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Data types à la carte
"... This paper describes a technique for assembling both data types and functions from isolated individual components. We also explore how the same technology can be used to combine free monads and, as a result, structure Haskell’s monolithic IO monad. 1 ..."
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This paper describes a technique for assembling both data types and functions from isolated individual components. We also explore how the same technology can be used to combine free monads and, as a result, structure Haskell’s monolithic IO monad. 1
To Properly Reflect Physicists’ Reasoning about Randomness, We Also Need a Maxitive (Possibility
 Measure”, Proceedings of the 2005 IEEE International Conference on Fuzzy Systems FUZZIEEE’2005
"... According to the traditional probability theory, events with a positive but very small probability can occur (although very rarely). For example, from the purely mathematical viewpoint, it is possible that the thermal motion of all the molecules in a coffee cup goes in the same direction, so this cu ..."
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According to the traditional probability theory, events with a positive but very small probability can occur (although very rarely). For example, from the purely mathematical viewpoint, it is possible that the thermal motion of all the molecules in a coffee cup goes in the same direction, so this cup will start lifting up. In contrast, physicists believe that events with extremely small probability cannot occur. In this paper, we show that to get a consistent formalization of this belief, we need, in addition to the original probability measure, to also consider a maxitive (possibility) measure. We also show that the resulting advanced and somewhat difficulttodescribed definition can be actually viewed as a particular case of something very natural: the general notion of boundedness. 1
Recursion and the infinitude claim ∗
"... We address certain recent suggestions that the existence of infinitely many grammatical expressions in human languages (the infinitude claim) is a universal of human language. We examine the arguments given for the infinitude claim, and show that they tacitly depend on the unwarranted assumption tha ..."
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We address certain recent suggestions that the existence of infinitely many grammatical expressions in human languages (the infinitude claim) is a universal of human language. We examine the arguments given for the infinitude claim, and show that they tacitly depend on the unwarranted assumption that the only way to represent the structural properties of a language is by means of a generative grammar with a recursive rule system. We explore some of the reasons why linguists have been so willing to accept language infinitude despite its inadequate support and its paucity of linguistic consequences. We suggest that the infinitude claim is motivated chiefly by an inadvisable adherence to the notion that languages are sets. It is not motivated by considerations of the creative aspect of language use, or opposition to associationist psychology, or the putative universality of iterable linguistic structure such as recursive embedding or unbounded coordination (which are in any case probably not universal). 1 Infinitude as a linguistic universal In a number of recent works, linguists have portrayed the infinitude of sentences in human languages as an established linguistic universal. Lasnik (2000) asserts, in the opening chapter of a textbook based on transcriptions of a series of introductory syntax lectures:
Towards Specifying Reactive Autonomic Systems with a Categorical Approach: A Case Study
"... Abstract. Software complexity is the main obstacle to further progress in the IT industry. One solution is the autonomic system with self * properties. Formal methods are proven approaches to ensuring the correct operation of complex interacting systems. However, the current formal methods do not a ..."
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Abstract. Software complexity is the main obstacle to further progress in the IT industry. One solution is the autonomic system with self * properties. Formal methods are proven approaches to ensuring the correct operation of complex interacting systems. However, the current formal methods do not adequately address the problem of verifying two of the most important features of autonomic systems, namely emergent behavior and evolving behavior. Category Theory (CT) has recently been proposed as a formal framework to provide a structure for isolating the management of evolving specifications and the analysis of changes. We propose a formal framework based on CT in this paper to specify reactive autonomic systems. Our approach is illustrated with a NASA case study.
A Unified SheafTheoretic Account Of NonLocality and Contextuality
, 2011
"... A number of landmark results in the foundations of quantum mechanics show that quantum systems exhibit behaviour that defies explanation in classical terms, and that cannot be accounted for in such terms even by postulating “hidden variables” as additional unobserved factors. Much has been written o ..."
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A number of landmark results in the foundations of quantum mechanics show that quantum systems exhibit behaviour that defies explanation in classical terms, and that cannot be accounted for in such terms even by postulating “hidden variables” as additional unobserved factors. Much has been written on these matters, but there is surprisingly little unanimity even on basic definitions or the interrelationships among the various concepts and results. We use the mathematical language of sheaves and monads to give a very general and mathematically robust description of the behaviour of systems in which one or more measurements can be selected, and one or more outcomes observed. We say that an empirical model is extendable if it can be extended consistently to all sets of measurements, regardless of compatibility. A hiddenvariable model is factorizable if, for each value of the hidden variable, it factors as a product of distributions on the basic measurements. We prove that an empirical model is extendable if and only if there is a factorizable hiddenvariable model which realizes it. From this we are able to prove generalized versions of wellknown NoGo theorems. At the conceptual level, our equivalence result says that the existence of incompatible measurements is the essential ingredient in nonlocal and contextual behavior in quantum mechanics.
The expression lemma ⋆
"... Abstract. Algebraic data types and catamorphisms (folds) play a central role in functional programming as they allow programmers to define recursive data structures and operations on them uniformly by structural recursion. Likewise, in objectoriented (OO) programming, recursive hierarchies of objec ..."
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Abstract. Algebraic data types and catamorphisms (folds) play a central role in functional programming as they allow programmers to define recursive data structures and operations on them uniformly by structural recursion. Likewise, in objectoriented (OO) programming, recursive hierarchies of object types with virtual methods play a central role for the same reason. There is a semantical correspondence between these two situations which we reveal and formalize categorically. To this end, we assume a coalgebraic model of OO programming with functional objects. The development may be helpful in deriving refactorings that turn sufficiently disciplined functional programs into OO programs of a designated shape and vice versa. Key words: expression lemma, expression problem, functional object, catamorphism, fold, the composite design pattern, program calculation, distributive law, free monad, cofree comonad. 1
MSc in Logic
, 2010
"... under the supervision of Prof.dr J. F. A. K. van Benthem, and submitted to the Board of ..."
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under the supervision of Prof.dr J. F. A. K. van Benthem, and submitted to the Board of
AN ALTERNATIVE DEFINITION OF THE COMPLETION OF METRIC SPACES
, 810
"... Abstract. In this article, the author proposes another way to define the completion of a metric space, which is different from the classical one via the dense property, and prove the equivalence between two definitions. This definition is based on considerations from category theory, and can be gene ..."
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Abstract. In this article, the author proposes another way to define the completion of a metric space, which is different from the classical one via the dense property, and prove the equivalence between two definitions. This definition is based on considerations from category theory, and can be generalized to arbitrary categories. 1.
A category theory explanation for systematicity
"... Classical and Connectionist theories of cognitive architecture “explain ” systematicity, whereby the capacity for some cognitive behaviors is intrinsically linked to the capacity for others, as a consequence of syntactically and functionally combinatorial representations, respectively. However, both ..."
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Classical and Connectionist theories of cognitive architecture “explain ” systematicity, whereby the capacity for some cognitive behaviors is intrinsically linked to the capacity for others, as a consequence of syntactically and functionally combinatorial representations, respectively. However, both theories depend on ad hoc assumptions to exclude specific architectures—grammars, or Connectionist networks—that do not account for systematicity. By analogy with the Ptolemaic (i.e., geocentric) theory of planetary motion, although either theory can be made to be consistent with the data, both nonetheless fail to explain it (Aizawa, 2003b). Category theory provides an alternative explanation based on the formal concept of adjunction, which consists of a pair of structure preserving maps, called functors. A functor generalizes the notion of a map between representational states to include a map between state transformations (processes). In a formal sense, systematicity is a necessary consequence of a “higherorder ” theory of cognitive architecture, in contrast to the “firstorder ” theories derived from Classicism or Connectionism. Category theory offers a reconceptualization for cognitive science, analogous to the one that Copernicus provided for astronomy, where representational states are no longer the center of the cognitive universe—replaced by the relationships between the maps that transform them.