E. Melis, `AI-techniques in proof planning', in European Conference on Artificial Intelligence, pp. 494--498, (1998).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....other problem domains. 4 Limit theorems Developers of automated reasoning systems have been interested in proving automatically some basic limit theorems. Some promising advances have been made in this area (but without the use of analogy) by Bledsoe and his group [Bledsoe et al., 1972] and by Melis [Melis, 1998]. Here, we use LIM as a source theorem, and LIM Theta as a target theorem, and then try to use Melis and Whittle s analogy mechanism in order to prove LIM Theta given the proof of LIM . First, let us define a limit [Bledsoe et al., 1972] 8a : real; 8l : real: lim x a f(x) l 8ffl : real: ....

Melis, E. (1998). AI-techniques in proof planning. In 13th European Conference on Artificial Intelligence, pages 494-- 498. Wiley & Sons.

....additional parameters have profound consequences for natural Gentzen formats and other central tools of current Proof Theory. More realistic formats may also lead to new logics. For higher useful proof structures, cf. the more intelligent theorem proving in Melis 1997, Siekmann, Kohlhase Melis 1998. 7 For a survey, cf. Moortgat and Buszkowski s chapters in the Handbook of Logic and Language, and for further logical theory, van Benthem 1991. 6 Noam Chomsky, and logic might profit from a similar set up, too. This line may not be all that far fetched. A form of parametrization occurs in ....

E. Melis, 1998, 'AI-Techniques in Proof Planning', European Conference on Artificial Intelligence (ECAI-98), Kluwer, Brighton, 494-498.

....are consistent, the detection of inconsistencies for the restriction of the search space, the computation of an answer constraint formula that expresses the collected constraints, and finally the delivery of witnesses for the problem variables. As a fundamental extension of our description in [8] this paper presents the reasons and the realization of the extensions of o# the shelf constraint solvers that are necessary for its use in proof planning. We shall explain these extensions in general terms and with respect to their implementation in the concurrent constraint language Oz [17] In ....

E. Melis, `AI-techniques in proof planning', in European Conference on Artificial Intelligence, pp. 494--498, (1998).

....x and to backtrack in search, if no proof can be found with the chosen witness. This approach yields unmanageable search spaces. We have chosen the approach to introduce M x as a place holder for the term t and to search for the instantiation of M x when all constraints on t are known only. Melis [10] motivates the use of domain speci c constraint solvers to nd witnesses for existentially quanti ed variables. The key idea is to delay the instantiations as long as possible and let the constraint solver incrementally restrict the admissible object values. 1.2 The Integration Constraint ....

E. Melis. AI-Techniques in Proof Planning. In European Conference on Articial Intelligence. Kluwer Academic, 1998.

....jg(x2) Gamma L2 j ffl 2 ) 8ffl(0 ffl 9ffi(0 ffi 8x(jx Gamma aj 0 jx Gamma aj ffi j(f(x) g(x) Gamma (L1 L2)j ffl) The construction of a real number ffi yields a solution that may depend on other real numbers ffl; ffi 1 and ffi 2 . Now knowledge based proof planning [7] is an appropriate framework to integrate external reasoning systems, in particular constraint solvers, that can help in those constructions. We have [7] motivated the use of constraint solving for restricting the search for instantiations of existentially quantified variables and briefly ....

....of a real number ffi yields a solution that may depend on other real numbers ffl; ffi 1 and ffi 2 . Now knowledge based proof planning [7] is an appropriate framework to integrate external reasoning systems, in particular constraint solvers, that can help in those constructions. We have [7] motivated the use of constraint solving for restricting the search for instantiations of existentially quantified variables and briefly discussed the integration of a constraint solver into proof planning. The integration is schematically shown in Figure 1, where the functions tell and ask in ....

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E. Melis. AI-techniques in proof planning. In European Conference on Artificial Intelligence, pages 494--498, Brighton, 1998. Kluwer.

....to the output streams of mega s processes. In the following subsection we shall present the details of this visualization and the motivation underlying its design. 3.1. Hierarchical Proof Plan Data Structure Finding a proof with mega can be viewed as a process that interleaves proof planning [Mel98, CS98] plan execution, and veri cation, all of which is datadriven by the so called Proof Plan Data Structure (PDS) This hierarchical data structure represents a (possibly partial) proof at different levels of abstraction, called proof plans . Its nodes correspond to steps of the derivation ....

E. Melis. AI-techniques in proof planning. In European Conference on Articial Intelligence (ECAI-98), pages 494-498, Brighton, 1998. Kluwer.

....at the level of methods most of which are more abstract than the logical inference steps of ATPs. Even more important for restricting the search space is the meta level reasoning that guides the selection of methods. Currently, this meta level reasoning is encoded by a list of control rules [13] or by variations of the difference reduction search heuristic called rippling [7] However, when the class of theorems to be proved with the same methods and control rules grows, the search heuristics might prove to be inappropriate for new theorems. There are at least two reasons for this ....

E. Melis. AI-techniques in proof planning. In Proc. of European Conference on Artificial Intelligence, pages 494--498. Kluwer, 1998.

....streams of Omega mega s processes. In the following subsection we shall present the details of this visualization and the motivation underlying its design. 3.1. Hierarchical Proof Plan Data Structure Finding a proof with Omega mega can be viewed as a process that interleaves proof planning [Mel98, CS98] plan execution, and verification, all of which is datadriven by the so called Proof Plan Data Structure (PDS) This hierarchical data structure represents a (possibly partial) proof at different levels of abstraction, called proof plans. Its nodes correspond to steps of the derivation ....

E. Melis. AI-techniques in proof planning. In European Conference on Artificial Intelligence (ECAI-98), pages 494--498, Brighton, 1998. Kluwer.

....to the output streams of mega s processes. In the following subsection we shall present the details of this visualization and the motivation underlying its design. 3.1. Hierarchical Proof Plan Data Structure Finding a proof with mega can be viewed as a process that interleaves proof planning [Mel98, CS98] plan execution, and veri cation, all of which is datadriven by the so called Proof Plan Data Structure (PDS) This hierarchical data structure represents a (partial) proof at di erent levels of abstraction (called proof plans) Its nodes are justi ed by methods. Conceptually, each justi ....

E. Melis. AI-techniques in proof planning. In European Conference on Articial Intelligence (ECAI-98), pages 494-498, Brighton, 1998. Kluwer.

....variables, and more abstract proof presentation were not investigated. In order to make progress I introduced knowledge based proof planning in which a general purpose proof planner makes use of mathematical domain knowledge including methods, control rules, and domain specific external reasoners [18]. This common principle of AI was not generally accepted in traditional theorem proving but has proven very fruitful when it comes to realistic maths. Furthermore, I introduced explicit and modular mathematical search control knowledge into theorem proving that allows for meta level reasoning ....

E. Melis. AI-techniques in proof planning. In European Conference on Artificial Intelligence, pages 494--498, Brighton, 1998. Kluwer.

....theorems, to understand the proofs produced (semi )automatically, and to improve their capabilities, its proof planner should produce a output comparable to an expert teacher s proof presentation. 3 Proof Planning Proof planning is a relatively new paradigm in automated theorem proving [3, 11]. As opposed to traditional automated theorem proving which searches for a proof at the level of a logic calculus, proof planning automatically constructs a plan by searching at a higher level of abstraction and by using explicit control knowledge. A resulting plan represents a proof at the more ....

E. Melis. AI-techniques in proof planning. In European Conference on Artificial Intelligence (ECAI-98), pages 494--498, Brighton, 1998. Kluwer.

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E. Melis. AI-techniques in proof planning. In European Conference on Artificial Intelligence, pages 494--498, Brighton, 1998. Kluwer.

....der Dom ane vorhanden ist. Ein wesentlicher Teil der Arbeit im Omega mega Projekt war daher die Entwicklung von geeigneten Repr asentationsmechanismen f ur das faktische, das methodische und das Steuerungswissen, das aus bekannten mathematischen Beweisen extrahiert wurde (Cheikhrouhou, 1997; Melis, 1998). 4.1 Techniken zur Beherrschung des Suchraumes Hierarchisches Planen in Omega mega kann entweder durch Expansion von abstrakten Methoden oder durch Verschieben weniger wichtiger Teilziele realisiert werden. Das hierarchische Planen kann jederzeit unterbrochen werden und liefert, je nach ....

Melis, E. (1998). AI-techniques in proof planning. In European Conference on Artificail Intelligence, 1998.

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