Metacognitive Question Answering from Euclid’s Elements Text by
John Kontos and Joseph Armaos
Artificial Intelligence Group
Department of History and Philosophy of Science,
University of Athens
(ikontos2003@yahoo.com, iarmaos@otenet.gr )
Extended Abstract
The present paper presents a system for meta-cognitive question answering from texts as applied to the Euclid’s “Elements” text. The system can answer “why” questions from geometric proof texts by representing and processing its own state of automatic proof analysis and generating explanations and supporting diagrams. The system is implemented in Prolog and uses a question grammar combined with a text grammar. These two grammars use rules, lexicons, ontologies and the state of the system to answer the input questions. The semantics of the question grammar are expressed in terms of combinations of predicates that analyze chains of text chunks recognized by the text grammar. Inference of implicit facts is performed directly from the text. The system facilitates adaptation to the language of different users.
The question answering system presented is based on an approach that aims at the creation of systems which accept natural language questions and generate answers with information and meta-information extracted from a text and a record of its own state either directly or after applying deductive inference. The record of the state of the system contains information concerning the progress of the deductive inference that is used for generating automatically explanations of the answers given to the input questions.
The deductive inference from texts by computer is traditionally performed in two stages. In the first stage the text is translated by computer or by hand into some formal representation. In the second stage reasoning is performed with this formal representation. On the contrary we avoid the translation step and deduction is performed directly from the texts following the ARISTA method of text analysis. This method was first proposed in [1] and further elaborated in [2,3,4 and 5].
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In traditional methods of syntactic analysis one syntactic rule must be written for every particular sequence of phrasal constituents. If we apply such a method for the syntactic analysis of Greek a plethora of syntactic rules will be needed for the same constituent set due to the word order freedom of this language. On the contrary the method followed in the present system allows the statement of a single syntactic rule for the parsing of two or more equivalent syntactic structures that differ only in the relative position of the words involved.
Our parsing method consists of the automatic translation of all the words of every sentence into a number of logical facts written in Prolog for further processing.
These facts take the form of a logical predicate with three arguments for each word of a sentence. The first argument specifies the number of the sentence that contains a given word, the second argument specifies the position of the word in the sentence and the third specifies the word itself.
The sentences found in the corpus of geometric proofs in Euclid’s elements consist of either a single predicative statement or a pair of statements in the hypothesis
– apodosis relation. The recognition and analysis of logical chains of such sentences is used to answer “why” questions with the extraction of meta-information that supports the meta-cognitive character of the question answering process.
An illustrative example from the first Proposition of the Elements is given below together with the explanatory answers generated by the system for a few illustrative questions. The first Proposition may be stated in English: “On a given finite straight line to construct an equilateral triangle”. The equilateral triangle abc is created by the point of intersection of two equal circles and their centers a and b where these centers lie on each others circumference and the points a and b are the two endpoints of the given finite straight line. Three illustrative questions are answered by the system by analyzing automatically the corresponding Euclidean proof as follows:
Question 1: “why is side ac equal to side ab?”
Answer
1: “because they are radii of the same circle 1”
Question 2: “why is side bc equal to side ba?”
Answer 2: “because they are radii of the same circle 2”
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Question
3: “why is side ca equal to side cb?”
Answer 3: “because each of ca and cb is equal to ab”
Note that Answer 3 is generated by using the common notion “things which are equal to the same thing are also equal t o one another” following the metacognitive decision of the system that no statement in the proof can be used to complete the answer to the question. In conclusion our system can be applied either to the testing of mathematical reasoning models [6] or to metacognitive tutoring [7].
References
[1] Kontos, J. (1992) “ARISTA: Knowledge Engineering with Scientific Texts”.
Information and Software Technology, Vol. 34, No 9, 611-616.
[2] Kontos J. and Malagardi I. (1999). “Information Extraction and Knowledge
Acquisition from Texts using Bilingual QuestionAnswering”. Journal of Intelligent and Robotic Systems, Vol 26, No. 2, pp. 103-122, October.
[3] Kontos J., et al (2002)”. “ARISTA Causal Knowledge Discovery from Texts”.
Proceedings of the 5th International Conference on Discovery Science DS 2002,
Luebeck, Germany, pp. 348-355.
[4] Kontos, J., et al (2003). “The AROMA System for Intelligent Text Mining”.
HERMIS, International Journal of Computers mathematics and its Applications, Vol.
4, pp.163-173, LEA.
[5] Kontos, J., et al (2005). “Question Answering and Rhetoric Analysis of Biomedical
Texts in the AROMA System”.
Proceedings of 7 th Hellenic European Research on
Computer Mathematics & its Applications Conference. Athens. On-line publication: http://www.aueb.gr/pympe/hercma/proceedings2005.
[6] Shapiro S. et al (In Press). ”Metacognition in SnePS”. AI Magazine.
[7] Aleven, V. et al (2006). ”Toward Meta-cognitive Tutoring: A Method of Help-
Seeking with a Cognitive Tutor”. International Journal of Artificial Intelligence in
Education, Vol 16, pp. 101-130, IOS Press.
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