We present a personal view and strategy for algorithm-supported mathematical theory exploration and draw some conclusions for the desirable functionality of future mathematical software systems. The main points of emphasis are: The use of schemes for bottom-up mathematical invention, the algorithmic generation of conjectures from failing proofs for top-down mathematical invention, and the possibility to program new reasoners within the logic on which the reasoners work ("meta-programming").

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.

We describe an approach for formally verifying the linkage between a symbolic state enumeration system and a theorem proving system. This involves the following three stages of proof. Firstly we prove theorems about the correctness of the translation part of the symbolic state system. It interfaces between low level decision diagrams and high level description languages. We ensure that the semantics of a program is preserved in those of its translated form. Secondly we prove linkage theorems: theorems that justify introducing a result from a state enumeration system into a proof system. Finally we combine the translator correctness and linkage theorems. The resulting new linkage theorems convert results to a high level language from the low level decision diagrams that the result was actually proved about in the state enumeration system.They justify importing low-level external verification results into a theorem prover. We use a linkage between the HOL system and a simplified version of the MDG system to illustrate the ideas and consider a small example that integrates two applications from MDG and HOL to illustrate the linkage theorems.

We consider the entire process of mathematical theory exploration: invention of mathematical concepts (axioms and definitions), consistency check for axioms, invention and proof of mathematical propositions, invention of mathematical problems, invention and verification of methods (in particular algorithms) for solving problems. We aim at computer-supporting this process in all phases and at building up and managing verified formal mathematical knowledge bases as a result of this process. Our logic frame is predicate logic in a special version and implementation called ”Theorema”. In the course, we will consider several case studies, notably the theory of Groebner bases, and will discuss the heuristics of the process of mathematical theory exploration and possibilities for computer-support in this process. We distinguish between ”top-down ” and ”bottom-up ” approaches to mathematical theory exploration. In both approaches, formula schemes play an important role. A formula scheme is a formula in which a couple of unspecified function and predicate symbols occur. Depending on the context, a formula scheme can be used as definition scheme, axiom scheme, proposition scheme, problem scheme, or algorithm schemed and may help both in partially automating the bottom-up and the top-down approach to mathematical theory exploration.

Observing is the process of obtaining new knowledge, expressed in language, by bringing the senses in contact with reality. Reasoning, in contrast, is the process of obtaining new knowledge from given knowledge, by applying certain general transformation rules that depend only on the form of

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