arXiv subjects

Cameron Calk

Publications and source records attributed to Cameron Calk.

6 recordsLinked to original sources

arXiv2025

Stone Duality Proofs for Colorless Distributed Computability Theorems

We introduce a new topological encoding of executions of round-based, full-information distributed protocols via spectral spaces. Such protocols constitute a model of distributed computations which are functorially presented and englobe message adversaries. We give a characterization of the solvability of colorless tasks against compact adversaries. Colorless tasks are an important class of distributed tasks, examples thereof including the ubiquitous agreement tasks. Therefore, our result is a significant step toward unifying topological methods in distributed computing. The main insight of this work is in considering global states obtained after finite executions of a distributed protocol not as abstract simplicial complexes as was previously done, but as spectral spaces, considering the Alexandrov topology on the associated face posets. Given an adversary $\mathcal{M}$ with a set of inputs $\mathcal{I}$, we define a limit object $\Pi^\infty_{\mathcal{M}}(\mathcal{I})$ by a projective limit in the category of spectral spaces. This encodes all distributed information about the adversary, allowing us to derive a new distributed computability theorem using Stone duality: there exists an algorithm solving a colorless task $(\mathcal{I},\mathcal{O},\Delta)$ against the compact adversary $\mathcal{M}$ if and only if there exists a spectral map $\Pi^\infty_{\mathcal{M}}(\mathcal{I})\rightarrow\mathcal{O}$ compatible with $\Delta$. From this characterization, we derive the known colorless computability theorems for (colored or uncolored) Iterated Immediate Snapshot. Quite surprisingly, colored and uncolored models have the same distributed computability power, i.e. they solve the same tasks. Our new proofs give topological reasons for this equivalence, previously known through algorithmic reductions.

cs.DC↗

arXiv2023

Higher Catoids, Higher Quantales and their Correspondences

We introduce $\omega$-catoids as generalisations of (strict) $\omega$-categories and in particular the higher path categories generated by computads or polygraphs in higher-dimensional rewriting. We also introduce $\omega$-quantales that generalise the $\omega$-Kleene algebras recently proposed for algebraic coherence proofs in higher-dimensional rewriting. We then establish correspondences between $\omega$-catoids and convolution $\omega$-quantales. These are related to J\'onsson-Tarski-style dualities between relational structures and lattices with operators. We extend these correspondences to $(\omega,p)$-catoids, catoids with a groupoid structure above some dimension, and convolution $(\omega,p)$-quantales, using Dedekind quantales above some dimension to capture homotopic constructions and proofs in higher-dimensional rewriting. We also specialise them to finitely decomposable $(\omega, p)$-catoids, an appropriate setting for defining $(\omega, p)$-semirings and $(\omega, p)$-Kleene algebras. These constructions support the systematic development and justification of $\omega$-Kleene algebra and $\omega$-quantale axioms, improving on the recent approach mentioned, where axioms for $\omega$-Kleene algebras have been introduced in an ad hoc fashion.

cs.LO↗

arXiv2023

Persistent homology of partially ordered spaces

In this work, we explore links between natural homology and persistent homology for the classification of directed spaces. The former is an algebraic invariant of directed spaces, a semantic model of concurrent programs. The latter was developed in the context of topological data analysis, in which topological properties of point-cloud data sets are extracted while eliminating noise. In both approaches, the evolution homological properties are tracked through a sequence of inclusions of usual topological spaces. Exploiting this similarity, we show that natural homology may be considered a persistence object, and may be calculated as a colimit of uni-dimensional persistent homologies along traces. Finally, we suggest further links and avenues of future work in this direction.

math.AT↗

arXiv2020

Algebraic coherent confluence and higher globular Kleene algebras

We extend the formalisation of confluence results in Kleene algebras to a formalisation of coherent confluence proofs. For this, we introduce the structure of higher globular Kleene algebra, a higher-dimensional generalisation of modal and concurrent Kleene algebra. We calculate a coherent Church-Rosser theorem and a coherent Newman's lemma in higher Kleene algebras by equational reasoning. We instantiate these results in the context of higher rewriting systems modelled by polygraphs.

cs.LO↗

arXiv2020

Beyond formulas-as-cographs: an extension of Boolean logic to arbitrary graphs

We propose a graph-based extension of Boolean logic called Boolean Graph Logic (BGL). Construing formula trees as the cotrees of cographs, we may state semantic notions such as evaluation and entailment in purely graph-theoretic terms, whence we recover the definition of BGL. Naturally, it is conservative over usual Boolean logic. Our contributions are the following: (1) We give a natural semantics of BGL based on Boolean relations, i.e. it is a multivalued semantics, and show adequacy of this semantics for the corresponding notions of entailment. (2) We show that the complexity of evaluation is NP-complete for arbitrary graphs (as opposed to ALOGTIME-complete for formulas), while entailment is $\Pi^p_2$-complete (as opposed to coNP-complete for formulas). (3) We give a 'recursive' algorithm for evaluation by induction on the modular decomposition of graphs. (Though this is not polynomial-time, cf. point (2) above). (4) We characterise evaluation in a game-theoretic setting, in terms of both static and sequentical strategies, extending the classical notion of positional game forms beyond cographs. (5) We give an axiomatisation of BGL, inspired by deep-inference proof theory, and show soundness and completeness for the corresponding notions of entailment. One particular feature of the graph-theoretic setting is that it escapes certain no-go theorems such as a recent result of Das and Strassburger, that there is no linear axiomatisation of the linear fragment of Boolean logic (equivalently the multiplicative fragment of Japaridze's Computability Logic or Blass' game semantics for Mutliplicative Linear Logic).

cs.LO↗

arXiv2018

Time-reversal homotopical properties of concurrent systems

Directed topology was introduced as a model of concurrent programs, where the flow of time is described by distinguishing certain paths in the topological space representing such a program. Algebraic invariants which respect this directedness have been introduced to classify directed spaces. In this work we study the properties of such invariants with respect to the reversal of the flow of time in directed spaces. Known invariants, natural homotopy and homology, have been shown to be unchanged under this time-reversal. We show that these can be equipped with additional algebraic structure witnessing this reversal. Specifically, when applied to a directed space and to its reversal, we show that these refined invariants yield dual objects. We further refine natural homotopy by introducing a notion of relative directed homotopy and showing the existence of a long exact sequence of natural homotopy systems.

math.CT↗