HOL(y)Hammer: Online ATP Service for HOL Light

HOL(y)Hammer is an online AI/ATP service for formal (computer-understandable) mathematics encoded in the HOL Light system. The service allows its users to upload and automatically process an arbitrary formal development (project) based on HOL Light, and to attack arbitrary conjectures that use the concepts defined in some of the uploaded projects. For that, the service uses several automated reasoning systems combined with several premise selection methods trained on all the project proofs. The projects that are readily available on the server for such query answering include the recent versions of the Flyspeck, Multivariate Analysis and Complex Analysis libraries. The service runs on a 48-CPU server, currently employing in parallel for each task 7 AI/ATP combinations and 4 decision procedures that contribute to its overall performance. The system is also available for local installation by interested users, who can customize it for their own proof development. An Emacs interface allowing parallel asynchronous queries to the service is also provided. The overall structure of the service is outlined, problems that arise and their solutions are discussed, and an initial account of using the system is given

Exploring Semantic Hierarchies to Improve Resolution Theorem Proving on Ontologies

A resolution-theorem-prover (RTP) evaluates the validity (truthfulness) of conjectures against a set of axioms in a knowledge base. When given a conjecture, an RTP attempts to resolve the negated conjecture with axioms from the knowledge base until the prover nds a contradiction. If the RTP nds a contradiction between the axioms and a negated conjecture, the conjecture is proven. The order in which the axioms within the knowledge-base are evaluated signicantly impacts the runtime of the program, as the search-space increases exponentially with the number of axioms. Ontologies, knowledge bases with semantic (and predominantly hierarchical) structures, describe objects and their relationships to other objects. For example, a \u27Sedan\u27 class might exist in a sample ontology with \u27Automobile\u27 as a parent class and \u27Minivan\u27 as a sibling class. Currently, hierarchical structures within an ontology are not taken into account when evaluating the relevance of each axiom. Instead, each predicate is automatically assigned a weight based on a heuristic measure (such as the number of terms or the frequency of predicates relevant to the conjecture) and axioms with higher weights are evaluated rst. My research aims to intelligently select relevant axioms within a knowledge-base given a structured relationship between predicates. I have used semantic hierarchies passed to a weighting function to assign weights to each predicate. The research aims to design heuristics based upon the semantics of the predicates, rather than solely the syntax of the statements. I developed weighting functions based upon various parameters relevant to the ontological structure of predicates contained in the ontology, such as the size and depth of a hierarchy based upon the structure. The functions I have designed calculate weights for each predicate and thus each axiom in attempts to select relevant axioms when proving a theorem. I have conducted an experimental study to determine if my methods show any improvements over current reasoning methods. Results for the experiments conducted show promising results for generating weights based on semantic hierarchies and encourage further research

Learning-Assisted Automated Reasoning with Flyspeck

The considerable mathematical knowledge encoded by the Flyspeck project is combined with external automated theorem provers (ATPs) and machine-learning premise selection methods trained on the proofs, producing an AI system capable of answering a wide range of mathematical queries automatically. The performance of this architecture is evaluated in a bootstrapping scenario emulating the development of Flyspeck from axioms to the last theorem, each time using only the previous theorems and proofs. It is shown that 39% of the 14185 theorems could be proved in a push-button mode (without any high-level advice and user interaction) in 30 seconds of real time on a fourteen-CPU workstation. The necessary work involves: (i) an implementation of sound translations of the HOL Light logic to ATP formalisms: untyped first-order, polymorphic typed first-order, and typed higher-order, (ii) export of the dependency information from HOL Light and ATP proofs for the machine learners, and (iii) choice of suitable representations and methods for learning from previous proofs, and their integration as advisors with HOL Light. This work is described and discussed here, and an initial analysis of the body of proofs that were found fully automatically is provided

An Overview of Methods for Large-Theory Automated Theorem Proving

Contains fulltext : 91550.pdf (preprint version ) (Open Access)First Workshop on Automated Theory Engineering Co-Located with the 23rd International Conference on Automated Deduction Wrocław, Poland, July 31, 2011, 31 juli 201