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In this paper we generalize the notion of method for proofplanning. While we adopt the general structure of methods introducedby Alan Bundy, we make an essential advancement in that we strictlyseparate the declarative knowledge from the procedural knowledge. Thischange of paradigm not only leads to representations easier to under-stand, it also enables modeling the important activity of formulatingmeta-methods, that is, operators that adapt the declarative part of exist-ing methods to suit novel situations. Thus this change of representationleads to a considerably strengthened planning mechanism.After presenting our declarative approach towards methods we describethe basic proof planning process with these. Then we define the notion ofmeta-method, provide an overview of practical examples and illustratehow meta-methods can be integrated into the planning process.
Extending the planADbased paradigm for auto-mated theorem proving, we developed in previ-ous work a declarative approach towards rep-resenting methods in a proof planning frame-work to support their mechanical modification.This paper presents a detailed study of a classof particular methods, embodying variations ofa mathematical technique called diagonaliza-tion. The purpose of this paper is mainly two-fold. First we demonstrate that typical math-ematical methods can be represented in ourframework in a natural way. Second we illus-trate our philosophy of proof planning: besidesplanning with a fixed repertoire of methods,metaADmethods create new methods by modify-ing existing ones. With the help of three differ-ent diagonalization problems we present an ex-ample trace protocol of the evolution of meth-ods: an initial method is extracted from a par-ticular successful proof. This initial method isthen reformulated for the subsequent problems,and more general methods can be obtained byabstracting existing methods. Finally we comeup with a fairly abstract method capable ofdealing with all the three problems, since it cap-tures the very key idea of diagonalization.
Most automated theorem provers suffer from the problem thatthey can produce proofs only in formalisms difficult to understand even forexperienced mathematicians. Effort has been made to reconstruct naturaldeduction (ND) proofs from such machine generated proofs. Although thesingle steps in ND proofs are easy to understand, the entire proof is usuallyat a low level of abstraction, containing too many tedious steps. To obtainproofs similar to those found in mathematical textbooks, we propose a newformalism, called ND style proofs at the assertion level , where derivationsare mostly justified by the application of a definition or a theorem. Aftercharacterizing the structure of compound ND proof segments allowing asser-tion level justification, we show that the same derivations can be achieved bydomain-specific inference rules as well. Furthermore, these rules can be rep-resented compactly in a tree structure. Finally, we describe a system calledPROVERB , which substantially shortens ND proofs by abstracting them tothe assertion level and then transforms them into natural language.
Planning Argumentative Texts
(1999)
This paper presents PROVERB a text planner forargumentative texts. PROVERB's main feature isthat it combines global hierarchical planning and un-planned organization of text with respect to local de-rivation relations in a complementary way. The formersplits the task of presenting a particular proof intosubtasks of presenting subproofs. The latter simulateshow the next intermediate conclusion to be presentedis chosen under the guidance of the local focus.
We present two techniques for reasoning from cases to solve classification tasks: Induction and case-based reasoning. We contrast the two technologies (that are often confused) and show how they complement each other. Based on this, we describe how they are integrated in one single platform for reasoning from cases: The Inreca system.
The introduction of sorts to first-order automated deduc-tion has brought greater conciseness of representation and a considerablegain in efficiency by reducing search spaces. This suggests that sort in-formation can be employed in higher-order theorem proving with similarresults. This paper develops a sorted (lambda)-calculus suitable for automatictheorem proving applications. It extends the simply typed (lambda)-calculus by ahigher-order sort concept that includes term declarations and functionalbase sorts. The term declaration mechanism studied here is powerfulenough to subsume subsorting as a derived notion and therefore gives ajustification for the special form of subsort inference. We present a set oftransformations for sorted (pre-) unification and prove the nondetermin-istic completeness of the algorithm induced by these transformations.
Higher-Order Tableaux
(1999)
Even though higher-order calculi for automated theorem prov-ing are rather old, tableau calculi have not been investigated yet. Thispaper presents two free variable tableau calculi for higher-order logicthat use higher-order unification as the key inference procedure. Thesecalculi differ in the treatment of the substitutional properties of equival-ences. The first calculus is equivalent in deductive power to the machine-oriented higher-order refutation calculi known from the literature, whereasthe second is complete with respect to Henkin's general models.
Many mathematical proofs are hard to generate forhumans and even harder for automated theoremprovers. Classical techniques of automated theoremproving involve the application of basic rules, of built-in special procedures, or of tactics. Melis (Melis 1993)introduced a new method for analogical reasoning inautomated theorem proving. In this paper we showhow the derivational analogy replay method is relatedand extended to encompass analogy-driven proof planconstruction. The method is evaluated by showing theproof plan generation of the Pumping Lemma for con-text free languages derived by analogy with the proofplan of the Pumping Lemma for regular languages.This is an impressive evaluation test for the analogicalreasoning method applied to automated theorem prov-ing, as the automated proof of this Pumping Lemmais beyond the capabilities of any of the current auto-mated theorem provers.
This paper addresses the decomposition of proofs as a means of constructingmethods in plan-based automated theorem proving. It shows also, howdecomposition can beneficially be applied in theorem proving by analogy.Decomposition is also useful for human-style proof presentation. We proposeseveral decomposition techniques that were found to be useful in automatedtheorem proving and give examples of their application.