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13
Gradual computerisation/formalisation of mathematical texts into Mizar
 From Insight to Proof: Festschrift in Honour of Andrzej Trybulec
"... Abstract. We explain in this paper the gradual computerisation process of an ordinary mathematical text into more formal versions ending with a fully formalised Mizar text. The process is part of the MathLang–Mizar project and is divided into a number of steps (called aspects). The first three aspec ..."
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Abstract. We explain in this paper the gradual computerisation process of an ordinary mathematical text into more formal versions ending with a fully formalised Mizar text. The process is part of the MathLang–Mizar project and is divided into a number of steps (called aspects). The first three aspects (CGa, TSa and DRa) are the same for any MathLang–TP project where TP is any proof checker (e.g., Mizar, Coq, Isabelle, etc). These first three aspects are theoretically formalised and implemented and provide the mathematician and/or TP user with useful tools/automation. Using TSa, the mathematician edits his mathematical text just as he would use L ATEX, but at the same time he sees the mathematical text as it appears on his paper. TSa also gives the mathematician easy editing facilities to help assign to parts of the text, grammatical and mathematical roles and to relate different parts through a number of mathematical, rethorical and structural relations. MathLang would then automatically produce CGa and DRa versions of the text, checks
A Compendium of Continuous Lattices in MIZAR  Formalizing recent mathematics
, 2002
"... This paper reports on the Mizar formalization of the theory of continuous lattices as presented in A Compendium of Continuous Lattices, [25]. By the Mizar formalization we mean a formulation of theorems, de nitions, and proofs written in the Mizar language whose correctness is veri ed by the Mizar ..."
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This paper reports on the Mizar formalization of the theory of continuous lattices as presented in A Compendium of Continuous Lattices, [25]. By the Mizar formalization we mean a formulation of theorems, de nitions, and proofs written in the Mizar language whose correctness is veri ed by the Mizar processor. This eort was originally motivated by the question of whether or not the Mizar system was suciently developed for the task of expressing advanced mathematics. The current state of the formalization57 Mizar articles written by 16 authors indicates that in principle the Mizar system has successfully met the challenge. To our knowledge it is the most sizable eort aimed at mechanically checking some substantial and relatively recent eld of advanced mathematics. However, it does not mean that doing mathematics in Mizar is as simple as doing mathematics traditionally (if doing mathematics is simple at all). The work of formalizing the material of [25] has: (i) prompted many improvements of the Mizar proof checking system; (ii) caused numerous revisions of the the Mizar data base; and (iii) contributed to the \to do" list of further changes to the Mizar system.
Development of the theory of continuous lattices in mizar
 In Kerber and Kohlhase
, 2001
"... Abstract. This paper reports on Mizar formalization of the theory of continuous lattices included in the A Compendium of Continuous Lattices, [7]. Mizar formalization means a formalization of theorems, definitions, and proofs in the Mizar language such that it is accepted by the Mizar system. This e ..."
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Abstract. This paper reports on Mizar formalization of the theory of continuous lattices included in the A Compendium of Continuous Lattices, [7]. Mizar formalization means a formalization of theorems, definitions, and proofs in the Mizar language such that it is accepted by the Mizar system. This effort was originally motivated by the question whether the Mizar system is sufficiently developed as to allow expressing advanced mathematics. The current state of the formalization, which includes 49 Mizar articles written by 14 authors, suggests that the answer is positive. The work of the team of authors in cooperation with the Library Committee1 and system designers resulted in improvements of the system towards a more convenient technology for doing mechanically checked mathematics. It revealed, also, that the substantial element of the convenience is the incorporation of computer algebra into Mizar system. 1
Transformation Methods in LDS
 In Logic, Language and Reasoning. An Essay in Honor of Dov Gabbay
, 1997
"... this paper we shall, instead, use a fragment of this family of logics as a casestudy to illustrate a set of methods originating in the LDS program. In particular, we aim to illuminate the following aspects: (I) By virtue of the extra power of labels and labelling algebras, traditional proof systems ..."
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this paper we shall, instead, use a fragment of this family of logics as a casestudy to illustrate a set of methods originating in the LDS program. In particular, we aim to illuminate the following aspects: (I) By virtue of the extra power of labels and labelling algebras, traditional proof systems can be transformed so as to become applicable over a much wider territory whilst retaining a uniform structure. Different logics can be obtained by defining different labelling algebras, which therefore act as "parameters", and the transition from one logic to another can be captured as a parameterchanging process which leaves the structure of deductions unchanged
Declarative/LogicBased Computational Cognitive Modeling
"... This chapter is an esemplastic systematization of declarative computational cognitive modeling, a field that cuts across cognitive modeling based on cognitive architectures (such as ACTR, Soar, and Clarion), humanlevel artificial intelligence (AI), logic itself, and psychology of reasoning (espec ..."
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This chapter is an esemplastic systematization of declarative computational cognitive modeling, a field that cuts across cognitive modeling based on cognitive architectures (such as ACTR, Soar, and Clarion), humanlevel artificial intelligence (AI), logic itself, and psychology of reasoning (especially of the computational kind). The hallmarks of declarative computational cognitive modeling are the following two intertwined constraints: (1) The central units of information used in the approach are (at least in significant part) declarative in nature, and the central process carried out over these units is inference. (2) The approach to modeling the mind is topdown, rather than bottomup. (These two points are interconnected because once one commits to (1), (2) becomes quite unavoidable, since bottomup processing in the brain, as reflected in relevant formalisms (e.g., artificial neural networks), is based on units of information that are numerical, not declarative.) The systematization of declarative computational cognitive modeling is achieved by using formal logic, and hence declarative computational cognitive modeling, from the formal perspective, becomes logicbased computational cognitive modeling (LCCM). The chapter covers some prior research that falls under LCCM; this research has been carried out by
Part I The art of logic 1 Chapter 1
"... After some preliminary grammatical considerations in this chapter, we collect the material on truthfunctional logic in chapter 2; the material on quantifiers and identity in chapter 3 (proofs), chapter 5 (symbolization), and chapter 6 (semantics); some applications in chapter 7 (logical theory, ari ..."
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After some preliminary grammatical considerations in this chapter, we collect the material on truthfunctional logic in chapter 2; the material on quantifiers and identity in chapter 3 (proofs), chapter 5 (symbolization), and chapter 6 (semantics); some applications in chapter 7 (logical theory, arithmetic, set theory), and a discussion of definitions in chapter 8. Chapter 13 is an unfulfilled promise of a discussion of the chief theorems about modern logic.
Toward Aligning Computer Programming with Clear Thinking via the Reason Programming Language
"... Logic has long set itself the task of helping humans think clearly. Certain computer programming languages, most prominently the Logo language, have been billed as helping young people become clearer thinkers. It is somewhat doubtful that such languages can succeed in this regard, but at any rate it ..."
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Logic has long set itself the task of helping humans think clearly. Certain computer programming languages, most prominently the Logo language, have been billed as helping young people become clearer thinkers. It is somewhat doubtful that such languages can succeed in this regard, but at any rate it seems sensible to explore an approach to programming that guarantees an intimate link between the thinking required to program, and the kind of clear thinking that logic has historically sought to cultivate. Accordingly, I have invented a new computer programming language, Reason, one firmly based in the declarative programming paradigm, and specifically aligned with the core skills constituting clear thinking. Reason thus offers the intimate link in
NATURAL DEDUCTION IN PREDICATE CALCULUS A TOOL FOR ANALYSING PROOF IN A DIDACTIC PERSPECTIVE
"... Abstract: In this paper, we intend to provide theoretical arguments for the importance of taking account in quantification matters while analysing proofs in a didactic perspective, not only at tertiary level, where various research are still available, but also at secondary level and we argue that n ..."
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Abstract: In this paper, we intend to provide theoretical arguments for the importance of taking account in quantification matters while analysing proofs in a didactic perspective, not only at tertiary level, where various research are still available, but also at secondary level and we argue that natural deduction in predicate calculus is a relevant logical reference for this purpose. Following Quine, we emphasize on an example the interest of formalizing mathematical statements in Predicate Calculus in a purpose of conceptual clarification. In a second part of the paper, we give some short insights about the theory of quantification before exposing the system of Copi for natural deduction. The last section is devoted to analysing a proof using the logical tools offered by natural deduction in predicate calculus. I. About he importance of quantification in elementary geometry It is widely recognized that in tertiary mathematical education, quantification matters are central and source of strong difficulties, even for gifted students (Dubinsky & Yparaki, 2000, Selden & Selden, 1995, Epp 2004, Chellougui, 2003) ; but it seems that, in secondary mathematical education, a low interest is paid on quantification
Chapter 1 Preliminaries
"... This course assumes you know how to use truth functions and quantifiers as tools; such is part of the art of logic. Our principal task here will be to study these very tools; we shall be engaged in part of the science of logic. 1A.1 Aim of the course ..."
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This course assumes you know how to use truth functions and quantifiers as tools; such is part of the art of logic. Our principal task here will be to study these very tools; we shall be engaged in part of the science of logic. 1A.1 Aim of the course