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Flexible encoding of mathematics on the computer
 In MKM 2004, volume 3119 of LNCS
, 2004
"... Abstract. This paper reports on refinements and extensions to the MathLang framework that add substantial support for natural language text. We show how the extended framework supports multiple views of mathematical texts, including natural language views using the exact text that the mathematician ..."
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Abstract. This paper reports on refinements and extensions to the MathLang framework that add substantial support for natural language text. We show how the extended framework supports multiple views of mathematical texts, including natural language views using the exact text that the mathematician wants to use. Thus, MathLang now supports the ability to capture the essential mathematical structure of mathematics written using natural language text. We show examples of how arbitrary mathematical text can be encoded in MathLang without needing to change any of the words or symbols of the texts or their order. In particular, we show the encoding of a theorem and its proof that has been used by Wiedijk for comparing many theorem prover representations of mathematics, namely the irrationality of √ 2 (originally due to Pythagoras). We encode a 1960 version by Hardy and Wright, and a more recent version by Barendregt. 1 On the way to a mathematical vernacular for computers Mathematicians now use computer software for a variety of tasks: typing mathematical texts, performing calculation, analyzing theories, verifying proofs. Software tools like
Toward an objectoriented structure for mathematical text
 MATHEMATICAL KNOWLEDGE MANAGEMENT, 4TH INT’L CONF., PROCEEDINGS. VOLUME 3863 OF LECTURE NOTES IN ARTIFICIAL INTELLIGENCE
, 2006
"... Computerizing mathematical texts to allow software access to some or all of the texts ’ semantic content is a long and tedious process that currently requires much expertise. We believe it is useful to support computerization that adds some structural and semantic information, but does not require j ..."
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Computerizing mathematical texts to allow software access to some or all of the texts ’ semantic content is a long and tedious process that currently requires much expertise. We believe it is useful to support computerization that adds some structural and semantic information, but does not require jumping directly from the wordprocessing level (e.g., L ATEX) to full formalization (e.g., Mizar, Coq, etc.). Although some existing mathematical languages are aimed at this middle ground (e.g., MathML, OpenMath, OMDoc), we believe they miss features needed to capture some important aspects of mathematical texts, especially the portion written with natural language. For this reason, we have been developing MathLang, a language for representing mathematical texts that has weak type checking and support for the special mathematical use of natural language. MathLang is currently aimed at only capturing the essential grammatical and binding structure of mathematical text without requiring full formalization. The development of MathLang is directly driven by experience encoding real mathematical texts. Based on this experience, this paper presents the changes that yield our latest version of MathLang. We have restructured and simplified the core of the language, replaced our old notion of “context” by a new system of blocks and local scoping, and made other changes. Furthermore, we have enhanced our support for the mathematical use of nouns and adjectives with objectoriented features so that nouns now correspond to classes, and adjectives to mixins.
What is a structural representation
, 2001
"... We outline a formal foundation for a \structural " (or \symbolic") object/event representation, the necessity of which is acutely felt in all sciences, including mathematics and computer science. The proposed foundation incorporates two hypotheses: 1) the object's formative hi ..."
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We outline a formal foundation for a \structural &quot; (or \symbolic&quot;) object/event representation, the necessity of which is acutely felt in all sciences, including mathematics and computer science. The proposed foundation incorporates two hypotheses: 1) the object's formative history must be an integral part of the object representation and 2) the process of object construction is irreversible, i.e. the \trajectory &quot; of the object's formative evolution does not intersect itself. The last hypothesis is equivalent to the generalized axiom of (structural) induction. Some of the main diculties associated with the transition from the classical numeric to the structural representations appear to be related precisely to the development of a formal framework satisfying these two hypotheses. The concept of (inductive) class representationwhich has inspired the development of this approach to structural representationdiers fundamentally from the known concepts of class. In the proposed, evolving transformations system (ETS), model, the class is dened by the transformation systema nite set of weighted transformations acting on the class progenitor and the generation of the class elements is associated with the corresponding generative process which also induces the class typicality measure. Moreover, in the ETS model, a fundamental role of the object's class in the object's representation is claried: the representation of an object must include the class. From the point of view of ETS model, the classical discrete representations, e.g. strings and graphs, appear now as incomplete special cases, the proper completion of which should incorporate the corresponding formative histories, i.e. those of the corresponding strings or graphs. 1 Concepts which have proved useful for ordinary things easily assume so great an authority over us, that we forget their terrestrial origin and accept them as unalterable facts. They then become labeled as \conceptual necessities&quot;, a priori situations, etc. The road of scientic progress is frequently blocked for long periods by such errors.
What Is a Structural Measurement Process?
, 2001
"... Numbers have emerged historically as by far the most popular form of representation. All our basic scientific paradigms are built on the foundation of these, numeric, or quantitative, concepts. Measurement, as conventionally understood, is the corresponding process for (numeric) representation of ..."
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Numbers have emerged historically as by far the most popular form of representation. All our basic scientific paradigms are built on the foundation of these, numeric, or quantitative, concepts. Measurement, as conventionally understood, is the corresponding process for (numeric) representation of objects or events, i.e., it is a procedure or device that realizes the mapping from the set of objects to the set of numbers. Any (including a future) measurement device is constructed based on the underlying mathematical structure that is thought appropriate for the purpose. It has gradually become clear to us that the classical numeric mathematical structures, and hence the corresponding (including all present) measurement devices, impose on "real" events/objects a very rigid form of representation, which cannot be modified dynamically in order to capture their combinative, or compositional, structure. To remove this fundamental limitation, a new mathematical structureevolving transformation system (ETS)was proposed earlier. This mathematical structure specifies a radically new form of object representation that, in particular, allows one to capture (inductively) the compositional, or combinative, structure of objects or events. Thus, since the new structure also captures the concept of number, it o#ers one the possibility of capturing simultaneously both the qualitative (compositional) and the quantitative structure of events. In a broader scientific context, we briefly discuss the concept of a fundamentally new, biologically inspired, "measurement process", the inductive measurement process, based on the ETS model. In simple terms, all existing measurement processes "produce" numbers as their outputs, while we are proposing a measurement process whos...
Constructing the real numbers in HOL
, 1992
"... This paper describes a construction of the real numbers in the HOL theoremprover by strictly definitional means using a version of Dedekind's method. It also outlines the theory of mathematical analysis that has been built on top of it and discusses current and potential applications in verifi ..."
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This paper describes a construction of the real numbers in the HOL theoremprover by strictly definitional means using a version of Dedekind's method. It also outlines the theory of mathematical analysis that has been built on top of it and discusses current and potential applications in verification and computer algebra. Keywords: Mathematical Logic; Deduction and Theorem Proving 1 The real numbers For some mathematical tasks, the natural numbers N = f0; 1; 2; : : :g are sufficient. However for many purposes it is convenient to use a more extensive system, such as the integers (Z) or the rational (Q ), real (R) or complex (C ) numbers. In particular the real numbers are normally used for the measurement of physical quantities which (at least in abstract models) are continuously variable, and are therefore ubiquitous in scientific applications. 1.1 Properties of the real numbers We can characterize the reals as the unique `complete ordered field'. More precisely, the reals are a set ...
Writing PVS proof strategies
 Design and Application of Strategies/Tactics in Higher Order Logics (STRATA 2003), number CP2003212448 in NASA Conference Publication
, 2003
"... Abstract. PVS (Prototype Verification System) is a comprehensive framework for writing formal logical specifications and constructing proofs. An interactive proof checker is a key component of PVS. The capabilities of this proof checker can be extended by defining proof strategies that are similar t ..."
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Abstract. PVS (Prototype Verification System) is a comprehensive framework for writing formal logical specifications and constructing proofs. An interactive proof checker is a key component of PVS. The capabilities of this proof checker can be extended by defining proof strategies that are similar to LCFstyle tactics. Commonly used proof strategies include those for discharging typechecking proof obligations, simplification and rewriting using decision procedures, and various forms of induction. We describe the basic building blocks of PVS proof strategies and provide a pragmatic guide for writing sophisticated strategies. 1
On the Formalization of the Evolving Transformation System Model
, 2004
"... The central concept of the Evolving Transformation System (ETS) model is structural object representation constructed by the process of inductive inference. The model was proposed in 1990 by Lev Goldfarb to be applied to any pattern learning or classification problem. A formal exposition of the mod ..."
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The central concept of the Evolving Transformation System (ETS) model is structural object representation constructed by the process of inductive inference. The model was proposed in 1990 by Lev Goldfarb to be applied to any pattern learning or classification problem. A formal exposition of the model is presented in this thesis. It defines the concepts that encapsulate the idea of structural representation and includes lemmas and theorems that link these concepts together into a single model. The chosen form of definitions is related to several general postulates about structural representation. The main feature of this formalization of the ETS model is the presence of an infinite hierarchy of representational levels. At each level, object representations are constructed from primitive constructive transformations (building blocks). Primitive transformations of the next level correspond to complex contextdependent additive transformations of the previous one. This hierarchy allows to reduce the complexity of representation of an object by constructing its higherlevel representation through
Representational formalisms: What they are and why we haven’t had any, submitted to a special issue of Pattern Recognition (2007) http://www.cs.unb.ca/~goldfarb/ETS special issue/Repr formalisms.pdf
, 2006
"... Abstract. Currently, the only discipline that has dealt with scientific representations— albeit nonstructural ones—is mathematics (as distinct from logic). I suggest that it is this discipline, only vastly expanded based on a new, structural, foundation, that will also deal with structural represen ..."
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Abstract. Currently, the only discipline that has dealt with scientific representations— albeit nonstructural ones—is mathematics (as distinct from logic). I suggest that it is this discipline, only vastly expanded based on a new, structural, foundation, that will also deal with structural representations. Logic (including computability theory) is not concerned with the issues of various representations useful in natural sciences. Artificial intelligence was supposed to address these issues but has, in fact, hardly advanced them at all. How do we, then, approach the development of representational formalisms? It appears that the only reasonable starting point is the primordial point at which all of mathematics began, i.e. we should start with the generalization of the process of construction of natural numbers, replacing the identical structureless units, out of which numbers are built, by structural ones, each signifying an atomic “transforming ” event. This paper is conceived as a companion to [1], and is a revised version of [2]. Mathematics is the science of the infinite, its goal is the symbolic comprehension of the infinite with human, that is finite, means.
What is a Structural Representation? (Second Version)
, 2004
"... We outline a formalism for "structural", or "symbolic", representation, the necessity of which is acutely felt in all sciences. One can develop an initial intuitive understanding of the proposed representation by simply generalizing the process of construction of natural numbe ..."
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We outline a formalism for "structural", or "symbolic", representation, the necessity of which is acutely felt in all sciences. One can develop an initial intuitive understanding of the proposed representation by simply generalizing the process of construction of natural numbers: replace the identical structureless units out of which numbers are built by several structural ones, attached consecutively. Now, however, the resulting constructions embody the corresponding formative/generative histories, since we can see what was attached and when.