Simplifications of the recursion scheme

Gladstone, M. D.

doi:10.2307/2272468

Cited by 23 publications

(10 citation statements)

References 2 publications

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“…By a result of Gladstone [6] we obtain all primitive recursive functions if we start from the base functions and close under composition and pure iteration (even allowing only one parameter).…”

Section: Primitive Recursive Functions In Cartesian Categoriesmentioning

confidence: 97%

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Partial Recursive Functions and Finality

Plotkin

2013

Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky

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Abstract. We seek universal categorical conditions ensuring the representability of all partial recursive functions. In the category Pfn of sets and partial functions, the natural numbers provide both an initial algebra and a final coalgebra for the functor 1 + −. We recount how finality yields closure of the partial functions on natural numbers under Kleene's µ-recursion scheme. Noting that Pfn is not cartesian, we then build on work of Paré and Román, obtaining weak initiality and finality conditions on natural numbers algebras in monoidal categories that ensure the (weak) representability of all partial recursive functions. We further obtain some positive results on strong representability. All these results adapt to Kleisli categories of cartesian categories with natural numbers algebras. However, in general, not all partial recursive functions need be strongly representable.

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Section: Primitive Recursive Functions In Cartesian Categoriesmentioning

confidence: 97%

“…By a result of Gladstone [6], all primitive recursive functions can be obtained starting from the base functions (zero, successor, and the projections) and closing under composition and pure iteration (even allowing only one parameter). We therefore have: Proposition 2.…”

Section: The Category Of Sets and Partial Functionsmentioning

confidence: 99%

Partial Recursive Functions and Finality

Plotkin

2013

Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky

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“…Related work: There are several alternative definitions of the primitive recursion scheme [22,15,16,21,10]. In some of these works, for instance [16,10], primitive recursion was replaced by pure iteration.…”

Section: Introductionmentioning

confidence: 99%

“…In some of these works, for instance [16,10], primitive recursion was replaced by pure iteration. Pure iteration is a linear scheme, in the sense that arguments of functions are used exactly once.…”

Section: Introductionmentioning

confidence: 99%

“…Pure iteration is a linear scheme, in the sense that arguments of functions are used exactly once. Gladstone [16] gave a definition of primitive recursion using the standard initial functions and composition, but replaced primitive recursion by pure iteration. Here we refine this definition by replacing also the initial functions (by linear initial functions) and the composition scheme (by a linear composition scheme).…”

Section: Introductionmentioning

confidence: 99%

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Linear Recursive Functions

Alves

Fernández

Florido

et al.

Rewriting, Computation and Proof

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Abstract. With the recent trend of analysing the process of computation through the linear logic looking glass, it is well understood that the ability to copy and erase data is essential in order to obtain a Turingcomplete computation model. However, erasing and copying do not need to be explicitly included in Turing-complete computation models: in this paper we show that the class of partial recursive functions that are syntactically linear (that is, partial recursive functions where no argument is erased or copied) is Turing-complete.

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Classes of One‐Argument Recursive Functions

Georgieva¹

1976

Mathematical Logic Qtrly

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Grrwdlngen d. Jlalh. CLASSES OF ONE-ARGUMENT RECURSIVE FTJNCTIONS by NADEJDA V. GEOR~IEVA in Sofia (Bulgaria) GRZEGORCZYK'S [l] sequence 8" (n 2 0) of increasing classes of functions has the following properties: 8 n + l contains gn, gn+l + &; each gn+l(x, x), where gn(x, y) is the function by which Bn is defined, majorizes the classe 8; of all the one argument functions in 8"; US" is the class of all primitive recursive functions.RITCHIE has proved in 121 that 82, n 2 2, is the smallest class of functions containing s(~), E(x), [x1j2] and fn(x, x) (ACKERMANN'S function), which is closed under addition, multiplication, composition and limited iteration.Here, we define and study a sequence Sn (n 2 0) of classes of one-argument functions the definition of which seems more simple than this of 8; and we prove that 3'n+1 = >l for n 2 2.In the first section below, we define the classes S,,, n 2 0, and develop some of their properties. In the second section, we consider $Po, S1 and di", more precisely. In the last section we establish the equality of 9'n+l and 8;+* for n 2 2. 1. The classes gnLet us setting f&) = S(x) = x + 1 and f r + l ( O ) = f r ( 1 ) , f r + l ( x + 1 ) = f r ( f r t l ( X ) ) > for r 5 0 .Here are including for reference two of monotonicity properties of the fr(x), which were established in [5]. We shall refer to them simply as [5].f r ( x ) < f r + l ( X ) , for T 2 0 and fr(z) < fr(x + l ) , for r 5 0 .Although we have got these properties of f&), now we shall prove three lemmas more. Lemma 1.1. For all x and r 2 2, 2fr(z) < fr(fr(x)).Proof. We shall proceed by induction on r, beginning at T = 2. 2 * f 2 ( z ) = 2(2Z + 3) < 2(2z + 3) + 3 = f z ( f 2 ( x ) ) . * f r + l ( O ) = * f r ( 1 ) < f r ( f r ( 1 ) ) = f r ( f r + l ( O ) ) < fr+x(fr+l(O))and 2 * f r t l ( x + 1) = 2 * f r ( f r + 1 ( 2 ) ) < f r ( f r ( f r t l ( X ) ) ) = f A f r t l ( x + 1)) < f r t l ( f r + l ( x + 1)).Assume the result holds for r. Using [5] we see that:Lemma 1.2. For each k 2 0 and r 2 1 there is an integer xo, 5 w h that fr(s + k ) < Proof. We have x + k < f2(xk -I) 5 frtl(xkl), for x > 3k -1 andThe function f(x) is defined from g(z) and j(s) by limited iteration, if f(0) = 0, j(x) for all x. This wiU be written sometimes for short < fr+l(zk), for all z 2 20. therefore fi(z + k ) < f r ( f r + l ( xk -1)) = f r t i ( xk).f(x + 1) = g(f(z)) and f ( x ) as m) = Sz(O), f(4 5 i(4. and f:' 2(1) = f r ( f r + l ( X ) ) = f r + l (~ + 1)-Lemma 1.3. For r 2 0 and x 2 0, g+'(l) = f r + l ( x ) .Proof. The definition of the function fr(x) and the induction on x give f r ( l ) = f , + l ( 0 )

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Simplifications of the recursion scheme

Cited by 23 publications

References 2 publications

Partial Recursive Functions and Finality

Partial Recursive Functions and Finality

Linear Recursive Functions

Classes of One‐Argument Recursive Functions

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