## SEKI Report

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#### Dokumenttyp

- Preprint (85)
- Wissenschaftlicher Artikel (6)
- Bericht (1)

#### Schlagworte

- Knowledge acquisition (3)
- resolution (3)
- Case-Based Reasoning (2)
- Deduction (2)
- HOT (2)
- MOLTKE-Projekt (2)
- Term rewriting systems (2)
- Wissensakquisition (2)
- analogy (2)
- combined systems with sha (2)

93,11

This report contains a collection of abstracts for talks given at the "Deduktionstreffen" held at Kaiserslautern, October 6 to 8, 1993. The topics of the talks range from theoretical aspects of term rewriting systems and higher order resolution to descriptions of practical proof systems in various applications. They are grouped together according the following classification: Distribution and Combination of Theorem Provers, Termination, Completion, Functional Programs, Inductive Theorem Proving, Automatic Theorem Proving, Proof Presentation. The Deduktionstreffen is the annual meeting of the Fachgruppe Deduktionssysteme in the Gesellschaft für Informatik (GI), the German association for computer science.

93,2

93,3

We study deterministic conditional rewrite systems, i.e. conditional rewrite systemswhere the extra variables are not totally free but 'input bounded'. If such a systemR is quasi-reductive then !R is decidable and terminating. We develop a critical paircriterion to prove confluence if R is quasi-reductive and strongly deterministic. In thiscase we prove that R is logical, i.e./!R==R holds. We apply our results to proveHorn clause programs to be uniquely terminating.This research was supported by the Deutsche Forschungsgemeinschaft, SFB 314, Project D4

93,9

We investigate restricted termination and confluence properties of term rewritADing systems, in particular weak termination and innermost termination, and theirinterrelation. New criteria are provided which are sufficient for the equivalenceof innermost / weak termination and uniform termination of term rewriting sysADtems. These criteria provide interesting possibilities to infer completeness, i.e.termination plus confluence, from restricted termination and confluence properADties.Using these basic results we are also able to prove some new results aboutmodular termination of rewriting. In particular, we show that termination ismodular for some classes of innermost terminating and locally confluent termrewriting systems, namely for nonADoverlapping and even for overlay systems. Asan easy consequence this latter result also entails a simplified proof of the factthat completeness is a decomposable property of soADcalled constructor systems.Furthermore we show how to obtain similar results for even more general cases of(nonADdisjoint) combined systems with shared constructors and of certain hierarADchical combinations of systems with constructors. Interestingly, these modularityresults are obtained by means of a proof technique which itself constitutes a modADular approach.

93,18

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.

94,5

Automatic proof systems are becoming more and more powerful.However, the proofs generated by these systems are not met withwide acceptance, because they are presented in a way inappropriatefor human understanding.In this paper we pursue two different, but related, aims. First wedescribe methods to structure and transform equational proofs in away that they conform to human reading conventions. We developalgorithms to impose a hierarchical structure on proof protocols fromcompletion based proof systems and to generate equational chainsfrom them.Our second aim is to demonstrate the difficulties of obtaining suchprotocols from distributed proof systems and to present our solutionto these problems for provers using the TEAMWORK method. Wealso show that proof systems using this method can give considerablehelp in structuring the proof listing in a way analogous to humanbehaviour.In addition to theoretical results we also include descriptions onalgorithms, implementation notes, examples and data on a variety ofexamples.

95,8

The well-known and powerful proof principle by well-founded induction says that for verifying \(\forall x : P (x)\) for some property \(P\) it suffices to show \(\forall x : [[\forall y < x :P (y)] \Rightarrow P (x)] \) , provided \(<\) is a well-founded partial ordering on the domainof interest. Here we investigate a more general formulation of this proof principlewhich allows for a kind of parameterized partial orderings \(<_x\) which naturallyarises in some cases. More precisely, we develop conditions under which theparameterized proof principle \(\forall x : [[\forall y <_x x : P (y)] \Rightarrow P (x)]\) is sound in thesense that \(\forall x : [[\forall y <_x x : P (y)] \Rightarrow P (x)] \Rightarrow \forall x : P (x)\) holds, and givecounterexamples demonstrating that these conditions are indeed essential.

95,9

We study the combination of the following already known ideas for showing confluence ofunconditional or conditional term rewriting systems into practically more useful confluence criteria forconditional systems: Our syntactic separation into constructor and non-constructor symbols, Huet's intro-duction and Toyama's generalization of parallel closedness for non-noetherian unconditional systems, theuse of shallow confluence for proving confluence of noetherian and non-noetherian conditional systems, theidea that certain kinds of limited confluence can be assumed for checking the fulfilledness or infeasibilityof the conditions of conditional critical pairs, and the idea that (when termination is given) only primesuperpositions have to be considered and certain normalization restrictions can be applied for the sub-stitutions fulfilling the conditions of conditional critical pairs. Besides combining and improving alreadyknown methods, we present the following new ideas and results: We strengthen the criterion for overlayjoinable noetherian systems, and, by using the expressiveness of our syntactic separation into constructorand non-constructor symbols, we are able to present criteria for level confluence that are not criteria forshallow confluence actually and also able to weaken the severe requirement of normality (stiffened withleft-linearity) in the criteria for shallow confluence of noetherian and non-noetherian conditional systems tothe easily satisfied requirement of quasi-normality. Finally, the whole paper also gives a practically usefuloverview of the syntactic means for showing confluence of conditional term rewriting systems.

95,10

Problems stemming from the study of logic calculi in connection with an infer-ence rule called "condensed detachment" are widely acknowledged as prominenttest sets for automated deduction systems and their search guiding heuristics. Itis in the light of these problems that we demonstrate the power of heuristics thatmake use of past proof experience with numerous experiments.We present two such heuristics. The first heuristic attempts to re-enact aproof of a proof problem found in the past in a flexible way in order to find a proofof a similar problem. The second heuristic employs "features" in connection withpast proof experience to prune the search space. Both these heuristics not onlyallow for substantial speed-ups, but also make it possible to prove problems thatwere out of reach when using so-called basic heuristics. Moreover, a combinationof these two heuristics can further increase performance.We compare our results with the results the creators of Otter obtained withthis renowned theorem prover and this way substantiate our achievements.