Cardinality of relations with applications

Berghammer, Rudolf; Danilenko, Nikita; Höfner, P.; Stucke, Insa

doi:10.1016/j.disc.2016.06.019

Cited by 6 publications

(4 citation statements)

References 21 publications

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“…There are several directions of extension. First, relation algebras can be extended to deal with cardinalities of relations [32,10,5]; for a path this would simply be the number of edges it contains. This facilitates, for example, proofs about the complexity of graph algorithms.…”

Section: Discussionmentioning

confidence: 99%

Relational Characterisations of Paths

Berghammer¹,

Furusawa²,

Guttmann³

et al. 2018

Preprint

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Binary relations are one of the standard ways to encode, characterise and reason about graphs. Relation algebras provide equational axioms for a large fragment of the calculus of binary relations. Although relations are standard tools in many areas of mathematics and computing, researchers usually fall back to point-wise reasoning when it comes to arguments about paths in a graph. We present a purely algebraic way to specify different kinds of paths in relation algebras. We study the relationship between paths with a designated root vertex and paths without such a vertex. Since we stay in first-order logic this development helps with mechanising proofs. To demonstrate the applicability of the algebraic framework we verify the correctness of three basic graph algorithms. All results of this paper are formally verified using the interactive proof assistant Isabelle/HOL.

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Section: Discussionmentioning

confidence: 99%

Relational Characterisations of Paths

Berghammer¹,

Furusawa²,

Guttmann³

et al. 2018

Preprint

Self Cite

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“…A dual characterisation can be obtained for total morphisms, i.e., that R : X → Y is total iff it holds |R T p| ≥ 1 for every point p : X → 1. Both statements are proved in [2]. Furthermore, in the context of graph theory, if R : X → X is an adjacency relation, the second property means that the maximum (in-)degree in the modelled graph is smaller than k. This theorem translates as follows in Coq:…”

Section: Decomposition As a Union Of Univalent Morphismsmentioning

confidence: 94%

“…We follow the presentation from [2]. A forest is an acyclic directed graph such that each vertex has at most one parent.…”

Section: Number Of Edges In Treesmentioning

confidence: 99%

Cardinalities of Finite Relations in Coq

Brunet¹,

Pous²,

Stucke³

2016

Interactive Theorem Proving

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In this paper we present an extension of a Coq library for relation algebras and related algebraic structures. So far, the library did not provide any tools about the cardinalities of relations. Thus we add an algebraic axiomatisation of cardinalities. Its point-free nature makes it possible to reason about cardinal purely algebraically, which is well-suited for mechanisation. We present several applications, in the area of graph theory and program verification.

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“…For future work, we aim to provide a more comprehensive construction of the base PSL, by mining previously unseen code structures. capturing the semantic details, through syntactic interpretation that a single variable is being used to manipulate information, rather than a collection of variables, could be critical in understanding cardinality constraints of a particular problem [6].…”

Section: Program-derived Semantic Graphmentioning

confidence: 99%

Software Language Comprehension using a Program-Derived Semantics Graph

Iyer,

Sun,

Wang

et al. 2020

Preprint

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Traditional code transformation structures, such as an abstract syntax tree, may have limitations in their ability to extract semantic meaning from code. Others have begun to work on this issue, such as the state-of-the-art Aroma system and its simplified parse tree (SPT). Continuing this research direction, we present a new graphical structure to capture semantics from code using what we refer to as a program-derived semantic graph (PSG). The principle behind the PSG is to provide a single structure that can capture program semantics at many levels of granularity. Thus, the PSG is hierarchical in nature. Moreover, because the PSG may have cycles due to dependencies in semantic layers, it is a graph, not a tree. In this paper, we describe the PSG and its fundamental structural differences to the Aroma's SPT. Although our work in the PSG is in its infancy, our early results indicate it is a promising new research direction to explore to automatically extract program semantics.

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Cardinality of relations with applications

Cited by 6 publications

References 21 publications

Relational Characterisations of Paths

Relational Characterisations of Paths

Cardinalities of Finite Relations in Coq

Software Language Comprehension using a Program-Derived Semantics Graph

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