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Integrating TPS and ΩMEGA
 JOURNAL OF UNIVERSAL COMPUTER SCIENCE
, 1999
"... This paper reports on the integration of the higherorder theorem proving environment Tps [Andrews et al., 1996] into the mathematical assistant Ωmega [Benzmuller et al., 1997]. Tps can be called from mega either as a black box or as an interactive system. In black box mode, the user has control ov ..."
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Cited by 7 (4 self)
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This paper reports on the integration of the higherorder theorem proving environment Tps [Andrews et al., 1996] into the mathematical assistant Ωmega [Benzmuller et al., 1997]. Tps can be called from mega either as a black box or as an interactive system. In black box mode, the user has control over the parameters which control proof search in Tps; in interactive mode, all features of the Tpssystem are available to the user. If the subproblem which is passed to Tps contains concepts defined in Ωmega's database of mathematical theories, these definitions are not instantiated but are also passed to Tps. Using a special theory which contains proof tactics that model the NDcalculus variant of Tps within mega, any complete or partial proof generated in Tps can be translated one to one into an mega proof plan. Proof transformation is realised by proof plan expansion in Ωmega's 3dimensional proof data structure, and remains transparent to the user.
Proof Reuse for Deductive Program Verification
"... We present a proof reuse mechanism for deductive program verification calculi. It reuses proofs incrementally (one proof step at a time) and is employs a similarity measure for the points (formulas, terms, programs) where a rule is applied. ..."
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Cited by 7 (4 self)
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We present a proof reuse mechanism for deductive program verification calculi. It reuses proofs incrementally (one proof step at a time) and is employs a similarity measure for the points (formulas, terms, programs) where a rule is applied.
Proof Planning: A Practical Approach To Mechanized Reasoning In Mathematics
, 1998
"... INTRODUCTION The attempt to mechanize mathematical reasoning belongs to the first experiments in artificial intelligence in the 1950 (Newell et al., 1957). However, the idea to automate or to support deduction turned out to be harder than originally expected. This can not at least be seen in the mul ..."
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Cited by 6 (3 self)
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INTRODUCTION The attempt to mechanize mathematical reasoning belongs to the first experiments in artificial intelligence in the 1950 (Newell et al., 1957). However, the idea to automate or to support deduction turned out to be harder than originally expected. This can not at least be seen in the multitude of approaches that were pursued to model different aspects of mathematical reasoning. There are different dimension according to which these systems can be classified: input language (e.g., ordersorted firstorder logic), calculus (e.g., resolution), interaction level (e.g., batch mode), proof output (e.g., refutation graph), and the purpose (e.g., automated theorem proving) as well as many more subtle points concerning the fine tuning of the proof search. In this contribution the proof planning approach will be presented. Since it is not the mainstream approach to mechanized reasoning, it seems to be worth to look at it in a more principled way and to contrast it to other appro
Adaptation of Declaratively Represented Methods in Proof Planning
 SEKI REPORT SR9512, FACHBEREICH INFORMATIK, UNIVERSIT AT DES SAARLANDES
, 1995
"... The reasoning power of humanoriented planbased reasoning systems is primarily derived from their domainspecific problem solving knowledge. Such knowledge is, however, intrinsically incomplete. In order to model the human ability of adapting existing methods to new situations we present in this wo ..."
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Cited by 4 (2 self)
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The reasoning power of humanoriented planbased reasoning systems is primarily derived from their domainspecific problem solving knowledge. Such knowledge is, however, intrinsically incomplete. In order to model the human ability of adapting existing methods to new situations we present in this work a declarative approach for representing methods, which can be adapted by socalled metamethods. Since apparently the success of this approach relies on the existence of general and strong metamethods, we describe several metamethods of general interest in detail by presenting the problem solving process of two familiar classes of mathematical problems. These examples should illustrate our philosophy of proof planning as well: besides planning with the current repertoire of methods, the repertoire of methods evolves with experience in that new ones are created by metamethods which modify existing ones.
ModelGuided Proof Planning
, 2002
"... Proof planning is a form of theorem proving in which the proving procedure is viewed as a planning process. The plan operators in proof planning are called methods. In this paper we propose a strategy for heuristically restricting the set of methods to be applied in proof search. It is based on the ..."
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Cited by 3 (3 self)
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Proof planning is a form of theorem proving in which the proving procedure is viewed as a planning process. The plan operators in proof planning are called methods. In this paper we propose a strategy for heuristically restricting the set of methods to be applied in proof search. It is based on the idea that the plausibility of a method can be estimated by comparing the model class of proof lines newly generated by the method with that of the assumptions and of the theorem. For instance, in forward reasoning when a method produces a new assumption whose model class is not a superset of the model class of the given premises, the method will lead to a situation which is semantically not justified and will not lead to a valid proof in later stages. A semantic restriction strategy is to reduce the search space by excluding methods whose application results in a semantic mismatch. A semantic selection strategy heuristically chooses the method that is likely to make most progress towards filling the gap between the assumptions and the theorem. Each candidate method is evaluated with respect to the subset and superset relation with the given premises. All models considered are taken from a finite reference subset of the full model class. In this contribution we present the modelguided approach as well as first experiments with it.
A multimodi Proof Planner
 UNIVERSITY OF KOBLENZLANDAU
, 1998
"... Proof planning is a novel knowledgebased approach for proof construction, which supports the incorporation of mathematical knowledge and the common mathematical proof techniques of a particular mathematical field. This paradigm is adopted in the\Omega mega proof development system, to provide supp ..."
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Cited by 3 (3 self)
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Proof planning is a novel knowledgebased approach for proof construction, which supports the incorporation of mathematical knowledge and the common mathematical proof techniques of a particular mathematical field. This paradigm is adopted in the\Omega mega proof development system, to provide support for the user. A considerable part of the proof construction and even sometimes the whole work can be undertaken by a proof planner. In the\Omega mega project we are investigating the aspect of computation under bounded resources in mathematical theorem proving. The relevant resources are, in addition to time and memory space, user availability as well as the frequency of user interaction. At this issue, the proof planner of\Omega mega is conceived in such a way that it has a resourceadaptive behaviour. This property of the planner is achieved by a planner modus which defines the planner behaviour depending on which and how many resources are available. In this paper, we describe the...
Proving Ground Completeness of Resolution by Proof Planning
, 1997
"... A lot of the human ability to prove hard mathematical theorems can be ascribed to a problemspecific problem solving knowhow. Such knowledge is intrinsically incomplete. In order to prove related problems human mathematicians, however, can go beyond the acquired knowledge by adapting their knowhow ..."
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Cited by 2 (1 self)
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A lot of the human ability to prove hard mathematical theorems can be ascribed to a problemspecific problem solving knowhow. Such knowledge is intrinsically incomplete. In order to prove related problems human mathematicians, however, can go beyond the acquired knowledge by adapting their knowhow to new related problems. These two aspects, having rich experience and extending it by need, can be simulated in a proof planning framework: the problemspecific reasoning knowledge is represented in form of declarative planning operators, called methods; since these are declarative, they can be mechanically adapted to new situations by socalled metamethods. In this contribution we apply this framework to two prominent proofs in theorem proving, first, we present methods for proving the ground completeness of binary resolution, which essentially correspond to key lemmata, and then, we show how these methods can be reused for the proof of the ground completeness of lock resolution.
Integrating TPS with ΩMEGA
, 1998
"... We report on the integration of Tps as an external reasoning component into the mathematical assistant system Ωmega. Thereby Tps can be used both as an automatic theorem prover for higher order logic as well as interactively employed from within the Ωmega environment. Tps proofs can be directly inc ..."
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Cited by 2 (1 self)
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We report on the integration of Tps as an external reasoning component into the mathematical assistant system Ωmega. Thereby Tps can be used both as an automatic theorem prover for higher order logic as well as interactively employed from within the Ωmega environment. Tps proofs can be directly incorporated into Ωmega on a tactic level enabling their visualization and verbalization. Using an example we show how Tps proofs can be inserted into Ωmega's knowledge base by expanding them to calculus level using both Ωmega's tactic mechanism and the first order theorem prover Otter. Furthermore we demonstrate how the facts from Ωmega's knowledge base can be used to build a Tps library.
Incremental Proof Planning by MetaRules
 Proc. FLAIRS97, Daytona Beach, ISBN
, 1997
"... We propose a new approach to automated tactical theorem proving and proof planning: By using metarules to control the search for a proof, heuristic knowledge is declaratively collected. This eases the user's understanding of the system's search for a proof thus making userinteractions easier. The ..."
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Cited by 1 (0 self)
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We propose a new approach to automated tactical theorem proving and proof planning: By using metarules to control the search for a proof, heuristic knowledge is declaratively collected. This eases the user's understanding of the system's search for a proof thus making userinteractions easier. The system's heuristics can be modified by simply using different sets of metarules. Via metarules contextual preconditions for tactics can be formulated in a transparent way. The metarule interpreter interleaves proof planning, plan execution (tactic application), and reasoning about the tactic's results. In our framework the planner has access to more information, because the metarule interpreter knows about the history of the proof resp. the proof plan by being able to access the (partial) prooftree. 1 Introduction Albeit we focus here on the domain of automated induction theorem proving, our method is not limited to this domain. An automated induction theorem proving system has various...
Planning Equivalence Proofs
 In Denzinger et al. [1998
, 1998
"... Different axiomatizations of mathematical concepts prove to be useful in a mathematical knowledge base, since each axiomatization of a concept is more or less helpful for the task at hand. To keep the knowledge base consistent, the equivalence of distinct definitions for some concept must be formall ..."
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Cited by 1 (1 self)
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Different axiomatizations of mathematical concepts prove to be useful in a mathematical knowledge base, since each axiomatization of a concept is more or less helpful for the task at hand. To keep the knowledge base consistent, the equivalence of distinct definitions for some concept must be formally proven. Especially in algebra, where various axiomatizations of an algebraic structure often occur, the proofs of equivalent definitions are canonical in major simplification steps. Starting out with a proof for the equivalence of two distinct definitions for the group concept, we report in this paper on the first attempts of proof planning the equivalence of different definitions for algebraic structures in the\Omega mega system. We present a proof method that implements the common simplification steps of such theorems and give some relevant methods for closing the subgoals of this method. 1 Introduction For many proofs of mathematical theorems it is often desirable to choose an appropr...