## What Is Computational Knowledge And How Do We Acquire It? (0)

Venue: | Submitted, IFIP Working Conference on Programming Concepts and Methods (PROCOMET '98 |

Citations: | 1 - 1 self |

### BibTeX

@INPROCEEDINGS{Stevenson_whatis,

author = {D. E. Stevenson},

title = {What Is Computational Knowledge And How Do We Acquire It?},

booktitle = {Submitted, IFIP Working Conference on Programming Concepts and Methods (PROCOMET '98},

year = {}

}

### OpenURL

### Abstract

. The goal of foundational thinking in computer science is to understand the methods and practices of working programmers; we might even be able to improve upon those practices. The investigation outlined here applies the methods of constructive mathematics 'a l`a A. N. Kolmogoroff, L. E. J. Brouwer and Errett Bishop to contemporary computer science. The major approach is to use Kolmogoroff's interpretation of the predicate calculus. This investigation includes an attempt to merge contemporary thoughts on computability and computing semantics with the language of mental constructions proposed by Brouwer. This necessarily forces us to ask about the psychology of language. I present a definition of algorithms that links language, constructive mathematics, and logic. Using the concept of an abstract family of algorithms (Hennie) and principles of constructivity, a definition of problem solving. The constructive requirements for an algorithm are developed and presented. Given this framewor...

### Citations

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Citation Context ... Passive, verbal associations ffl Feedback ffl Information These ideas help generate what are called models or paradigms in science. Surely, language of science is the language of models and paradigms=-=[17]-=-. The sum total of all these considerations lead to the issue of language. 3.3. Language. There are all types of natural language based on intended use, including poetic, theologic, romantic, : : : . ... |

692 |
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(Show Context)
Citation Context ...d what it means to be a problem, can I then what it means to solve a problem? The short answer is that this was all settled by Turing[26] in the 1930s. Turing's work was brilliantly extended by Kleene=-=[15]-=- so that by the 1950s most of the upper level concepts were solved or at least understood. Another flurry of activity followed Chomsky[8]. And let's not forget Petri. But the long answer is something ... |

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(Show Context)
Citation Context ...rth a consistent set of principles for constructive programming, there are certain concepts that I inherit from other aspects of constructivity. A succinct list of rules gleaned from Bishop's writings=-=[2, 3, 5, 6, 4, 7]-=- are shown in Figure 6 and expanded below: 1. Objects. Objects in programming are figments of the imagination of humans but representations stored in a computer. The first view is from Brouwer and the... |

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Citation Context ...in the 1930s. Turing's work was brilliantly extended by Kleene[15] so that by the 1950s most of the upper level concepts were solved or at least understood. Another flurry of activity followed Chomsky=-=[8]-=-. And let's not forget Petri. But the long answer is something much different. The long answer is that, while there are highly theoretical results in many areas, the methods of describing problems and... |

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(Show Context)
Citation Context ...r Construction There are several ways to present semantics. The three most common --- as the reader is well aware --- are (1) operational, (2) axiomatic, and (3) denotational. See, for example, Gunter=-=[13]-=-. It is now common to used category-theoretic notation to describe denotational theories as we did in Equation (1). To provide a constructive basis for semantics, however, we return to the original co... |

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Citation Context ...omputational basis of science and engineering. Such a basis would have to be constructive and not classical[22]. This is not a particularly popular stand: the formulation of the -calculus is classical=-=[1]-=-. There are many arguments against a constructionist view, not the least of which is it is harder to work under constructive regimes. However, for the most part, mathematics is constructive: any proof... |

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(Show Context)
Citation Context ...mpilers didn't have the time or space to deal with other issues. But this is the 21st Century, not the 1960s. The usual model for programming is something based on the -calculus, going back to McCarty=-=[18]. This mod-=-el represents the "unwinding" version of the -calculus. Hence the metalanguage rules have those operations, stacks, etc. The thesis presented here is that algorithms have both state and witn... |

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(Show Context)
Citation Context ...anipulations. Therefore, we must define the metalanguage rules so that both are presented: (1) the usual state machine and (2) a modified proof in the spirit of the static-dynamic principle. Using LCF=-=[19]-=- and NuPrl[9] as exemplars, the system could be seen as a mapping hA i ; \Sigma i ; W i i = hA 0 ; \Sigma 0 ; W 0 i where A are algorithms, \Sigma are states, and W = hProof, Hypotheses, Tacticali. Ta... |

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Citation Context ... OE is an adequate formalization of an operator F if every feature of F is represented by OE. 2 His leadership was cut short by his death in 1936; killed by a student over his Jewish heritage. 3 Ranta=-=[20]-=- suggests Waismann as a key to understanding analytic/linguistic philosophy 16 D. E. STEVENSON M2 A operation OE is faithful to an operator F if every basic concept of OE corresponds to a basic concep... |

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Citation Context ...tation is that it can fail. This is even true in classical mathematics: 1=0 = 1 because the uniqueness of the inverse of multiplication fails to hold. See Suppes' venerable book, Introduction to Logic=-=[24]-=-. 1 is not a crisis --- it is a reality to be dealt with. 2 D. E. STEVENSON These are not pie-in-the-sky problems, these are everyday, down-to-earth problems that affect on how our discipline is pract... |

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(Show Context)
Citation Context ...entrated on the issue of computational science and engineering, defined as the study of the computational basis of science and engineering. Such a basis would have to be constructive and not classical=-=[22]-=-. This is not a particularly popular stand: the formulation of the -calculus is classical[1]. There are many arguments against a constructionist view, not the least of which is it is harder to work un... |

24 |
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(Show Context)
Citation Context ...rth a consistent set of principles for constructive programming, there are certain concepts that I inherit from other aspects of constructivity. A succinct list of rules gleaned from Bishop's writings=-=[2, 3, 5, 6, 4, 7]-=- are shown in Figure 6 and expanded below: 1. Objects. Objects in programming are figments of the imagination of humans but representations stored in a computer. The first view is from Brouwer and the... |

20 |
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(Show Context)
Citation Context ...approach distills to one simple question: "What does it mean to say that program p solves problem P ?" which I write as p ` P . This is not a unique approach to the author---I took it from K=-=olmogoroff[16]. Even tho-=-ugh Markov and Kolmogoroff were important early contributors to what might be called "constructive science", they had radically different ideas and approaches to computation. While the Marko... |

17 |
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(Show Context)
Citation Context ...e vocabulary on computable solutions. A procedure is some entity (to be dissected later) that takes inputs, eventually halts, and produces an answer. I denote the statement p halts as p # after Hennie=-=[14] and -=-p does not halt as p ". In equational form I write pa = b and pa = ? respectively, where a is the input. For relations, I specify that the range is over the boolean set f0; 1g. Totality and parti... |

16 |
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(Show Context)
Citation Context ... x." That is, philosophers of science do not do science, they do philosophy of science. This is probably true, but philosophers within science x undoubtedly do science x. For example, Stephen J. =-=Gould[12]-=- certainly is a true biologist and he even does biology --- but he also criticizes the conduct of biological research. Mathematics has certainly had its share of philosophers over the past 200 years: ... |

8 |
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(Show Context)
Citation Context ...rth a consistent set of principles for constructive programming, there are certain concepts that I inherit from other aspects of constructivity. A succinct list of rules gleaned from Bishop's writings=-=[2, 3, 5, 6, 4, 7]-=- are shown in Figure 6 and expanded below: 1. Objects. Objects in programming are figments of the imagination of humans but representations stored in a computer. The first view is from Brouwer and the... |

8 |
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(Show Context)
Citation Context ...es to these constructions and hence, language is born in the subject matter. It is the language and its associated objects that must be learned. The shared constructions are built by three basic means=-=[27]: 1. Activ-=-e, ostensive acts; 2. Passive, verbal/nominal association; and 3. Feedback using information content. Active, ostensive acts are "show me" acts. These can be reasonably subtle. Passive acts ... |

5 |
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(Show Context)
Citation Context ... (data structure) and transition (algorithm). Let S = (state space; transition system) = (St; T r) be the set of all systems that solve the original problem. This, of course, is not a new formulation =-=[28]-=-. How do I know that S = (St; T r) solves a problem P , Let p 2 S; then we want to know if p ` P ? This is a question of (1) the problem space and (2) translation of the problem space to the encoded s... |

4 |
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(Show Context)
Citation Context ...ience. Firstly, what does it mean to be a problem? Once I understand what it means to be a problem, can I then what it means to solve a problem? The short answer is that this was all settled by Turing=-=[26]-=- in the 1930s. Turing's work was brilliantly extended by Kleene[15] so that by the 1950s most of the upper level concepts were solved or at least understood. Another flurry of activity followed Chomsk... |

3 |
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(Show Context)
Citation Context ... cue from analysis[21, p. 42], I call the setsa a cover. 8.2. Sets. SincesA is constructive, there is a function that passes on membership insA. To reduce notation, assume that A is that function. In =-=[25], th-=-e convention is that Aa is true whensa is an object in A as in (ii) above. Actually, this convention seems to be handy about one-half the time; the other half A��a =sa would be better. Staying wit... |

1 |
Principles of constructive euclidean geometry
- Stevenson
(Show Context)
Citation Context ...uctionist and live within the framework Euclid presented. And if that constructionist framework holds, what does it say about programming practices? The work here is an outgrowth of the Euclidean work=-=[23]-=-. The Elements is generally presented as the seminal text in axiomatic thinking: not so. The following observations/conclusions from [23] are important here: ffl Most likely, the ancients were neutral... |