Automated verification of refinement laws

Demonic refinement algebras are variants of Kleene algebras. Introduced by von Wright as a light-weight variant of the refinement calculus, their intended semantics are positively disjunctive predicate transformers, and their calculus is entirely within first-order equational logic. So, for the first time, off-the-shelf automated theorem proving (ATP) becomes available for refinement proofs. We used ATP to verify a toolkit of basic refinement laws. Based on this toolkit, we then verified two classical complex refinement laws for action systems by ATP: a data refinement law and Back's atomicity refinement law. We also present a refinement law for infinite loops that has been discovered through automated analysis. Our proof experiments not only demonstrate that refinement can effectively be automated, they also compare eleven different ATP systems and suggest that program verification with variants of Kleene algebras yields interesting theorem proving benchmarks. Finally, we apply hypothesis learning techniques that seem indispensable for automating more complex proofs

Automated Reasoning and Presentation Support for Formalizing Mathematics in Mizar

This paper presents a combination of several automated reasoning and proof presentation tools with the Mizar system for formalization of mathematics. The combination forms an online service called MizAR, similar to the SystemOnTPTP service for first-order automated reasoning. The main differences to SystemOnTPTP are the use of the Mizar language that is oriented towards human mathematicians (rather than the pure first-order logic used in SystemOnTPTP), and setting the service in the context of the large Mizar Mathematical Library of previous theorems,definitions, and proofs (rather than the isolated problems that are solved in SystemOnTPTP). These differences poses new challenges and new opportunities for automated reasoning and for proof presentation tools. This paper describes the overall structure of MizAR, and presents the automated reasoning systems and proof presentation tools that are combined to make MizAR a useful mathematical service.Comment: To appear in 10th International Conference on. Artificial Intelligence and Symbolic Computation AISC 201

Complexity, Requirements and Design

So why do we get worried about complex systems and what can we do about it? Complexity worries us because the world is unpredictable, large scale, multi component and densely interconnected. We perceived interactions as complex since we have difficulty in generalising over multiple events especially when events are poorly ordered. However interactional complexity is tractable by mathematical modeling as (misnamed) chaos theory has shown. Interactional complexity is being modeled with increasing accuracy by computational theories and simulations of physical and biological systems, viz. the IPCC world climate model. The second form is semantic complexity which implicates the difficulties we have in understanding intent of people. Here sadly there is no short term tractable solution. The Dagstuhl process of discussion leading to incremental (maybe radical) advances in understanding is one answer

Kink Chains from Instantons on a Torus

We describe how the procedure of calculating approximate solitons from instanton holonomies may be extended to the case of soliton crystals. It is shown how sine-Gordon kink chains may be obtained from CP1 instantons on a torus. These kink chains turn out to be remarkably accurate approximations to the true solutions. Some remarks on the relevance of this work to Skyrme crystals are also made.Comment: latex 17 pages, DAMTP 94-7

The Vampire and the FOOL

This paper presents new features recently implemented in the theorem prover Vampire, namely support for first-order logic with a first class boolean sort (FOOL) and polymorphic arrays. In addition to having a first class boolean sort, FOOL also contains if-then-else and let-in expressions. We argue that presented extensions facilitate reasoning-based program analysis, both by increasing the expressivity of first-order reasoners and by gains in efficiency

Requirements Engineering Domain Dimensions

This doc gives my initial ideas on the dimensions/criteria for different genres of applications (or domains if you prefer), following my summary presentation at the Dagstuhl workshop

Premise Selection and External Provers for HOL4

Learning-assisted automated reasoning has recently gained popularity among the users of Isabelle/HOL, HOL Light, and Mizar. In this paper, we present an add-on to the HOL4 proof assistant and an adaptation of the HOLyHammer system that provides machine learning-based premise selection and automated reasoning also for HOL4. We efficiently record the HOL4 dependencies and extract features from the theorem statements, which form a basis for premise selection. HOLyHammer transforms the HOL4 statements in the various TPTP-ATP proof formats, which are then processed by the ATPs. We discuss the different evaluation settings: ATPs, accessible lemmas, and premise numbers. We measure the performance of HOLyHammer on the HOL4 standard library. The results are combined accordingly and compared with the HOL Light experiments, showing a comparably high quality of predictions. The system directly benefits HOL4 users by automatically finding proofs dependencies that can be reconstructed by Metis

Baby Skyrme models for a class of potentials

We consider a class of (2+1) dimensional baby Skyrme models with potentials that have more than one vacum. These potentials are generalisation of old and new baby Skyrme models;they involve more complicated dependence on phi_3.We find that when the potential is invariant under phi_3 -> -phi_3 the configuration corresponding to the baby skyrmions lying "on top of each other" are the minima of the energy. However when the potential breaks this symmetry the lowest field configurations correspond to separated baby skyrmions. We compute the energy distributions for skyrmions of degrees between one and eight and discuss their geometrical shapes and binding energies. We also compare the 2-skyrmion states for these potentials. Most of our work has been performed numerically with the model being formulated in terms of three real scalar fields (satisfying one constraint).Comment: LaTeX, 14 pages, 10 figure

Numerical Investigation of Monopole Chains

We present numerical results for chains of SU(2) BPS monopoles constructed from Nahm data. The long chain limit reveals an asymmetric behavior transverse to the periodic direction, with the asymmetry becoming more pronounced at shorter separations. This analysis is motivated by a search for semiclassical finite temperature instantons in the 3D SU(2) Georgi-Glashow model, but it appears that in the periodic limit the instanton chains either have logarithmically divergent action or wash themselves out.Comment: 14 pages, 6 figures; v2 minor changes, published versio

Splitting Proofs for Interpolation

We study interpolant extraction from local first-order refutations. We present a new theoretical perspective on interpolation based on clearly separating the condition on logical strength of the formula from the requirement on the com- mon signature. This allows us to highlight the space of all interpolants that can be extracted from a refutation as a space of simple choices on how to split the refuta- tion into two parts. We use this new insight to develop an algorithm for extracting interpolants which are linear in the size of the input refutation and can be further optimized using metrics such as number of non-logical symbols or quantifiers. We implemented the new algorithm in first-order theorem prover VAMPIRE and evaluated it on a large number of examples coming from the first-order proving community. Our experiments give practical evidence that our work improves the state-of-the-art in first-order interpolation.Comment: 26th Conference on Automated Deduction, 201