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The study of categories that abstract the structural properties of relations has been extensively developed over the years, resulting in a rich and diverse body of work. This paper strives to provide a modern presentation of these ``categories for relations'', including their enriched version, further showing how they arise as Kleisli categories of symmetric monoidal monads. The resulting taxonomy aims at bringing clarity and organisation to the many related concepts and frameworks occurring in the literature.

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Non-well-founded material sets have been modelled in Martin-Löf type theory by Lindström using setoids. In this paper we construct models of non-wellfounded material sets in Homotopy Type Theory (HoTT) where equality is interpreted as the identity type. The first model satisfies Scott's Anti-Foundation Axiom (SAFA) and dualises the construction of iterative sets. The second model satisfies Aczel's Anti-Foundation Axiom (AFA), and is constructed by adaption of Aczel-Mendler's terminal coalgebra theorem to type theory, which requires propositional resizing. In an bid to extend coalgebraic theory and anti-foundation axioms to higher type levels, we formulate generalisations of AFA and SAFA, and construct a hierarchy of models which satisfies the SAFA generalisations. These generalisations build on the framework of Univalent Material Set Theory, previously developed by two of the authors. Since the model constructions are based on M-types, the paper also includes a characterisation of the identity type of M-types as indexed M-types. Our results are formalised in the proof-assistant Agda.

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Digital circuits, despite having been studied for nearly a century and used at scale for about half that time, have until recently evaded a fully compositional theoretical in which arbitrary circuits may be freely composed together without consulting their internals. Recent work remedied this theoretical shortcoming by showing how digital circuits can be presented compositionally as morphisms in a freely generated symmetric traced category. However, this was done informally; in this paper we refine and expand the previous work in several ways, culminating in the presentation of three sound and complete semantics for digital circuits: denotational, operational and algebraic. For the denotational semantics, we establish a correspondence between stream functions with certain properties and circuits constructed syntactically. For the operational semantics, we present the reductions required to model how a circuit processes a value, including the addition of a new reduction for eliminating non-delay-guarded feedback; this leads to an adequate notion of observational equivalence for digital circuits. Finally, we define a new family of equations for translating circuits into bisimilar circuits of a 'normal form', leading to a complete algebraic semantics for sequential circuits.

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We study the expressive power of first-order logic with counting quantifiers, especially the $k$-variable and quantifier-rank-$q$ fragment, using homomorphism indistinguishability. Recently, Dawar, Jakl, and Reggio~(2021) proved that two graphs satisfy the same $k$-variable and quantifier-rank-$q$ sentences if and only if they are homomorphism indistinguishable over the class of graphs admitting a $k$-pebble forest cover of depth $q$. After reproving this result using elementary means, we provide a graph-theoretic analysis of this graph class. This allows us to separate it from the intersection of the class of all graphs of treewidth at most $k-1$ and the class of all graphs of treedepth at most $q$, provided that $q$ is sufficiently larger than $k$. We are able to lift this separation to a (semantic) separation of the respective homomorphism indistinguishability relations. We do this by showing that the graph classes of all graphs of treedepth at most $q$ and of graphs admitting a $k$-pebble forest cover of depth $q$ are homomorphism distinguishing closed, as conjectured by Roberson~(2022). In order to prove Roberson's conjecture for the class of graphs admitting a $k$-pebble forest cover of depth $q$ we characterise the class in terms of a monotone Cops-and-Robber game.The crux is to prove that if Cop has a winning strategy then Cop also has a winning strategy that is monotone.To that end, we show how to transform Cop's winning strategy into a pre-tree-decomposition, […]

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Timed automata (TAs) are an extension of finite automata that can measure and react to the passage of time, providing the ability to handle real-time constraints using clocks. In 2009, Franck Cassez showed that the timed opacity problem, where an attacker can observe some actions with their timestamps and attempts to deduce information, is undecidable for TAs. Moreover, he showed that the undecidability holds even for subclasses such as event-recording automata. In this article, we consider the same definition of opacity, by restricting either the system or the attacker. Our first contribution is to prove the inter-reducibility of two variants of opacity: full opacity (for which the observations should be the same regardless of the visit of a private location) and weak opacity (for which it suffices that the attacker cannot deduce whether the private location was visited, but for which it is harmless to deduce that it was not visited); we also prove further results including a connection with timed language inclusion. Our second contribution is to study opacity for several subclasses of TAs: with restrictions on the number of clocks, the number of actions, the nature of time, or a new subclass called observable event-recording automata. We show that opacity is mostly decidable in these cases, except for one-action TAs and for one-clock TAs with $ε$-transitions, for which undecidability remains. Our third (and arguably main) contribution is to propose a new definition of […]

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