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.
In this report we present a case study of employing goal-oriented heuristics whenproving equational theorems with the (unfailing) Knut-Bendix completion proce-dure. The theorems are taken from the domain of lattice ordered groups. It will bedemonstrated that goal-oriented (heuristic) criteria for selecting the next critical paircan in many cases significantly reduce the search effort and hence increase per-formance of the proving system considerably. The heuristic, goalADoriented criteriaare on the one hand based on so-called "measures" measuring occurrences andnesting of function symbols, and on the other hand based on matching subterms.We also deal with the property of goal-oriented heuristics to be particularly helpfulin certain stages of a proof. This fact can be addressed by using them in a frame-work for distributed (equational) theorem proving, namely the "teamwork-method".
The team work method is a concept for distributing automated theoremprovers and so to activate several experts to work on a given problem. We haveimplemented this for pure equational logic using the unfailing KnuthADBendixcompletion procedure as basic prover. In this paper we present three classes ofexperts working in a goal oriented fashion. In general, goal oriented experts perADform their job "unfair" and so are often unable to solve a given problem alone.However, as a team member in the team work method they perform highly effiADcient, even in comparison with such respected provers as Otter 3.0 or REVEAL,as we demonstrate by examples, some of which can only be proved using teamwork.The reason for these achievements results from the fact that the team workmethod forces the experts to compete for a while and then to cooperate by exADchanging their best results. This allows one to collect "good" intermediate resultsand to forget "useless" ones. Completion based proof methods are frequently reADgarded to have the disadvantage of being not goal oriented. We believe that ourapproach overcomes this disadvantage to a large extend.
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.
In 1978, Klop demonstrated that a rewrite system constructed by adding the untyped lambda calculus, which has the Church-Rosser property, to a Church-Rosser first-order algebraic rewrite system may not be Church-Rosser. In contrast, Breazu-Tannen recently showed that argumenting any Church-Rosser first-order algebraic rewrite system with the simply-typed lambda calculus results in a Church-Rosser rewrite system. In addition, Breazu-Tannen and Gallier have shown that the second-order polymorphic lambda calculus can be added to such rewrite systems without compromising the Church-Rosser property (for terms which can be provably typed). There are other systems for which a Church-Rosser result would be desirable, among them being X^t+SP+FIX, the simply-typed lambda calculus extended with surjective pairing and fixed points. This paper will show that Klop's untyped counterexample can be lifted to a typed system to demonstrate that X^t+SP+FIX is not Church-Rosser.
Over the past thirty years there have been significant achievements in the field of auto-mated theorem proving with respect to the reasoning power of the inference engines.Although some effort has also been spent to facilitate more user friendliness of the de-duction systems, most of them failed to benefit from more recent developments in therelated fields of artificial intelligence (AI), such as natural language generation and usermodeling. In particular, no model is available which accounts both for human deductiveactivities and for human proof presentation. In this thesis, a reconstructive architecture issuggested which substantially abstracts, reorganizes and finally translates machine-foundproofs into natural language. Both the procedures and the intermediate representationsof our architecture find their basis in computational models for informal mathematicalreasoning and for proof presentation. User modeling is not incorporated into the currenttheory, although we plan to do so later.
This paper presents a new way to use planning in automated theorem provingby means of distribution. To overcome the problem that often subtasks fora proof problem can not be detected a priori (which prevents the use of theknown planning and distribution techniques) we use a team of experts that workindependently with different heuristics on the problem. After a certain amount oftime referees judge their results using the impact of the results on the behaviourof the expert and a supervisor combines the selected results to a new startingpoint.This supervisor also selects the experts that can work on the problem inthe next round. This selection is a reactive planning task. We outline whichinformation the supervisor can use to fulfill this task and how this informationis processed to result in a plan or to revise a plan. We also show that the useof planning for the assignment of experts to the team allows the system to solvemany different examples in an acceptable time with the same start configurationand without any consultation of the user.Plans are always subject to changeShin'a'in proverb
We present a method for learning heuristics employed by an automated proverto control its inference machine. The hub of the method is the adaptation of theparameters of a heuristic. Adaptation is accomplished by a genetic algorithm.The necessary guidance during the learning process is provided by a proof prob-lem and a proof of it found in the past. The objective of learning consists infinding a parameter configuration that avoids redundant effort w.r.t. this prob-lem and the particular proof of it. A heuristic learned (adapted) this way canthen be applied profitably when searching for a proof of a similar problem. So,our method can be used to train a proof heuristic for a class of similar problems.A number of experiments (with an automated prover for purely equationallogic) show that adapted heuristics are not only able to speed up enormously thesearch for the proof learned during adaptation. They also reduce redundancies inthe search for proofs of similar theorems. This not only results in finding proofsfaster, but also enables the prover to prove theorems it could not handle before.