Recursion Theory, its Generalisations and Applications


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Fundamental methods for connecting recursively enumerable degrees R. Soare; 2. Posner; 3. Degrees of Generic Sets C. Jockusch; 4. The Degrees of Unsolvability: Some recent results M. Lerman; 5. Some Constructions in -Recursion theory R. Shore; 6. The Recursion theory of the continuous functionals D. For, Van Wijngaarden and Dijkstra had stumbled upon the possibility to syntactically express recursive-procedure activations.

Any other occurrence of the procedure identifier within the procedure body denotes activation of the procedure. The fact that the alternative, of preventing recursive-procedure activations by means of several language restrictions, would be cumbersome, was also mentioned by Van Wijngaarden 7. It is, in hindsight, clear that Van Wijngaarden and Dijkstra were primarily reasoning along linguistic lines and not in terms of specific machine features. As we shall see, simplicity for them meant less language restrictions. According to Naur's recollections:. I got charmed with the boldness and simplicity of this [one-sentence] suggestion and decided to follow it in spite of the risk of subsequent trouble over the question cf.

Appendix 5, Bauer's point 2. Naur's reference to Bauer shows that he was well aware of ALCOR 's strong will to prohibit introducing recursive procedures in the language, in accordance with the November meeting 8. Van Wijngaarden's team in Amsterdam spent most of the s building computing machines and corresponding machine languages.

In hindsight, the Amsterdammers' lack of expertise in translation technology and machine-independent languages may have been a blessing in disguise, as Dijkstra's words from indicate:. The combination of no prior experience in compiler writing and a new machine [the X1 ] without established ways of use greatly assisted us in approaching the problem of implementing ALGOL60 with a fresh mind.

If certain aesthetic criteria were met, they would consider the language to be elegant and, hence, relevant.

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Only after such a language was defined, did they seek an automatic-translation technique. Their working style differed radically from the efficiency-driven approach followed by Bauer, Samelson, Strachey, Wilkes, and others. An important aesthetic criterion for the Dutch was generality: any unnecessary language restriction had to be avoided at all costs, in the interest of obtaining a simple language. He suggested considering any English text that respects five restrictions:. Dijkstra remarked that the readability of any text respecting restrictions is not necessarily hindered and one can read such a text while being completely ignorant of the restrictions.

But, constructing a correct English text in conformance with restrictions is problematic, due to a lack of intuition when trying to comprehend the restrictions. In the interest of clarity and program correctness, Dijkstra did not want a language such as ALGOL60 to contain restrictions similar to either.

For instance, a procedure that can call another procedure but not itself was an unnecessary language restriction for Dijkstra. By discarding it, a more general and hence simpler language was obtained; i. The switch is a vector of statement labels, declared at the beginning of a block. Its declaration looks syntactically like an assignment statement. Both the procedure and switch declaration have a hybrid nature; i. On the one hand, the procedure and switch declarations both reserve an identifier for a special sort of object and that object is defined statically; i.

On the other hand, however, while a constant number can be used in an assignment statement which dynamically assigns a value, a procedure or switch declaration can not be used in such a dynamic manner. This, in turn, would allow one to remove the value-defining function of the procedure and switch declarations. According to Dijkstra:.

Dijkstra's emphasis on general language constructs and corresponding dynamic implementations would, as many observed, have a negative effect on computation time. In Strachey's words:.

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I think the question of simplifying or reducing a language in order to make the object program more efficient is extremely important. I disagree fundamentally with Dijkstra, about the necessity of having everything as general as possible in all possible occasions as I think that this is a purely theoretical approach [ ] 9. Indeed, for most people at the symposium efficiency was the prime concern. A closer look at Dijkstra's ideology, however, shows that his agenda was not to neglect efficiency issues per se, but to focus on the more general objective of increasing programming comfort.

In Dijkstra's words:. Those who on the ground of this remark now doubt the honest fervour with which the following is written, should remember that, in the last instance, a machine serves one of its highest purposes when its activities significantly contribute to our comfort. In other words, to better understand the real underlying problems of programming, Dijkstra suggested to temporarily ignore i.

While a decrease in execution time or memory footprint may, indeed, contribute to an increase in programming comfort, other criteria, such as program correctness, could contribute much more. I am convinced that these problems [of program correctness] will prove to be much more urgent than, for example, the exhaustive exploitation of specific machine features, if not now, then at any rate in the near future. As a final example, by countering Strachey and Wilkes's efficiency-driven proposal to explicitly delimit the use of recursive procedures, Dijkstra clearly explained where he stood on these matters:.

And this influence could quite easily be so considerable, that the possible gain in efficiency that can still be booked by excluding recursiveness, will become negligible. To clarify, Strachey and Wilkes were in favor of static solutions enforced by language restrictions in the interest of machine efficiency. Instead, they wanted all procedures to be nonrecursive by default and only recursive if explicitly delimited by the programmer. By doing so, it would be possible to write compilers that could generate far more efficient machine code.

Dijkstra, by contrast, wanted to avoid such case distinctions in order to obtain a simple language and corresponding implementation technique. To do so, he advocated for general language constructs and corresponding dynamic solutions. Dijkstra acknowledged that his general recursive-programming approach led to inefficient machine code, but he also stressed that this would probably be resolved in the nearby future cf. In hindsight, it seems that Dijkstra had anticipated the advent of the hardware stack.

To conclude, Dijkstra's philosophy is in many ways similar to the Gorn-Brown-Carr philosophy of And, his appeal for dynamic constructs is, albeit for different reasons, similar to the work of Newell, Shaw, and Simon.


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Furthermore, Dijkstra believed that efficiency problems would be resolved in the nearby future, or at least become negligible. According to Dijkstra, generalization of a programming language allowed for simplification in compiler building and this would in the long term prevail over the short-term engineering problems that concerned people such as Bauer, Samelson, Strachey, and Wilkes.

Dijkstra's urge to generalize was not only felt at the level of language definition. Also when designing the Dijkstra-Zonneveld compiler did he seek general principles. For example, Dijkstra generalized the manner in which a stack was used in two essential ways. First, he showed how the functionality of the stack can be further generalized in time, by not only using it to evaluate arithmetic expressions, but also expressions containing procedure calls. Second, he showed how several different kinds of items operators, operands, states, priorities, etc.

By the late s, the stack had been invented over and over again by several independent researchers The top-most horizontal arrow shows how B is placed on the stack, assuming that B is ready made.

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The generalization lies in the fact that a stack can be used for both scenarios: whether B is ready made or has to be computed, by temporarily using a number of next stack locations, the net result remains the same. To arrange the procedure in such a way that it operates in the first free places of the stack, Dijkstra subsequently, in his paper, explained how one run-time stack could do the job As a by-product of Dijkstra's generalization, recursive-procedure activations became feasible:. Thus the subroutine has developed into a defining element that can be used completely recursively.

Dijkstra's generalized stack was able to store parameters and local variables of activated procedures. And, to make these elements accessible, Dijkstra had devised a mechanism to delve deep down in his stack, a mechanism which was not needed for the evaluation of simple arithmetic expressions. Dijkstra was not the first implementor of recursive activations Furthermore, Turing had, by , already thought through the idea of using a stack for recursive activations but had not implemented it It is Dijkstra's generalizing style which stands out when a comparison is made with the work of his contemporaries.

In his paper, Dijkstra described, what he called, the condensation of his meditations after having implemented ALGOL Some examples from his paper are presented below. When it encounters a number, it is copied to the top of the stack. When it encounters an operator, the corresponding operation is performed at the top of the stack. Dijkstra remarked that the function of an operator is, thus, a double one.

On the one hand, it indicates that copying words to the stack has to be interrupted. On the other hand, it also specifies the operation that has to be performed at the top of the stack. Dijkstra suggested to separate these two functions, by treating arithmetic operators in the same way as numbers: the operator should also be copied onto the stack, and its evaluation should be performed by a new and separate word E for Evaluate.

In accordance with these new conventions, the postfix notation would then be. Whenever the word read is unequal to E , the word is copied onto the stack. Whenever the word read is equal to E , it is not copied; instead, the operation on the top of the stack is performed. As the reader can check, such an execution would finally result in the number 2 being the top element of the stack, as desired. Further on in his paper, Dijkstra introduced variables which could be placed on the stack as well.

For instance, the expression. Upon execution, x would first be placed on top of the stack. After encountering the first occurrence of E , the variable x would be substituted by the value it denotes, which depends on the state of the process at that moment. For example, the top element of the stack, x , could be substituted by 3 , implying that the final result would be a stack with as top element the value 7. In short, Dijkstra treated variables as operators and, as explained previously, he treated operators as numbers. Dijkstra's further generalization was led by the observation that a number is only a special kind of expression.

That is, while in the previous examples the final addition to the stack was merely a number, it should, more generally, be possible to devise examples in which the final result is a general expression.

Recursion Theory, its Generalisations and Applications - F R Drake - Häftad () | Bokus

To achieve this generalization, Dijkstra allowed the word E to be placed on the stack as well by introducing a new word P for Postponement. Upon evaluation of P , it would be replaced by the word E. As an example, consider the following text with three variables x , y , and plinus :.

Likewise, if the value of the variable plinus had been - , then the resulting string of words would have corresponded to the expression x - y. In other words, the net effect of the previous illustration is that the expression x plinus y has been partially evaluated with as result another expression. Dijkstra continued in his paper by showing that the distinction between numbers and instructions was superfluous as well. His generalizations furthermore led him to introduce a second stack, which he called the stack of activations.

In his words:. We could try to merge our two stacks into one.

In general, however, this is not the case and trying to merge the two stacks into a single one would give a highly unnatural construction. In summary, while Dijkstra had only needed one stack to implement ALGOL60 , he needed two stacks to implement his more general stack-based programming language. In both cases it was his quest to generalize which stands out. Dijkstra's single most important ALGOL60 -related contribution, in terms of generalization, is without doubt his design of an intermediate machine-independent object language That is, a language which serves the purpose of describing the behavior of a general stack-based machine and not the X1 machine in particular.

Dijkstra situated this language in between ALGOL60 and the machine instructions of the X1 , and thereby created -what we would today call- a separation of concerns, which heavily simplified the implementation of the Dijkstra-Zonneveld compiler. The separation of concerns is twofold: a translation stage followed by an interpretation stage. The translation, from ALGOL60 to the object language, is accomplished in a machine-independent manner and, hence, without any appeal to machine efficiency. Candidates for this translation stage are jobs that can be done once and for all; i. In the second stage, the obtained object program is processed by an interpreter written in the machine code of the X1.

Only at this stage does dynamic physical memory management come into play. It is, again, Dijkstra's top-down perspective, from ALGOL60 to the machine, which stands out in comparison to the machine-specific approaches of Bauer, Samelson, and others In Dijkstra's own words:. To design his object language, Dijkstra had made several abstractions.

For instance, he had assumed the presence of a sufficiently large homogeneous store, and had thereby abstracted away from X1 's heterogeneous memory; i. Similar to Brown and Carr in and as mentioned before, Dijkstra was fully aware of the prolonged-computation times introduced by his abstractions. Moreover, he acknowledged the short-term limitations of his solution:. We are fully aware of [ ] a certain prolongation of the calculating time, and we can imagine that for some computer which is still in use today one cannot accept this delay.

Their success did not go unnoticed, as the British researcher Randell recollects:. For the next few years Dijkstra used our report to defend himself from the numerous further requests he was getting from people who wanted to visit him and find out about the X1 compiler. By breaking away from the efficiency regime, Dijkstra and Zonneveld had succeeded in building a general and fast ALGOL60 translator for a relatively small computer, the X1.

In the interest of machine efficiency, ALCOR had decided to minimize their run-time system as much as possible by following a strict static approach: each procedure was allocated a fixed working space prior to program execution. This meant that procedures could not be activated more than once during program execution and, hence, that recursive-procedure activations, in particular, were ruled out. At the Rome symposium, however, Bauer and Samelson expressed their regret in choosing a static solution:.

The waste of static storage, in conflict with our original cellar [i. These words are ironic because ALCOR was a strong proponent of practical engineering-based solutions. O pdf Recursion Theory, its Generalisations and Applications den daher a 1 inventory und sind ni i c, zu vcrricKteb pricing. Baumer pdf Recursion Theory, differentiation schon Perkussion machen zu wollen.

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Recursion Theory, its Generalisations and Applications Recursion Theory, its Generalisations and Applications
Recursion Theory, its Generalisations and Applications Recursion Theory, its Generalisations and Applications
Recursion Theory, its Generalisations and Applications Recursion Theory, its Generalisations and Applications
Recursion Theory, its Generalisations and Applications Recursion Theory, its Generalisations and Applications
Recursion Theory, its Generalisations and Applications Recursion Theory, its Generalisations and Applications
Recursion Theory, its Generalisations and Applications Recursion Theory, its Generalisations and Applications
Recursion Theory, its Generalisations and Applications Recursion Theory, its Generalisations and Applications

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