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Logic in Computer Science.

Theorema: Towards computer-aided mathematical theory exploration

We have developed a program for inductive theory formation, called IsaCoSy, which synthesises conjectures `bottom-up' from the available constants and free variables. The synthesis process is made tractable by only generating irreducible terms, which are then filtered through counter-example checking and passed to the automatic inductive prover IsaPlanner. The main technical contribution is the presentation of a constraint mechanism for synthesis. As theorems are discovered, this generates additional constraints on the synthesis process. We evaluate IsaCoSy as a tool for automatically generating the background theories one would expect in a mature proof assistant, such as the Isabelle system. The results show that IsaCoSy produces most, and sometimes all, of the theorems in the Isabelle libraries. The number of additional un-interesting theorems are small enough to be easily pruned by hand.

/pdf/conjecture-synthesis-for-inductive-theories-28o0qdl7ff.pdf

Conjecture Synthesis for Inductive Theories

We describe an approach to automatically invent/explore new mathematical theories, with the goal of producing results comparable to those produced by humans, as represented, for example, in the libraries of the Isabelle proof assistant. Our approach is based on 'schemes', which are formulae in higher-order logic. We show that it is possible to automate the instantiation process of schemes to generate conjectures and definitions. We also show how the new definitions and the lemmata discovered during the exploration of a theory can be used, not only to help with the proof obligations during the exploration, but also to reduce redundancies inherent in most theory-formation systems. We exploit associative-commutative (AC) operators using ordered rewriting to avoid AC variations of the same instantiation. We implemented our ideas in an automated tool, called IsaScheme, which employs Knuth-Bendix completion and recent automatic inductive proof tools. We have evaluated our system in a theory of natural numbers and a theory of lists.

Scheme-based theorem discovery and concept invention

The Theorema project aims at the development of a computer assistant for the working mathematician. Support should be given throughout all phases of mathematical activity, from introducing new mathematical concepts by definitions or axioms, through first (computational) experiments, the formulation of theorems, their justification by an exact proof, the application of a theorem as an algorithm, until to the dissemination of the results in form of a mathematical publication, the build up of bigger libraries of certified mathematical content and the like. This ambitious project is exactly along the lines of the QED manifesto issued in 1994 (see e.g. http://www.cs.ru.nl/~freek/qed/qed.html) and it was initiated in the mid-1990s by Bruno Buchberger. The Theorema system is a computer implementation of the ideas behind the Theorema project. One focus lies on the natural style of system input (in form of definitions, theorems, algorithms, etc.), system output (mainly in form of mathematical proofs) and user interaction. Another focus is theory exploration, i.e. the development of large consistent mathematical theories in a formal frame, in contrast to just proving single isolated theorems. When using the Theorema system, a user should not have to follow a certain style of mathematics enforced by the system (e.g. basing all of mathematics on set theory or certain variants of type theory), rather should the system support the user in her preferred flavour of doing math. The new implementation of the system, which we refer to as Theorema 2.0, is open-source and available through GitHub.

/pdf/theorema-2-0-computer-assisted-natural-style-mathematics-25wlilclpr.pdf

Theorema 2.0: Computer-Assisted Natural-Style Mathematics

Algorithm Synthesis by Lazy Thinking: Examples and Implementation in Theorema

Recently, as part of a general formal (i.e. logic based) methodology for mathematical knowledge management we also introduced a method for the automated synthesis of correct algorithms, which we called the lazy thinking method. For a given concrete problem specification (in predicate logic), the method tries out various algorithm schemes and derives specifications for the subalgorithms in the algorithm scheme. In this paper, we show that the lazy thinking method can be significantly streamlined by applying it, in a preprocessing step, to problem schemes before we apply the result of the preprocessing to concrete problems.

Algorithm Synthesis by Lazy Thinking: Using Problem Schemes

In this paper, we report a case study of computer supported exploration of the theory of natural numbers, using a theory exploration model based on knowledge schemes, proposed by Bruno Buchberger. We illustrate with examples from the exploration: (i) the invention of new concepts (functions, relations) in the theory, using knowledge schemes, (ii) the invention of new propositions, using proposition schemes, (iii) the invention of problems, using knowledge schemes, (iv) the introduction of new reasoning rules, by lifting knowledge to the inference level, after their correctness was proved.

/pdf/scheme-based-systematic-exploration-of-natural-numbers-2if5d7qptk.pdf

Scheme-Based Systematic Exploration of Natural Numbers

We investigate the proof complexity of a class of propositional formulas expressing a combinatorial principle known as the Kneser-Lovasz Theorem. This is a family of propositional tautologies, indexed by an nonnegative integer parameter k that generalizes the Pigeonhole Principle (obtained for k = 1).

/pdf/proof-complexity-and-the-kneser-lovasz-theorem-1lvvd2oyg5.pdf

Proof Complexity and the Kneser-Lovász Theorem

In the context of a scheme based exploration model proposed by Bruno Buchberger, we investigate the idea of decomposition, applied in the exploration of natural numbers. The free decomposition problem (i.e. whether an element can always be decomposed with respect to an operation) can be arbitrarily difficult, and we illustrate this in the theory of natural numbers. We consider a restriction, the decomposition in domains with a well-founded partial ordering: we introduce the notions of irreducible elements, reducible elements w.r.t. a composition operation, decomposition of domain elements into irreducible ones, and also the problem of irreducible decomposition which we then solve. Natural numbers can be classified as a decomposition domain, in which we know how to solve the decomposition problem. This leads to the prime decomposition theorem.

/pdf/decompositions-of-natural-numbers-from-a-case-study-in-2xhxd2jtsw.pdf

Adrian Crăciun

Papers

Algorithm Synthesis by Lazy Thinking: Examples and Implementation in Theorema

Algorithm Synthesis by Lazy Thinking: Using Problem Schemes

Scheme-Based Systematic Exploration of Natural Numbers

Proof Complexity and the Kneser-Lovász Theorem

Decompositions of Natural Numbers: From a Case Study in Mathematical Theory Exploration