From 09deefbbe6942eaff480389433680bd377d8a454 Mon Sep 17 00:00:00 2001 From: Joe Corneli Date: Sat, 20 May 2017 16:32:21 +0100 Subject: [PATCH] some editing --- org/arxana-redux-refs.bib | 50 +++++++++++++++++++++++++++++++++++++- org/arxana-redux.org | 61 ++++++++++++++++++++++++++++++----------------- 2 files changed, 88 insertions(+), 23 deletions(-) diff --git a/org/arxana-redux-refs.bib b/org/arxana-redux-refs.bib index 7004f8a..826ca1a 100644 --- a/org/arxana-redux-refs.bib +++ b/org/arxana-redux-refs.bib @@ -95,4 +95,52 @@ keywords = {argumentation, mathematics} url={http://cl-informatik.uibk.ac.at/cek/docs/14/ckjujvhg-cicm14-poster.pdf}, note={\url{http://cl-informatik.uibk.ac.at/cek/docs/14/ckjujvhg-cicm14-poster.pdf}}, keywords = {corpus statistics} -} \ No newline at end of file +} + +@inproceedings{sowa2003analogical, + title={Analogical reasoning}, + author={Sowa, John F and Majumdar, Arun K}, + editor={A. Aldo and W. Lex and B. Ganter}, + booktitle={Conceptual Structures for Knowledge Creation and Communication: 11th International Conference on Conceptual Structures, ICCS 2003, Dresden, Germany, July 21-25, 2003, Proceedings}, + pages={16--36}, + series={LNAI}, + number={2746}, + year={2003}, + organization={Springer}, + keywords = {analogical reasoning} +} + +@incollection{sowa2008conceptual, + title={Conceptual Graphs}, + author={Sowa, John F}, + booktitle={Handbook of Knowledge Representation}, + editor={van Harmelen, F. and Lifschitz, V. and Porter, B.}, + publisher={Elsevier}, + year={2008}, + pages={213--237}, + chapter={5}, + keywords = {conceptual graphs}} + +@article{lytinen1992conceptual, + title={Conceptual dependency and its descendants}, + author={Lytinen, Steven L}, + journal={Computers \& Mathematics with Applications}, + volume={23}, + number={2-5}, + pages={51--73}, + year={1992}, + publisher={Elsevier}, + keywords = {conceptual dependency} +} + +@article{schank1972conceptual, + title={Conceptual dependency: A theory of natural language understanding}, + author={Schank, Roger C}, + journal={Cognitive psychology}, + volume={3}, + number={4}, + pages={552--631}, + year={1972}, + publisher={Academic Press}, + keywords = {conceptual dependency, natural language} +} diff --git a/org/arxana-redux.org b/org/arxana-redux.org index 10bb31a..3c15bbc 100644 --- a/org/arxana-redux.org +++ b/org/arxana-redux.org @@ -27,7 +27,8 @@ making a presentation on Arxana and math texts. We plan to get some version of /Arxana/ working, then use it to mark up some mathematical text along the lines of /IAT/ and then come up with a demo which shows what such a /scholiumific representation/ is -good for. +good for. We draw on /CD/ theory to move /IAT/ style representations +into a computable form. ** Glossary of keywords @@ -41,6 +42,7 @@ good for. relate mathematical propositions. These predicates describe inferential structure, judgements of validity or usefulness, and reasoning tactics. + - scholium :: A scholium is a word for a marginal annotation or scholarly, explanatory, remark. Corneli and Krowne proposed a "scholia-based document model" \cite{corneli2005scholia} @@ -53,6 +55,7 @@ good for. familiar class of examples is provided by revision control systems, e.g., Git, which builds a network history of changes to documents in a directed acyclic graph (DAG). + - Arxana :: Arxana is the name given to a series of prototype implementations of the scholium-based document model with an Emacs front-end. Arxana is inspired in part by Ted @@ -63,16 +66,28 @@ good for. given Arxana a unique flavour. In particular, we propose to use a scholium model to model the logic of proofs, not just to represent mathematical texts for reading. -- CD :: Conceptual Dependence theory is a system for representing - knowledge about actions. The basis of this theory is a set of - /primitives/ which describe basic types of actions such as - "PTRANS — transfer of location" or "MBUILD — construction of a - thought or new information". Each primitive comes with a set - of /slots/ which accept objects of certain types. By filling - in the slots of a primitive, one obtains the most basic form - of a /conceptual dependency graph/. By combining such basic - graphs, one can build up more complicated CD graphs which - describe actions in greater detail. + +- CD :: Conceptual Dependence was introduced as a tool for + understanding natural language \cite{schank1972conceptual}. It + is also used to represent knowledge about actions + \cite{lytinen1992conceptual,sowa2008conceptual}. The basis of + the theory is a set of /primitives/ which describe basic types + of actions such as "PTRANS — transfer of location" or "MBUILD + — construction of a thought or new information". Each + primitive comes with a set of /slots/ which accept objects of + certain types. By filling in the slots of a primitive, one + obtains the most elementary form of a /conceptual dependency + graph/. By combining such basic graphs, one can build up more + complicated CD graphs which describe actions in greater + detail. A simple example from \cite{lytinen1992conceptual} + schematizes the sentence "John gave Mary a book" as an + s-expression: + +#+BEGIN_SRC lisp +(atrans actor (person name john) + object (phys-obj type book) + recip (person name mary)) +#+END_SRC ** Statement of the research problem @@ -134,13 +149,14 @@ goals and themes to a mathematical context. *** Registers of mathematical discourse The natural language which occurs in mathematical texts comes in -several different types. Since these types differ in ways such as -vocabulary, type of reasoning, subject matter, and degree of +several different types. Since these types differ in their +vocabulary, manner of reasoning, subject matter, and degree of literality, it simplifies the study of the subject to distinguish these types and examine them individually. -One type of mathematical language is what we may call the /formal -register/. This is the type of language which is used to state +**** The formal register +One type of mathematical language is what we may call the +/formal register/. This is the type of language which is used to state mathematical propositions. For example, "Every natural number can be expressed as the sum of four squares." or "In every triangle, the sum of the angles equals two right angles." In this register, vocabulary @@ -152,13 +168,13 @@ formal equivalents: \begin{align*} &(\forall n \in \mathbb{N}) (\exists a, b, c, d \in \mathbb{N}) n = a^2 + b^2 + c^2 + d^2 \\ - &(\forall \Delta abc \in \mathrm{trinagle}) + &(\forall \Delta abc \in \mathrm{triangle}) \angle abc + \angle bca + \angle cab = 2 * \perp \end{align*} - Indeed, in mathematical writing, such statements in natural language +Indeed, in mathematical writing, such statements in natural language and their formal equivalents both appear and may even be combined within a single sentence, such as "Whenever $(x, y)$ lies inside the -region, we have $x^2 + y^4 < 12$.". +region, we have $x^2 + y^4 < 12$." This register of discourse has been studied by de Bruijn, Ranta, GG and others and there exist parsers which can translate between natural @@ -169,6 +185,7 @@ we shall take this subject as given and allow ourselves to introduce formal restatements of propositions without worrying about how the translation might be accomplished. +**** The expository register Another type of mathematical language is what we will call the /expository register/. This is the sublanguage which is used to express how and why one is interested in a particular formal statement @@ -189,17 +206,17 @@ include: The expository register has several salient features: -The things which it discusses are mathematical objects and +1. The things which it discusses are mathematical objects and mathematical propositions (as opposed, say, to physical objects or psychological states). It is used to convey narratives of how one or more actors navigate their way through mathematical hyperreality. -It is metamathematical, discussing not only with statements but also +2. It is metamathematical, discussing not only with statements but also inference and proofs. For instance, in the first example "it is easy" says something about the proof of a statement. Likewise, it discusses types of proofs and proof strategies. -Whereas the formal register only has deductive reasoning of the +3. Whereas the formal register only has deductive reasoning of the strictest sort, in the expository register we also have inductive and abductive reasoning as well as reasoning by analogy, heuristics, and looser forms of approximate and plausible reasoning. Concomittantly, @@ -211,7 +228,7 @@ choice of actions and strategies. *** Related fields - In order to build a theory of the expository register, we will draw +In order to build a theory of the expository register, we will draw upon several well-studied topics in AI which deal with situations that have common features, namely the block world, board games, and story complehension. -- 2.11.4.GIT