P. Deussen. Halbgruppen und Automaten, volume 99 of Heidelberger Taschenb ucher. Springer, 1971.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....employing an extended symbol map for transferring single calculus level proof steps of the source proof 3 . Empirical investigations [12] however, have provided evidence that this idea does not appropriately cover the analogies drawn by mathematicians, not even all the analogies in the textbook [5]. In [14] we developed a new approach to theorem proving by analogy and explained why the plan level is an appropriate one for the analogical transfer. As shown in [11] this method can cope with the analogies in the mentioned textbook. In order to prove this approach to be successful, we now show ....

....L3 , L4 , L5 , L7 , L9 , L11 , L17 , Nit2, CIA , IA , and g 0 5a which are left to be proved in the target. 4 Conclusion and Related Work The presented analogy method was developed independently from the HeineBorel example by learning lessons from the analogy examples in the standard textbook [5]. Solving the Heine Borel analogy shows that substantial progress has been made by the analogy driven proof plan construction because it is able to cope with complicated real maths examples. Our approach advances ideas of previous systems for theorem proving by analogy and of derivational analogy ....

P. Deussen. Halbgruppen und Automaten, volume 99 of Heidelberger Taschenbucher. Springer, 1971.

....Introduction iii How to Read the Data Base iv Outlook and Open Problems viii References ix Alphabetic Index x Consecutive Index xiv ii Introduction In this paper we represent the mathematical factual knowledge of the first third of the textbook Halbgruppen und Automaten of P. Deussen [Deu71] As mathematical textbooks in general the book is not self contained, the additional knowledge concerns above all the natural numbers. We have formalized this knowledge following the first section of E. Landau s Grundlagen der Analysis [Lan30] We present the knowledge in the form of a ....

Peter Deussen. Halbgruppen und Automaten, volume 99 of Heidelberger Taschenbucher. Springer Verlag, Berlin, Germany, 1971.

....aspect is the proof of the theorem thm from the assumptions ass. To obtain an empirical test set and in order to gain practical experience with analogical reasoning in mathematical theorem proving we have studied the textbook Halbgruppen und Automaten (abbreviated as HUA in the following) [1], since it is particularly rich in proofs that are explicitly stated as analogous to previous proofs by the author. Furthermore the book already served as a test case for automated theorem proving for the Margraf Karl Refutation Procedure [2] This empirical study is the basis for our own approach ....

....mapping of the base case to the target case for a given representation fails in many cases: There is no such symbol map unless the actually given representation is reformulated such that the analogy becomes visible. Why Halbgruppen und Automaten The textbook Halbgruppen und Automaten [1] was chosen for this case study since it consists of the three chapters each of which is built upon the previous one, partially by analogies. This is the reason, why this particular textbook is so rich in explicit proofs by analogy and actually very much liked by students because of its uniform ....

P. Deussen. Halbgruppen und Automaten, volume 99 of Heidelberger Taschenbucher. Springer, 1971.

....employing an extended symbol map for transferring single proof steps of the source to the target. 20 Empirical investigations [94, 96] however, provided evidence that this idea does not appropriately cover many analogies drawn by mathematicians, not even all the analogies of a standard textbook [32]. The empirical studies have shown that: ffl Methods, rather than just single calculus steps, are transferred analogically in mathematical theorem proving by analogy. ffl For many of these analogies, symbol mapping is only insufficient. A change of a problem representation by, e.g. unfolding a ....

P. Deussen. Halbgruppen und Automaten, volume 99 of Heidelberger Taschenbucher. Springer, 1971.

....each single proof step from the source proof to a proof step in the target. Empirical investigations [Mel93a] however, have provided evidence that this idea does not appropriately cover the analogies drawn by mathematicians, not even all the analogies of a standard university level textbook [Deu71] The failure of step by step transfers at the calculus level has been experienced more widely. For instance, transformational tactics in Nuprl failed to succeed because formal proofs are often too brittle to apply such a transformation. 5 This evidence motivated the search for a more ....

P. Deussen. Halbgruppen und Automaten, volume 99 of Heidelberger Taschenbucher. Springer, 1971.

....integrate several planning strategies and about some control knowledge to guide the choice of planning strategies. Note that it is not essential to understand the mathematics behind the example. 3.1 Example Suppose we plan a proof of theorem 7.5. 7 from Deussen s book Halbgruppen und Automaten [9] that states that the mapping Phi from an F semimodul T 1 Planning Algorithm PLAN( Returns refinements of Parameters: sol procedure for picking solution candidates 1. Termination Check: If sol( returns a solution, return it and terminate. If it returns fail, fail. Otherwise continue. ....

P. Deussen. Halbgruppen und Automaten, volume 99 of Heidelberger Taschenbucher. Springer, 1971.

....daily work. Unlike existing proof checkers, however, we intend to incorporate strong automated deductive tools. The requirements for the Omega Gamma mkrp system have been derived from our experience in proving an interrelated collection of theorems of a typical textbook on semigroups and automata [12] with the first order theorem prover mkrp [36] The main conclusion we have drawn from these experiments is that although current automated theorem provers have obviously reached the power to solve non trivial problems, they do not provide sufficient assistance for systematically proving ....

....when proving theorems. They are, rather, analogous to the tasks of a calculator or of a computer algebra system. 2. 2 Experiences with mkrp Throughout the development of the mkrp system, testing was carried out on numerous theorems of a mathematical textbook on semigroups and automata [12]. During a time span of two years, about one third of the text book was fully encoded and finally proved by the mkrp system. As the shortcomings of the system became more and more apparent, however, our ultimate goal of proving the entire book in a systematic way was then abandoned. This ....

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Peter Deussen. Halbgruppen und Automaten, volume 99 of Heidelberger Taschenbucher. Springer Verlag, Berlin, Germany, 1971.

....for analogy driven proof plan construction. The next section presents a set of meta methods that were actually used for the necessary reformulations in the analogy driven construction of proofs in an empirical case study [23] Some examples from a mathematical textbook Halbgruppen und Automaten [13] (abbreviated as HUA) and an example of Terry Boy de la Tour and Christoph Kreitz are given in section five. More proofs by analogy are investigated and presented in detail in the case study [23] 2 Notions and Notation We assume the reader to be familiar with plan based problem solving (see ....

....attempts to prove Theorem2 by analogy, and it is interesting to note that this example can be handled in exactly the same way as for example the proof of theorem 17.6 in HUA. 4 Meta methods The following selection of meta methods is the result of an empirical study of the mathematical textbook [13], where we isolated all those proofs that are explicitly mentioned as proofs by analogy. These examples are presented in a case study [23] that contains the Natural Deduction proofs. The following set of meta methods is sufficient for an automated construction of these proofs by analogy, however, ....

P. Deussen. Halbgruppen und Automaten, volume 99 of Heidelberger Taschenbucher. Springer, 1971.

....and employing an extended symbol map for transferring single proof steps of the source to the target 1 . Empirical investigations, however, provided evidence that this idea does not appropriately cover many analogies drawn by mathematicians, not even all the analogies of a standard textbook [ Deussen, 1971 ] we used for an empirical study [ Melis, 1993a ] I empirically analyzed mathematical theorem proving [ Melis, 1994a ] and found that: 1 Though Bledsoe transfers larger steps calling an automated prover but again uses symbol mapping. ffl Theorem proving by analogy is embedded into proof ....

....structure but may change methods and sequents as well as justifications. An example is Add Arguments (See [ Huang et al. 1994b; Melis, 1995 ] 5. Compositions of the reformulations above. The sequence of reformulations that were needed to prove theorem 5. 7 from the examined textbook [ Deussen, 1971 ] by analogy is shown in Figure 1. Here the source proof plan consists of the goal: 8f; x(f 2 F x 2 T 1 Phi(f x) f Phi(x) 6 , assumptions, and one method M S only. The target goal ; 8x; y(x; y 2 H 1 Phi(x Delta y) Phi(x) Delta Phi(y) is theorem 5.7. The ....

P. Deussen. Halbgruppen und Automaten, volume 99 of Heidelberger Taschenbucher. Springer, 1971.

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P. Deussen. Halbgruppen und Automaten, volume 99 of Heidelberger Taschenb ucher. Springer, 1971.

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