Enumerability · Decidability Computability

Hermes, Hans

doi:10.1007/978-3-642-46178-1

Cited by 30 publications

(15 citation statements)

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“…The if-then-else and the unconditional break are efficiently equivalent, relatively to the base language Loop, if-then-else ≈ break (10) For instance, the program for x(P ; break; Q ) (where Q is never executed) can be replaced by the following program, where the conditional instruction (6) Proof. Consider the following simulation of "if x then(P ) else(Q )" using the "if-then" instruction (the simulation in the other direction is trivial).…”

Section: Conditional Instructions In Loop+breakmentioning

confidence: 98%

“…Consider the execution of a Loop+dec+break program. The break instructions can be replaced by if-then and/or if-then-else instructions, see (10), page 77. After the replacement, the loops do not contain break's.…”

Section: Proofmentioning

confidence: 99%

“…functions is usually defined [10,17] by a set of initial functions (the constant, successor, and projection functions) and two rules -composition and primitive recursion. It can also be characterised as the set of functions that can be implemented in a simple register programming language called Loop, see for instance [13].…”

Section: Introductionmentioning

confidence: 99%

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The efficiency of primitive recursive functions: A programmer's view

Matos¹

2015

Theoretical Computer Science

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Using denotational semantics tools, Colson and others studied primitive recursive (p.r.) algorithms, proving the "ultimate obstinacy" property, which has as a consequence that many computations cannot be efficiently implemented in a language that faithfully expresses the classical definition of p.r. functions. As shown by Ritchie and Meyer, the class of p.r. functions can also be characterised by the programming language Loop. Our purpose is to show that the informal, but precise, operational description of Loop (and its variants) is sufficient to prove non-trivial time lower bounds of p.r. algorithms. In particular, we present a simple proof of a property which is similar to ultimate obstinacy, namely that every p.r. program that implements a non-trivial function must execute x i + c times the body of some loop instruction, where x i is the initial contents of some fixed input register, and c does not depend on the input values. This and other lower bounds were obtained without using the lambda calculus or systems with functional parameters (such as Gödel system T). If the conditional break and the decrement instructions are included in the Loop language, the ultimate obstinacy property does not hold. In this case, we use another approach for obtaining lower bounds. The unconditional break instruction does not avoid the obstinacy property; it is equivalent (in function and efficiency) to the conditional instruction if-then-else. The efficient implementation of other functions like step(x) (step(0) = 0; step(x) = 1 for x ≥ 1) and min(x, y) is also studied.

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Section: Conditional Instructions In Loop+breakmentioning

confidence: 98%

Section: Proofmentioning

confidence: 99%

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The efficiency of primitive recursive functions: A programmer's view

Matos¹

2015

Theoretical Computer Science

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“…A construction process has a finite number of initial obs from some constructed domain, and is a procedure for producing a sequence of obs from that domain, where each element in the sequence is either an initial ob or is obtained by a specified transformation from earlier obs in the sequence. (Construction processes are related to what HERMES calls rule systems in [5]. However, HERMES uses terminology appropriate to inference for all rule systems, which is misleading.)…”

Section: Q 3 Construction Processesmentioning

confidence: 99%

“…A nonstrict algorithm is one for which there are alternatives possible in constructing the output sequence for a given input. (A nonstrict algorithm is not an algorithm in the sense of [5].) I will call the simple algorithms being considered relaxed algorithms.…”

Section: Algorithmsmentioning

confidence: 99%

The Logic of Calculation

Kearns¹

1977

Mathematical Logic Qtrly

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Q 1. IntroductionIn this paper, I will give a precise characterization of (ordinary) combinatory logic in order to cont,rast it with the theory of combinatory logic with discriminators that was introduced in [6] and further developed in [7]. It will be shown that the primary difference between these two theories consists in the "Literal" way that combinatory logic with discriminators encodes functions and characterizes properties. This makes it possible to develop a logical system for combinatory: logic with discriminators in which syntactical properties can be expressed st,raightforwardly, avoiding the detour of G6del numbering. Q 2. Constructed domainsThe elements of constructed domains, also called algorithmic domains, are normally expressions; but I will use CURRY'S more abstract term 'ob' to designate them.A constructed domain contains a finite number of basic obs. It may also contain complex obs formed from a finite number of occurrences of previously constructed obs (possibly including a null ob), organized by a composition structure. A given composition structure is n-adic, uniting occurrences of n obs in a single ob. I n a complex ob there may be punctuating devices which are not counted as obs.An effectively determined ,subset of a constructed domain is a constructed domain.If 9 is a constructed domain, an n-adic function p is a transformation on 9 if q is effective, total, and takes n-tuples of obs in 9 to single obs in Q. An effective, total function from finite sequences of obs in 9 to single obs in 9 is also a transformation on 9. If the number of arguments of a transformation pl is not specified, it will be understood that q is binary. If pl is a binary transformation, then prl+l(X1, . . ., X,,,) is defined to be rp(q"(X,, . . ., &), X n + J . If d is a set of individuals and 9 is a constructed domain, then a binary relation R is a correspondence from € to 9 iff R is a one-one relation from 8 to a subset of 9. Q 3. Construction processesThe elements of formal languages and proofs in formal systems are produced by construction processes. A construction process has a finite number of initial obs from some constructed domain, and is a procedure for producing a sequence of obs from that domain, where each element in the sequence is either an initial ob or is obtained by a specified transformation from earlier obs in the sequence. (Construction processes are related to what HERMES calls rule systems in [5]. However, HERMES uses terminology appropriate to inference for all rule systems, which is misleading.) GI 5

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Second‐Order Intensional Logic

Cresswell¹

1972

Mathematical Logic Qtrly

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Enumerability · Decidability Computability

Cited by 30 publications

References 0 publications

The efficiency of primitive recursive functions: A programmer's view

The efficiency of primitive recursive functions: A programmer's view

The Logic of Calculation

Second‐Order Intensional Logic

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