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- Partial functions (2)
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- Fachbereich Informatik (24) (remove)

In this paper we present an extensional higher-order resolution calculus that iscomplete relative to Henkin model semantics. The treatment of the extensionality princi-ples - necessary for the completeness result - by specialized (goal-directed) inference rulesis of practical applicability, as an implentation of the calculus in the Leo-System shows.Furthermore, we prove the long-standing conjecture, that it is sufficient to restrict the orderof primitive substitutions to the order of input formulae.

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.

Even though it is not very often admitted, partial functionsdo play a significant role in many practical applications of deduction sys-tems. Kleene has already given a semantic account of partial functionsusing a three-valued logic decades ago, but there has not been a satisfact-ory mechanization. Recent years have seen a thorough investigation ofthe framework of many-valued truth-functional logics. However, strongKleene logic, where quantification is restricted and therefore not truth-functional, does not fit the framework directly. We solve this problemby applying recent methods from sorted logics. This paper presents atableau calculus that combines the proper treatment of partial functionswith the efficiency of sorted calculi.

The semantics of everyday language and the semanticsof its naive translation into classical first-order language consider-ably differ. An important discrepancy that is addressed in this paperis about the implicit assumption what exists. For instance, in thecase of universal quantification natural language uses restrictions andpresupposes that these restrictions are non-empty, while in classi-cal logic it is only assumed that the whole universe is non-empty.On the other hand, all constants mentioned in classical logic arepresupposed to exist, while it makes no problems to speak about hy-pothetical objects in everyday language. These problems have beendiscussed in philosophical logic and some adequate many-valuedlogics were developed to model these phenomena much better thanclassical first-order logic can do. An adequate calculus, however, hasnot yet been given. Recent years have seen a thorough investigationof the framework of many-valued truth-functional logics. UnfortuADnately, restricted quantifications are not truth-functional, hence theydo not fit the framework directly. We solve this problem by applyingrecent methods from sorted logics.

Even though it is not very often admitted, partial functionsdo play a significant role in many practical applications of deduction sys-tems. Kleene has already given a semantic account of partial functionsusing a three-valued logic decades ago. This approach allows rejectingcertain unwanted formulae as faulty, which the simpler two-valued onesaccept. We have developed resolution and tableau calculi for automatedtheorem proving that take the restrictions of the three-valued logic intoaccount, which however have the severe drawback that existing theo-rem provers cannot directly be adapted to the technique. Even recentlyimplemented calculi for many-valued logics are not well-suited, since inthose the quantification does not exclude the undefined element. In thiswork we show, that it is possible to enhance a two-valued theorem proverby a simple strategy so that it can be used to generate proofs for the the-orems of the three-valued setting. By this we are able to use an existingtheorem prover for a large fragment of the language.

Dynamic Lambda Calculus
(1999)

The goal of this paper is to lay a logical foundation for discourse theories by providing analgebraic foundation of compositional formalisms for discourse semantics as an analogon tothe simply typed (lambda)-calculus. Just as that can be specialized to type theory by simply providinga special type for truth values and postulating the quantifiers and connectives as constantswith fixed semantics, the proposed dynamic (lambda)-calculus DLC can be specialized to (lambda)-DRT byessentially the same measures, yielding a much more principled and modular treatment of(lambda)-DRT than before; DLC is also expected to eventually provide a conceptually simple basisfor studying higher-order unification for compositional discourse theories.Over the past few years, there have been a series of attempts [Zee89, GS90, EK95, Mus96,KKP96, Kus96] to combine the Montagovian type theoretic framework [Mon74] with dynamicapproaches, such as DRT [Kam81]. The motivation for these developments is to obtain a generallogical framework for discourse semantics that combines compositionality and dynamic binding.Let us look at an example of compositional semantics construction in (lambda)-DRT which is one ofthe above formalisms [KKP96, Kus96]. By the use of fi-reduction we arrive at a first-order DRTrepresentation of the sentence A i man sleeps. (i denoting an index for anaphoric binding.)