Related papers: Kleene Algebra
For every partial combinatory algebra (pca), we define a hierarchy of extensionality relations using ordinals. We investigate the closure ordinals of pca's, i.e. the smallest ordinals where these relations become equal. We show that the…
Right-linear (or left-linear) grammars are a well-known class of context-free grammars computing just the regular languages. They may naturally be written as expressions with (least) fixed points but with products restricted to letters as…
Describing complex phenomena by means of cellular automata (CA) has shown to be a very effective approach in pure and applied sciences. In fact, the number of published papers concerning this topic has tremendously increased over the last…
What is Sequence Algebra? This is a question that any teacher or student of mathematics or computer science can engage with. Sequences are in Calculus, Combinatorics, Statistics and Computation. They are foundational, a step up from number…
$\omega$-regular languages are a natural extension of the regular languages to the setting of infinite words. Likewise, they are recognised by a host of automata models, one of the most important being Alternating Parity Automata (APAs), a…
We introduce a class of algebras that can be used as recognisers for regular tree languages. We show that it is the only such class that forms a pseudo-variety and we prove the existence of syntactic algebras. Finally, we give a more…
We investigate completions of partial combinatory algebras (pcas), in particular of Kleene's second model $\mathcal{K}_2$ and generalizations thereof. We consider weak and strong notions of embeddability and completion that have been…
Centered kernel alignment (CKA) is a popular metric for comparing representations, determining equivalence of networks, and neuroscience research. However, CKA does not account for the underlying manifold and relies on numerous heuristics…
(1) Let 1\leq k\leq \omega. Call an atom structure \alpha weakly k neat representable, the term algebra is in \RCA_n\cap \Nr_n\CA_{n+k}, but the complex algebra is not representable. Call an atom structure neat if there is an atomic algebra…
In our previous articles, we have presented a class of endomorphisms of the Cuntz algebras which are defined by polynomials of canonical generators and their conjugates. We showed the classification of some case under unitary equivalence by…
It is well-known that Abstract State Machines (ASMs) can simulate "step-by-step" any type of machines (Turing machines, RAMs, etc.). We aim to overcome two facts: 1) simulation is not identification, 2) the ASMs simulating machines of some…
Pomset automata are an operational model of weak bi-Kleene algebra, which describes programs that can fork an execution into parallel threads, upon completion of which execution can join to resume as a single thread. We characterize a…
In this work we develop an algebraic theory of linear recurrence equations and systems with constant coefficients and reflection. We obtain explicit solutions and the Green's functions associated to different problems under general linear…
Keyphrases provide semantic metadata that summarize and characterize documents. This paper describes Kea, an algorithm for automatically extracting keyphrases from text. Kea identifies candidate keyphrases using lexical methods, calculates…
In this paper, we show that the equational theory of relational Kleene algebra with the \emph{graph loop} operator (a.k.a.~\emph{fixset}) is \textsc{PSpace}-complete. Here, the graph loop is the unary operator that restricts a binary…
A theorem prover without an extensive library is much less useful to its potential users. Algebra, the study of algebraic structures, is a core component of such libraries. Algebraic theories also are themselves structured, the study of…
Currently it is widely accepted that the language of science is mathematics. This book explores an alternative idea where the future of science is based on the language of algorithms and programs. How such a language can actually be…
Despite the impressive performance of large language models (LLMs) pretrained on vast knowledge corpora, advancing their knowledge manipulation-the ability to effectively recall, reason, and transfer relevant knowledge-remains challenging.…
For every regular cardinal kappa there exists a simple complete Boolean algebra with kappa generators.
This research started with an algebra for reasoning about rely/guarantee concurrency for a shared memory model. The approach taken led to a more abstract algebra of atomic steps, in which atomic steps synchronise (rather than interleave)…