On arithmetical first-order theories allowing encoding and decoding of lists

Cegielski, Patrick; Richard, Denis

doi:10.1016/s0304-3975(97)00281-8

Cited by 13 publications

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“…The generalization of Cantor's pairing function is mentioned in two relatively recent papers (Cegielski and Richard 1999;Lisi 2007) with a possible attribution in Cegielski and Richard (1999) to Skolem as a first reference.…”

Section: The Generalized Cantor K-tupling Bijectionmentioning

confidence: 99%

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Compact serialization of Prolog terms (with catalan skeletons, cantor tupling and Gödel numberings)

Tarau¹

2013

Theory and Practice of Logic Programming

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We describe a compact serialization algorithm mapping Prolog terms to natural numbers of bit-sizes proportional to the memory representation of the terms. The algorithm is a ‘no bit lost’ bijection, as it associates to each Prolog term a unique natural number and each natural number corresponds to a unique syntactically well-formed term.To avoid an exponential explosion resulting from bijections mapping term trees to natural numbers, we separate the symbol content and the syntactic skeleton of a term that we serialize compactly using a ranking algorithm for Catalan families.A novel algorithm for the generalized Cantor bijection between ${\mathbb{N}$ and ${\mathbb{N}$k is used in the process of assigning polynomially bounded Gödel numberings to various data objects involved in the translation.

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Section: The Generalized Cantor K-tupling Bijectionmentioning

confidence: 99%

“…It is easy to see that the generalized Cantor n-tupling function defined by equation 1is a polynomial of degree n in its arguments, and a conjecture, attributed in Cegielski and Richard (1999) to Rudolf Fueter (1923), states that it is the only one, up to a permutation of the arguments.…”

Section: The Generalized Cantor K-tupling Bijectionmentioning

confidence: 99%

Compact serialization of Prolog terms (with catalan skeletons, cantor tupling and Gödel numberings)

Tarau¹

2013

Theory and Practice of Logic Programming

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“…We design a generic mechanism to derive pairing functions by combining the data type transformation operation as with the bsplit and bmerge functions that apply a characteristic function encoded as a list of bits. In this case, the characteristic functions given by cycle [0] or cycle [1] would trigger an infinite search for a non-existing first 1 or 0 in bsplit and bmerge.…”

Section: Defining Pairing Bijections Genericallymentioning

confidence: 99%

“…* InfPair> map (bunpair 2) [0..10] [(0,0),(1,0),(0,1),(1,1),(2,0),(3,0), (2,1),(3,1),(0,2),(1,2),(0,3)] * InfPair> map (bpair 2) it [0, 1,2,3,4,5,6,7,8,9,10] We conclude with a similar result for lists: Proof: Observe that a characteristic function corresponding to a subset of N containing an infinite bloc of 0 or 1 digits necessarily ends with the bloc. Therefore, by erasing the bloc we can put such functions in a bijection with a finite subset of N. Given that there are only a countable number of finite subsets of N, the cardinality of the set of the remaining subsets' characteristic functions is 2 N .…”

Section: Defining Pairing Bijections Genericallymentioning

confidence: 99%

“…There are about 19200 Google documents referring to the original "Cantor pairing function" among which we mention the surprising result that, together with the successor function it defines a decidable subset of arithmetic [2]. An extensive study of various pairing functions and their computational properties is presented in [1], [14]. They are also related to 2D-space filling curves (Z-order, Gray-code and Hilbert curves) [4]- [7].…”

Section: Related Workmentioning

confidence: 99%

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Two mechanisms for generating infinite families of pairing bijections

Tarau

2013

Proceedings of the 2013 Research in Adaptive and Convergent Systems

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We describe two general mechanisms for producing pairing bijections (bijective functions defined from N 2 → N). The first mechanism, using n-adic valuations results in parameterized algorithms generating a countable family of distinct pairing bijections. The second mechanism, using characteristic functions of subsets of N provides 2 N distinct pairing bijections. Mechanisms to combine such pairing functions and their application to generate families of permutations of N are also described. The paper uses a small subset of the functional language Haskell to provide type checked executable specifications of all the functions defined in a literate programming style. The selfcontained Haskell code extracted from the paper is available at http: //logic.cse.unt.edu/tarau/research/2012/infpair.hs .

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The theory of hereditarily bounded sets

Jeřábek

2022

Mathematical Logic Qtrly

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We show that for any k∈ω$k\in \omega$, the structure false⟨Hk,∈false⟩$\langle H_k,{\in }\rangle$ of sets that are hereditarily of size at most k is decidable. We provide a transparent complete axiomatization of its theory, a quantifier elimination result, and tight bounds on its computational complexity. This stands in stark contrast to the structure Vω=⋃kHk$V_\omega =\bigcup _kH_k$ of hereditarily finite sets, which is well known to be bi‐interpretable with the standard model of arithmetic false⟨double-struckN,+,·false⟩$\langle \mathbb {N},+,\cdot \rangle$.

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On arithmetical first-order theories allowing encoding and decoding of lists

Cited by 13 publications

References 13 publications

Compact serialization of Prolog terms (with catalan skeletons, cantor tupling and Gödel numberings)

Compact serialization of Prolog terms (with catalan skeletons, cantor tupling and Gödel numberings)

Two mechanisms for generating infinite families of pairing bijections

The theory of hereditarily bounded sets

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