Results 1  10
of
36
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 ..."
Abstract

Cited by 36 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
under the supervision of Prof.dr J. F. A. K. van Benthem, and submitted to the Board of
AMS Short Course on Computational Topology, JMM2011, New Orleans. On the Topology of Discrete Planning with Uncertainty
, 2012
"... Abstract. This chapter explores the topology of planning with uncertainty in discrete spaces. The chapter defines the strategy complex of a finite discrete graph as the collection of all plans for accomplishing all tasks specified by goal states in the graph. Transitions in the graph may be nondeter ..."
Abstract
 Add to MetaCart
Abstract. This chapter explores the topology of planning with uncertainty in discrete spaces. The chapter defines the strategy complex of a finite discrete graph as the collection of all plans for accomplishing all tasks specified by goal states in the graph. Transitions in the graph may be nondeterministic or stochastic. One key result is that a system can attain any state in its graph despite control uncertainty if and only if its strategy complex is homotopic to a sphere of dimension two less than the number of states in the graph. 1. Planning with Uncertainty in Robotics The goal of Robotics is to animate the inanimate, so as to endow machines with the ability to act purposefully in the world. Roboticists, working in the subfield of planning, create software by which robots reason about future outcomes of potential actions. Using such planning software, robots combine individual actions into collections that together accomplish particular tasks in the world [29, 30]. Two fundamental and intertwined issues confound this seemingly straightforward approach. One is world complexity, the other is uncertainty.
Operational Theories and Categorical Quantum Mechanics
, 2013
"... A central theme in current work in quantum information and quantum foundations is to see quantum mechanics as occupying one point in a space of possible theories, and to use this perspective to understand the special features and properties which single it out, and the possibilities for alternative ..."
Abstract
 Add to MetaCart
A central theme in current work in quantum information and quantum foundations is to see quantum mechanics as occupying one point in a space of possible theories, and to use this perspective to understand the special features and properties which single it out, and the possibilities for alternative theories. Two formalisms which have been used in this context are operational theories, and categorical quantum mechanics. The aim of the present paper is to establish strong connections between these two formalisms. We show how models of categorical quantum mechanics have representations as operational theories. We then show how nonlocality can be formulated at this level of generality, and study a number of examples from this point of view, including Hilbert spaces, sets and relations, and stochastic maps. The local, quantum, and nosignalling models are characterized in these terms.