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Mechanizing Coinduction and Corecursion in Higherorder Logic
 Journal of Logic and Computation
, 1997
"... A theory of recursive and corecursive definitions has been developed in higherorder logic (HOL) and mechanized using Isabelle. Least fixedpoints express inductive data types such as strict lists; greatest fixedpoints express coinductive data types, such as lazy lists. Wellfounded recursion expresse ..."
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Cited by 41 (5 self)
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A theory of recursive and corecursive definitions has been developed in higherorder logic (HOL) and mechanized using Isabelle. Least fixedpoints express inductive data types such as strict lists; greatest fixedpoints express coinductive data types, such as lazy lists. Wellfounded recursion expresses recursive functions over inductive data types; corecursion expresses functions that yield elements of coinductive data types. The theory rests on a traditional formalization of infinite trees. The theory is intended for use in specification and verification. It supports reasoning about a wide range of computable functions, but it does not formalize their operational semantics and can express noncomputable functions also. The theory is illustrated using finite and infinite lists. Corecursion expresses functions over infinite lists; coinduction reasons about such functions. Key words. Isabelle, higherorder logic, coinduction, corecursion Copyright c fl 1996 by Lawrence C. Paulson Content...
Categorical Ontology of Complex Spacetime Structures: The Emergence of Life and Human Consciousness
"... Abstract A categorical ontology of space and time is presented for emergent biosystems, supercomplex dynamics, evolution and human consciousness. Relational structures of organisms and the human mind are naturally represented in nonabelian categories and higher dimensional algebra. The ascent of ma ..."
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Abstract A categorical ontology of space and time is presented for emergent biosystems, supercomplex dynamics, evolution and human consciousness. Relational structures of organisms and the human mind are naturally represented in nonabelian categories and higher dimensional algebra. The ascent of man and other organisms through adaptation, evolution and social coevolution is viewed in categorical terms as variable biogroupoid representations of evolving species. The unifying theme of localtoglobal approaches to organismic development, evolution and human consciousness leads to novel patterns of relations that emerge in superand ultra complex systems in terms of colimits of biogroupoids, and more generally, as compositions of local procedures to be defined in terms of locally Lie groupoids. Solutions to such localtoglobal problems in highly complex systems with ‘broken symmetry ’ may be found with the help of generalized van Kampen theorems in algebraic topology such as the Higher Homotopy van Kampen theorem (HHvKT). Primordial organism structures are predicted from the simplest metabolicrepair systems extended to selfreplication through autocatalytic reactions. The intrinsic dynamic ‘asymmetry ’ of genetic networks in organismic development and evolution is investigated in terms of categories of manyvalued, Łukasiewicz–Moisil logic algebras and then compared with those obtained for (noncommutative) quantum
Abstract Computerizing Mathematical Text with
"... Mathematical texts can be computerized in many ways that capture differing amounts of the mathematical meaning. At one end, there is document imaging, which captures the arrangement of black marks on paper, while at the other end there are proof assistants (e.g., Mizar, Isabelle, Coq, etc.), which c ..."
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Cited by 1 (0 self)
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Mathematical texts can be computerized in many ways that capture differing amounts of the mathematical meaning. At one end, there is document imaging, which captures the arrangement of black marks on paper, while at the other end there are proof assistants (e.g., Mizar, Isabelle, Coq, etc.), which capture the full mathematical meaning and have proofs expressed in a formal foundation of mathematics. In between, there are computer typesetting systems (e.g., LATEX and Presentation MathML) and semantically oriented systems (e.g., Content MathML, OpenMath, OMDoc, etc.). The MathLang project was initiated in 2000 by Fairouz Kamareddine and Joe Wells with the aim of developing an approach for computerizing mathematical texts and knowledge which is flexible enough to connect the different approaches to computerization, which allows various degrees of formalization, and which is compatible with different logical frameworks (e.g., set theory, category theory, type theory, etc.) and proof systems. The approach is embodied in a computer representation, which we call MathLang, and associated software tools, which are being developed by ongoing work. Three Ph.D. students (Manuel Maarek (2002/2007), Krzysztof Retel (since 2004), and Robert Lamar (since 2006)) and over a dozen master’s degree and undergraduate students have worked on MathLang. The project’s progress and design choices are driven by the needs for computerizing real representative mathematical texts chosen from various
Frege, Russell and Wittgenstein on the judgment stroke
, 2011
"... Frege is highly valued as a logician by Russell and Wittgenstein, the latter nonetheless concludes in his Tractatus that one of Frege’s central notions, the judgment stroke, is “logically quite meaningless”. In order to see why Wittgenstein thinks so, we will investigate the ‘indirect interpretation ..."
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Frege is highly valued as a logician by Russell and Wittgenstein, the latter nonetheless concludes in his Tractatus that one of Frege’s central notions, the judgment stroke, is “logically quite meaningless”. In order to see why Wittgenstein thinks so, we will investigate the ‘indirect interpretation thesis’, which says that Wittgenstein’s interpretation of Frege was strongly influenced by the reading Russell gives of the Begriffsschrift in Principia Mathematica and Principles of Mathematics. This is done by analyzing the different conceptions of logic, focusing on the representations of judgment and assertion in Frege, Russell and the early Wittgenstein. Stong similarities can be found between the interpretations of Russell and Wittgenstein, this makes the indirect interpretation thesis plausible, although Russell’s influence cannot be the only reason why Wittgenstein rejected the judgment stroke as a logical symbol.
Additional Key Words and Phrases: Set theory, specification, types
"... Most specification languages have a type system. Type systems are hard to get right, and getting them wrong can lead to inconsistencies. Set theory can serve as the basis for a specification language without types. This possibility, which has been widely overlooked, offers many advantages. Untyped s ..."
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Most specification languages have a type system. Type systems are hard to get right, and getting them wrong can lead to inconsistencies. Set theory can serve as the basis for a specification language without types. This possibility, which has been widely overlooked, offers many advantages. Untyped set theory is simple and is more flexible than any simple typed formalism. Polymorphism, overloading, and subtyping can make a type system more powerful, but at the cost of increased complexity, and such refinements can never attain the flexibility of having no types at all. Typed formalisms have advantages too, stemming from the power of mechanical type checking. While types serve little purpose in hand proofs, they do help with mechanized proofs. In the absence of verification, type checking can catch errors in specifications. It may be possible to have the best of both worlds by adding typing annotations to an untyped specification language. We consider only specification languages, not programming languages.
JACQUES HERBRAND: LIFE, LOGIC, AND AUTOMATED DEDUCTION
"... The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1 ..."
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The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1811–1832) duel; the deaths of consumption of Gotthold Eisenstein (1823–1852) (who sometimes lectured his few students from his bedside) and of Gustav Roch (1839–1866) in Venice; the drowning of the topologist Pavel Samuilovich Urysohn (1898–1924) on vacation; the burial of Raymond Paley (1907–1933) in an avalanche at Deception Pass in the Rocky Mountains; as well as the fatal imprisonment of Gerhard Gentzen (1909–1945) in Prague1 — these are tales most scholars of logic and mathematics have heard in their student days. Jacques Herbrand, a young prodigy admitted to the École Normale Supérieure as the best student of the year1925, when he was17, died only six years later in a mountaineering accident in La Bérarde (Isère) in France. He left a legacy in logic and mathematics that is outstanding.
Reciprocity Theorems, Deconvolution Interferometry, and Imaging of Borehole Seismic Data
, 2007
"... Interferometry recovers the impulse response of waves propagating between two sensors as if one of them acts as a source. The primary focus of this thesis is on providing a framework for interferometry based on perturbation theory that can be used for the direct reconstruction of the portion of the ..."
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Interferometry recovers the impulse response of waves propagating between two sensors as if one of them acts as a source. The primary focus of this thesis is on providing a framework for interferometry based on perturbation theory that can be used for the direct reconstruction of the portion of the data that is of interest for imaging and inversion methodologies. I derive general reciprocity theorems in perturbed acoustic media. These theorems show that the wavefield perturbations are extracted from crosscorrelating the perturbations detected by one receiver with unperturbed waves sensed by another. Apart from applications to interferometry, the representation theorems presented here can also be used for inversescattering and timelapse monitoring. I also present a theory describing interferometry by deconvolution, based on a series expansion of deconvolved waves in the wavefield perturbations. This expansion is used to give a scatteringbased interpretation of the physics of deconvolution interferometry. Deconvolution interferometry, like its correlation counterpart, also retrieves the impulse response between the receivers, but with boundary conditions that are different than those of the original measurement. Interferometry by deconvolution is particularly important for recovering the impulse response from