Learning Semantics for Automated Reasoning

Learning reasoning techniques from previous knowledge is a largely underdeveloped area of automated reasoning. As large bodies of form al knowledge are becoming available, state-of-the-art machine learning me thods, particularly the ones that are able to leverage semantics from the mathematic al libraries, provide a new avenue for problem-specific detection of relevant know ledge contained in the mathematical data. We present automated reasoning as a n ovel application area for machine learning and briefly describe promising res ults we have recently obtained. 1 Automated Reasoning and Machine Learning In the last fifteen years, the body of formally expressed math ematics has grown substantially. Interactive Theorem Provers (ITPs) like Coq, Isabelle, Mizar, an d HOL [15] have been used for advanced formal theory developments and verification of non-trivial theorems, like the Four Color Theorem and Jordan Curve Theorem, and also for advanced verification of software and hardware models. The large Mizar mathematical library (MML) 1 contains today nearly 1100 formal mathematical articles, covering a substantial part of standard undergradu ate mathematical knowledge. The library has about 50000 theorems, proved with about 2.5 million line s of mathematical proofs. Such proofs often contain nontrivial mathematical ideas, s ometimes precised over decades and centuries of development of mathematics and abstract forma l thinking. Having this kind of a “knowledge base of abstract human thinking” in a completely machine-processable and machineunderstandable way, presents very interesting opportunit ies for application and development of novel artificial intelligence methods that make use of the semantic knowledge in various ways. An example is the MaLARea meta-system [14] combining deductive pro of finding (automated theorem proving) with learning from new proofs in a closed feedback loop, boosting over time the performance of both the deductive and inductive components. A concrete a nd pressing task is the selection of relevant premises from the large formal knowledge bases, wh en one is presented with a new conjecture that needs to be proved. Providing good solution to t his problem is important both for the mathematicians, and also for the existing tools for automat ed theorem proving (ATP) that typically cannot be successfully used directly with tens or hundreds o f thousands of axioms. Experiments like the MPTP Challenge 2 and the LTB (Large Theory Batch) division of the CASC competi tion3 that smart premise selection can significantly boost the perform ance of existing ATP techniques in large domains [14]. We aim to solve the followingpremise selection problem in large real-world mathematics: given a large knowledge base P of thousands of premises and proofs, and a new conjecture x, find the http://www.mizar.org http://www.tptp.org/MPTPChallenge http://www.tptp.org/CASC