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- Partial functions (2)
- many-valued logic (2)
- Declarative and Procedural Knowledge (1)
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- Tactics (1)
- automated theorem proving (1)
- higher order tableau (1)

Coloring terms (rippling) is a technique developed for inductive theorem proving which uses syntactic differences of terms to guide the proof search. Annotations (colors) to terms are used to maintain this information. This technique has several advantages, e.g. it is highly goal oriented and involves little search. In this paper we give a general formalization of coloring terms in a higher-order setting. We introduce a simply-typed lambda calculus with color annotations and present an appropriate (pre-)unification algorithm. Our work is a formal basis to the implementation of rippling in a higher-order setting which is required e.g. in case of middle-out reasoning. Another application is in the construction of natural language semantics, where the color annotations rule out linguistically invalid readings that are possible using standard higher-order unification.

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, 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.

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.

Mechanised reasoning systems and computer algebra systems have apparentlydifferent objectives. Their integration is, however, highly desirable, since in manyformal proofs both of the two different tasks, proving and calculating, have to beperformed. Even more importantly, proof and computation are often interwoven andnot easily separable. In the context of producing reliable proofs, the question howto ensure correctness when integrating a computer algebra system into a mechanisedreasoning system is crucial. In this contribution, we discuss the correctness prob-lems that arise from such an integration and advocate an approach in which thecalculations of the computer algebra system are checked at the calculus level of themechanised reasoning system. This can be achieved by adding a verbose mode to thecomputer algebra system which produces high-level protocol information that can beprocessed by an interface to derive proof plans. Such a proof plan in turn can beexpanded to proofs at different levels of abstraction, so the approach is well-suited forproducing a high-level verbalised explication as well as for a low-level machine check-able calculus-level proof. We present an implementation of our ideas and exemplifythem using an automatically solved extended example.

Die Beweisentwicklungsumgebung Omega-Mkrp soll Mathematiker bei einer ihrer Haupttätigkeiten, nämlich dem Beweisen mathematischer Theoreme unterstützen. Diese Unterstützung muß so komfortabel sein, daß die Beweise mit vertretbarem Aufwand formal durchgeführt werden können und daß die Korrektheit der so erzeugten Beweise durch das System sichergestellt wird. Ein solches System wird sich nur dann wirklich durchsetzen, wenn die rechnergestützte Suche nach formalen Beweisen weniger aufwendig und leichter ist, als ohne das System. Um dies zu erreichen, ergeben sich verschiedene Anforderungen an eine solche Entwicklungsumgebung, die wir im einzelnen beschreiben. Diese betreffen insbesondere die Ausdruckskraft der verwendeten Objektsprache, die Möglichkeit, abstrakt über Beweispläne zu reden, die am Menschen orientierte Präsentation der gefundenen Beweise, aber auch die effiziente Unterstützung beim Füllen von Beweislücken. Das im folgenden vorgestellte Omega-Mkrp-System ist eine Synthese der Ansätze des vollautomatischen, des interaktiven und des planbasierten Beweisens und versucht erstmalig die Ergebnisse dieser drei Forschungsrichtungen in einem System zu vereinigen. Dieser Artikel soll eine Übersicht über unsere Arbeit an diesem System geben.

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.)

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.

This paper describes a tableau-based higher-order theorem prover HOT and an application to natural language semantics. In this application, HOT is used to prove equivalences using world knowledge during higher-order unification (HOU). This extended form of HOU is used to compute the licensing conditions for corrections.

This paper introduces a multi-valued variant of higher-order resolution and provesit correct and complete with respect to a natural multi-valued variant of Henkin'sgeneral model semantics. This resolution method is parametric in the number of truthvalues as well as in the particular choice of the set of connectives (given by arbitrarytruth tables) and even substitutional quantifiers. In the course of the completenessproof we establish a model existence theorem for this logical system. The workreported in this paper provides a basis for developing higher-order mechanizationsfor many non-classical logics.

The introduction of sorts to first-order automated deduction has broughtgreater conciseness of representation and a considerable gain in efficiency byreducing the search space. It is therefore promising to treat sorts in higherorder theorem proving as well.In this paper we present a generalization of Huet's Constrained Resolutionto an order-sorted type theory SigmaT with term declarations. This system buildscertain taxonomic axioms into the unification and conducts reasoning withthem in a controlled way. We make this notion precise by giving a relativizationoperator that totally and faithfully encodes SigmaT into simple type theory.

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.

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.

Even though it is not very often admitted, partial functions do play asignificant role in many practical applications of deduction systems. Kleenehas already given a semantic account of partial functions using three-valuedlogic decades ago, but there has not been a satisfactory mechanization. Recentyears have seen a thorough investigation of the framework of many-valuedtruth-functional logics. However, strong Kleene logic, where quantificationis restricted and therefore not truth-functional, does not fit the frameworkdirectly. We solve this problem by applying recent methods from sorted logics.This paper presents a resolution calculus that combines the proper treatmentof partial functions with the efficiency of sorted calculi.

This paper concerns a knowledge structure called method , within a compu-tational model for human oriented deduction. With human oriented theoremproving cast as an interleaving process of planning and verification, the body ofall methods reflects the reasoning repertoire of a reasoning system. While weadopt the general structure of methods introduced by Alan Bundy, we make anessential advancement in that we strictly separate the declarative knowledgefrom the procedural knowledge. This is achieved by postulating some stand-ard types of knowledge we have identified, such as inference rules, assertions,and proof schemata, together with corresponding knowledge interpreters. Ourapproach in effect changes the way deductive knowledge is encoded: A newcompound declarative knowledge structure, the proof schema, takes the placeof complicated procedures for modeling specific proof strategies. This change ofparadigm not only leads to representations easier to understand, it also enablesus modeling the even more important activity of formulating meta-methods,that is, operators that adapt existing methods to suit novel situations. In thispaper, we first introduce briefly the general framework for describing methods.Then we turn to several types of knowledge with their interpreters. Finally,we briefly illustrate some meta-methods.

In this paper we provide a semantical meta-theory that will support the development of higher-order calculi for automated theorem proving like the corresponding methodology has in first-order logic. To reach this goal, we establish classes of models that adequately characterize the existing theorem-proving calculi, that is, so that they are sound and complete to these calculi, and a standard methodology of abstract consistency methods (by providing the necessary model existence theorems) needed to analyze completeness of machine-oriented calculi.

This report presents the main ideas underlyingtheOmegaGamma mkrp-system, an environmentfor the development of mathematical proofs. The motivation for the development ofthis system comes from our extensive experience with traditional first-order theoremprovers and aims to overcome some of their shortcomings. After comparing the benefitsand drawbacks of existing systems, we propose a system architecture that combinesthe positive features of different types of theorem-proving systems, most notably theadvantages of human-oriented systems based on methods (our version of tactics) andthe deductive strength of traditional automated theorem provers.In OmegaGamma mkrp a user first states a problem to be solved in a typed and sorted higher-order language (called POST ) and then applies natural deduction inference rules inorder to prove it. He can also insert a mathematical fact from an integrated data-base into the current partial proof, he can apply a domain-specific problem-solvingmethod, or he can call an integrated automated theorem prover to solve a subprob-lem. The user can also pass the control to a planning component that supports andpartially automates his long-range planning of a proof. Toward the important goal ofuser-friendliness, machine-generated proofs are transformed in several steps into muchshorter, better-structured proofs that are finally translated into natural language.This work was supported by the Deutsche Forschungsgemeinschaft, SFB 314 (D2, D3)

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.

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.