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In this article we formally describe a declarative approach for encoding plan operatorsin proof planning, the so-called methods. The notion of method evolves from the much studiedconcept tactic and was first used by Bundy. While significant deductive power has been achievedwith the planning approach towards automated deduction, the procedural character of the tacticpart of methods, however, hinders mechanical modification. Although the strength of a proofplanning system largely depends on powerful general procedures which solve a large class ofproblems, mechanical or even automated modification of methods is nevertheless necessary forat least two reasons. Firstly methods designed for a specific type of problem will never begeneral enough. For instance, it is very difficult to encode a general method which solves allproblems a human mathematician might intuitively consider as a case of homomorphy. Secondlythe cognitive ability of adapting existing methods to suit novel situations is a fundamentalpart of human mathematical competence. We believe it is extremely valuable to accountcomputationally for this kind of reasoning.The main part of this article is devoted to a declarative language for encoding methods,composed of a tactic and a specification. The major feature of our approach is that the tacticpart of a method is split into a declarative and a procedural part in order to enable a tractableadaption of methods. The applicability of a method in a planning situation is formulatedin the specification, essentially consisting of an object level formula schema and a meta-levelformula of a declarative constraint language. After setting up our general framework, wemainly concentrate on this constraint language. Furthermore we illustrate how our methodscan be used in a Strips-like planning framework. Finally we briefly illustrate the mechanicalmodification of declaratively encoded methods by so-called meta-methods.

We show how to buildup mathematical knowledge bases usingframes. We distinguish three differenttypes of knowledge: axioms, definitions(for introducing concepts like "set" or"group") and theorems (for relating theconcepts). The consistency of such know-ledge bases cannot be proved in gen-eral, but we can restrict the possibilit-ies where inconsistencies may be impor-ted to very few cases, namely to the oc-currence of axioms. Definitions and the-orems should not lead to any inconsisten-cies because definitions form conservativeextensions and theorems are proved to beconsequences.

In most cases higher-order logic is based on the (gamma)-calculus in order to avoid the infinite set of so-called comprehension axioms. However, there is a price to be paid, namelyan undecidable unification algorithm. If we do not use the(gamma) - calculus, but translate higher-order expressions intofirst-order expressions by standard translation techniques, we haveto translate the infinite set of comprehension axioms, too. Ofcourse, in general this is not practicable. Therefore such anapproach requires some restrictions such as the choice of thenecessary axioms by a human user or the restriction to certainproblem classes. This paper will show how the infinite class ofcomprehension axioms can be represented by a finite subclass,so that an automatic translation of finite higher-order prob-lems into finite first-order problems is possible. This trans-lation is sound and complete with respect to a Henkin-stylegeneral model semantics.

Extending existing calculi by sorts is astrong means for improving the deductive power offirst-order theorem provers. Since many mathemat-ical facts can be more easily expressed in higher-orderlogic - aside the greater power of higher-order logicin principle - , it is desirable to transfer the advant-ages of sorts in the first-order case to the higher-ordercase. One possible method for automating higher-order logic is the translation of problem formulationsinto first-order logic and the usage of first-order the-orem provers. For a certain class of problems thismethod can compete with proving theorems directlyin higher-order logic as for instance with the TPStheorem prover of Peter Andrews or with the Nuprlproof development environment of Robert Constable.There are translations from unsorted higher-order lo-gic based on Church's simple theory of types intomany-sorted first-order logic, which are sound andcomplete with respect to a Henkin-style general mod-els semantics. In this paper we extend correspond-ing translations to translations of order-sorted higher-order logic into order-sorted first-order logic, thus weare able to utilize corresponding first-order theoremprover for proving higher-order theorems. We do notuse any (lambda)-expressions, therefore we have to add so-called comprehension axioms, which a priori makethe procedure well-suited only for essentially first-order theorems. However, in practical applicationsof mathematics many theorems are essentially first-order and as it seems to be the case, the comprehen-sion axioms can be mastered too.

In this paper we are interested in using a firstorder theorem prover to prove theorems thatare formulated in some higher order logic. Tothis end we present translations of higher or-der logics into first order logic with flat sortsand equality and give a sufficient criterion forthe soundness of these translations. In addi-tion translations are introduced that are soundand complete with respect to L. Henkin's gen-eral model semantics. Our higher order logicsare based on a restricted type structure in thesense of A. Church, they have typed functionsymbols and predicate symbols, but no sorts.

An important research problem is the incorporation of "declarative" knowledge into an automated theorem prover that can be utilized in the search for a proof. An interesting pro-posal in this direction is Alan Bundy's approach of using explicit proof plans that encapsulatethe general form of a proof and is instantiated into a particular proof for the case at hand. Wegive some examples that show how a "declarative" highlevel description of a proof can be usedto find proofs of apparently "similiar" theorems by analogy. This "analogical" information isused to select the appropriate axioms from the database so that the theorem can be proved.This information is also used to adjust some options of a resolution theorem prover. In orderto get a powerful tool it is necessary to develop an epistemologically appropriate language todescribe proofs, for which a large set of examples should be used as a testbed. We presentsome ideas in this direction.

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.