society, which stem from the inner life of organizations and which aim at minimizing the risk of violations of basic moral norms. Two such ‘non-ideal’ proposals are discussed: first, improvements in the protection of individual rights against organizations, and second, a better match between organizational forms, and the pressures that come with them, and organizations with different kinds of task. For example, some organizations should be shielded from financial pressures in order to fulfil their tasks, such as care for patients, without creating grave moral risks. This is a matter of market regulation, but also of the line between public and private organizations.

In recent years a new class of hybrid real-time applications has emerged. The peculiarity of these applications is that both the resource requirements and the arrival rates of service requests are not known in advance and each type of job has its own temporal requests. The traditional open loop scheduling algorithms perform poorly in those systems because those algorithms are not adjusted base on the system workloads. It has been shown that feedback control is very effective to support the specified performance of those unpredictable dynamic systems. In order to use feedback control scheduling efficiently, it is necessary to have a model that describes the behavior of the system adequately. In this paper, we propose an adaptive reservation based scheduler with a feedback scheme. After describing a precise mathematical model of the scheduler via state-space equations, we design a controller to adjust the resource assignation for different kinds of tasks in order to satisfy their temporal requests. The global stability of our scheduler is discussed as a piece-wise affine system (PWA). Experiments based on VxWorks/Goahead embedded Web server system demonstrate the effectiveness of the proposed scheduling algorithm when there are large bursts of total load in the system.

In Portugal, the training of educators and teachers of the early years includes various components, requiring the conclusion of an internship report on the supervised teaching practice. In the Escola Superior de Educacao de Santarem this report includes a research component. This qualitative study analyzes the role of research conducted by future teachers and educators in mathematics with regard to their professional development, aiming to identify aspects of didactical knowledge concerning the tasks, which are evident in these reports. The data are collected by documentary analysis of the reports made in the courses that enable to teach Childhood Education to the 2nd cycle of basic education, school years 2010-2011 to 2014-2015. The studies conducted involving various mathematical topics and the use of different kinds of tasks that rely essentially on the use of manipulatives and the promotion of various representations. The study evidence shows that the investigative component, focused on teaching and learning of mathematics, allows them to develop research skills and deepen their knowledge of tasks, in particular, the analysis and reflection of its implementation with students, their productions and work in the classroom. Future teachers and educators identify gains and suggest future action strategies.

Selective and nonselective neuronal NOS inhibitors impair cognitive function in the three panel runway and passive avoidance tasks in rats

Hadoop has become true industry standard for kernel of the distributed operating system in Big data. Hadoop has picked up its prevalence because of its capacity of processing huge data, breaking down and getting to substantial measure of information, rapidly and also it is more stable technology. “Hadoop” has HDFS, YARN and Map Reduceas its core components. To supplement the Hadoop modules there are additionally an assortment of different components that give particular administrations and are extensively used to make Hadoop more available and more usable which is known as Hadoop Ecosystem. Late HadoopEco system comprises of various layers, each layer performing different kind of tasks like storing your data, processing stored data, resource allocating and supporting different programming languages to develop various applications. Granular Computing (GrC) can be considered as a common name of theories, methodologies, techniques and tools that make use of granules, i.e. groups, classes, or clusters of a universe, in the process of solving problems. Basic ideas of crisp information granulation have appeared in related fields. This paper proposes a new Ecosystem for Hadoop which can process date in image format and analyze to give proper results.