Tests and Proofs for Enumerative Combinatorics

Dubois, Catherine; Giorgetti, Alain; Genestier, Richard

doi:10.1007/978-3-319-41135-4_4

Cited by 8 publications

(6 citation statements)

References 18 publications

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“…These works use different mathematical objects depending on the system. We use combinatorial maps in this work, but other related constructions are, for example, root maps defined in terms of permutations (Dubois et al 2016) and hypermaps (Dufourd 2009;Dufourd and Puitg 2000;Gonthier 2008), among others. In particular, one can see that the notion of a hypermap is a generalisation of a combinatorial map for undirected finite graphs.…”

Section: Related Workmentioning

confidence: 99%

On planarity of graphs in homotopy type theory

Prieto-Cubides,

Gylterud

2024

Math. Struct. Comp. Sci.

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In this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces and maps of graphs embedded in the sphere, in homotopy type theory (HoTT). This allows us to provide an elementary characterisation of planarity for locally directed finite and connected multigraphs that takes inspiration from topological graph theory, particularly from combinatorial embeddings of graphs into surfaces. A graph is planar if it has a map and an outer face with which any walk in the embedded graph is walk-homotopic to another. A result is that this type of planar maps forms a homotopy set for a graph. As a way to construct examples of planar graphs inductively, extensions of planar maps are introduced. We formalise the essential parts of this work in the proof assistant Agda with support for HoTT.

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Section: Related Workmentioning

confidence: 99%

On planarity of graphs in homotopy type theory

Prieto-Cubides,

Gylterud

2024

Math. Struct. Comp. Sci.

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“…It has rst been used to check properties of functional languages, as exemplied by SmallCheck in Haskell [50]. Then, it has been adapted to several proof assistants, e.g., to Isabelle in Quickcheck [9] and to Coq, in an extension of QuickChick named CUT (Coq Unit Testing) [16]. In a former work we have initiated a BET tool for WhyML [18].…”

Section: Random and Enumerative Testing Toolsmentioning

confidence: 99%

Towards random and enumerative testing for OCaml and WhyML properties

2022

Self Cite

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Deductive program verication greatly improves software quality, but proving formal specications is dicult, and this activity can only be partially automated. It is therefore relevant to supplement deductive verication tools, such as Why3, with the ability to test the properties to be veried. We present a methodological study and a prototype for the random and enumerative testing of properties written either in the Why3 input language WhyML or in the OCaml programming language used by Why3 to run programs written in WhyML. An originality is that we propose enumerative testing based on data generators themselves written in WhyML and formally veried with Why3. Another specicity is that the development eort is reduced by exploiting Why3's extraction mechanism to OCaml and an existing random testing tool for OCaml. These design choices are applied in a propotypal implementation of a tool, called AutoCheck. The prototype and the paper are designed with simplicity and usability in mind, in order to make them accessible to the widest audience. Starting from the most elementary cases, a tutorial illustrates the implemented features with many examples presented in increasing complexity order.

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“…These works use different mathematical objects depending on the system. We use combinatorial maps in this work, but other related constructions are, for example, root maps defined in terms of permutations by Dubois et al [18], and hypermaps by Dufourd and Gonthier [19,20,23], among others. In particular, one can see that the notion of a hypermap is a generalisation of a combinatorial map for undirected finite graphs.…”

Section: Related Workmentioning

confidence: 99%

On Planarity of Graphs in Homotopy Type Theory

Prieto-Cubides¹,

Gylterud²

2021

Preprint

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In this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces, and maps of graphs embedded in the sphere, in homotopy type theory. This allows us to provide an elementary characterisation of planarity for locally directed finite and connected multigraphs that takes inspiration from topological graph theory, particularly from combinatorial embeddings of graphs into surfaces. A graph is planar if it has a map and an outer face with which any walk in the embedded graph is walk-homotopic to another. A result is that this type of planar maps forms a homotopy set for a graph. As a way to construct examples of planar graphs inductively, extensions of planar maps are introduced. We formalise the essential parts of this work in the proof assistant Agda with support for homotopy type theory.CCS Concepts: • Theory of computation → Constructive mathematics; Type theory; • Mathematics of computing → Graphs and surfaces.

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Tests and Proofs for Enumerative Combinatorics

Cited by 8 publications

References 18 publications

On planarity of graphs in homotopy type theory

On planarity of graphs in homotopy type theory

Towards random and enumerative testing for OCaml and WhyML properties

On Planarity of Graphs in Homotopy Type Theory

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