## SEKI Report

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- Wissenschaftlicher Artikel (6) (entfernen)

#### Schlagworte

- abstract description (1)
- analogy (1)
- completeness (1)
- higher order logic (1)
- morphism (1)
- proof plans (1)
- second order logic (1)
- soundness (1)
- theorem proving (1)
- translation (1)

94,3

A method for efficiently handling associativity and commutativity (AC) in implementations of (equational) theorem provers without incorporating AC as an underlying theory will be presented. The key of substantial efficiency gains resides in a more suitable representation of permutation-equations (such as f(x,f(y,z))=f(y,f(z,x)) for instance). By representing these permutation-equations through permutations in the mathematical sense (i.e. bijective func- tions :{1,..,n} {1,..,n}), and by applying adapted and specialized inference rules, we can cope more appropriately with the fact that permutation-equations are playing a particular role. Moreover, a number of restrictions concerning application and generation of permuta- tion-equations can be found that would not be possible in this extent when treating permu- tation-equations just like any other equation. Thus, further improvements in efficiency can be achieved.

94,8

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.

93,14

Unification in an Extensional Lambda Calculus with Ordered Function Sorts and Constant Overloading
(1999)

We develop an order-sorted higher-order calculus suitable forautomatic theorem proving applications by extending the extensional simplytyped lambda calculus with a higher-order ordered sort concept and constantoverloading. Huet's well-known techniques for unifying simply typed lambdaterms are generalized to arrive at a complete transformation-based unificationalgorithm for this sorted calculus. Consideration of an order-sorted logicwith functional base sorts and arbitrary term declarations was originallyproposed by the second author in a 1991 paper; we give here a correctedcalculus which supports constant rather than arbitrary term declarations, aswell as a corrected unification algorithm, and prove in this setting resultscorresponding to those claimed there.

93,10

The hallmark of traditional Artificial Intelligence (AI) research is the symbolic representation and processing of knowledge. This is in sharp contrast to many forms of human reasoning, which to an extraordinary extent, rely on cases and (typical) examples. Although these examples could themselves be encoded into logic, this raises the problem of restricting the corresponding model classes to include only the intended models.There are, however, more compelling reasons to argue for a hybrid representa-tion based on assertions as well as examples. The problems of adequacy, availability of information, compactness of representation, processing complexity, and last but not least, results from the psychology of human reasoning, all point to the same conclusion: Common sense reasoning requires different knowledge sources and hybrid reasoning principles that combine symbolic as well as semantic-based inference. In this paper we address the problem of integrating semantic representations of examples into automateddeduction systems. The main contribution is a formal framework for combining sentential with direct representations. The framework consists of a hybrid knowledge base, made up of logical formulae on the one hand and direct representations of examples on the other, and of a hybrid reasoning method based on the resolution calculus. The resulting hybrid resolution calculus is shown to be sound and complete.

90,19

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.

89,12

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.