Jochen Burghardt
We describe an approach to learn, in a term-rewriting setting, function definitions from input/output equations. By confining ourselves to structurally recursive definitions we obtain a fairly fast learning algorithm that often yields definitions close to intuitive expectations. We provide a Prolog prototype implementation of our approach, and indicate open issues of further investigation.
Jochen Burghardt
E-generalization computes common generalizations of given ground terms w.r.t. a given equational background theory E. In 2005 [arXiv:1403.8118], we had presented a computation approach based on standard regular tree grammar algorithms, and a Prolog prototype implementation. In this report, we present algorithmic improvements, prove them correct and complete, and give some details of an efficiency-oriented implementation in C that allows us to handle problems larger by several orders of magnitude.
Jochen Burghardt
Using the synthesis approach of Manna and Waldinger, a formally specified and verified control circuitery for a production cell was developped. Building an appropriate formal language level, we could achieve a requirements specification to the informal description. We demonstrated that the paradigm of deductive synthesis can be applied to the development of complete verified systems, including hardware and mechanics. We defined two domain-specific logical operators that schematise frequent patterns in specification and proof and hence allow a more concise and expressive presentation. In Burghardt (1995), an english short version of this paper, without appendices, can be found.
Jochen Burghardt
Starting from an informal requirements description of a toy production cell used in an intra-project competition in 1994, we give a formal specification that is as close as possible to requirements. We use the deductive program synthesis approach by Manna and Waldinger (1980) to obtain a verified TTL-like circuitery to control the cell. The formal specification also covers mechanical aspects and thus allows to reason not only about software issues but also about issues of mechanical engineering. Besides an approach confined to first order predicate logic with explicit, continuous time, an attempt is presented to employ application specific user-defined logical operators to get a more concise specification as well as proof.
Jochen Burghardt, Birgit Heinz
We present an implementation of E-anti-unification as defined in Heinz (1995), where tree-grammar descriptions of equivalence classes of terms are used to compute generalizations modulo equational theories. We discuss several improvements, including an efficient implementation of variable-restricted E-anti-unification from Heinz (1995), and give some runtime figures about them. We present applications in various areas, including lemma generation in equational inductive proofs, intelligence tests, diverging Knuth-Bendix completion, strengthening of induction hypotheses, and theory formation about finite algebras.
Jochen Burghardt
We extend the notion of anti-unification to cover equational theories and present a method based on regular tree grammars to compute a finite representation of E-generalization sets. We present a framework to combine Inductive Logic Programming and E-generalization that includes an extension of Plotkin's lgg theorem to the equational case. We demonstrate the potential power of E-generalization by three example applications: computation of suggestions for auxiliary lemmas in equational inductive proofs, computation of construction laws for given term sequences, and learning of screen editor command sequences.