We present an adapted construction of algebraic circuits over the reals introduced by Cucker and Meer to arbitrary infinite integral domains and generalize the $\mathrm{AC}_{\mathbb{R}}$ and $\mathrm{NC}_{\mathbb{R}}^{}$ classes for this setting. We give a theorem in the style of Immerman’s theorem which shows that for these adapted formalisms, sets decided by circuits of constant depth and polynomial size are the same as sets definable by a suitable adaptation of first-order logic. Additionally, we discuss a generalization of the guarded predicative logic by Durand, Haak and Vollmer, and we show characterizations for the $\mathrm{AC}_{R}$ and $\mathrm{NC}_R^{}$ hierarchy. Those generalizations apply to the Boolean $\mathrm{AC}$ and $\mathrm{NC}$ hierarchies as well. Furthermore, we introduce a formalism to be able to compare some of the aforementioned complexity classes with different underlying integral domains.

Convergent rewriting systems on algebraic structures give methods to solve decision problems, to prove coherence results, and to compute homological invariants. These methods are based on higher-dimensional extensions of the critical branching lemma that proves local confluence from confluence of the critical branchings. The analysis of local confluence of rewriting systems on algebraic structures, such as groups or linear algebras, is complicated because of the underlying algebraic axioms. This article introduces the structure of algebraic polygraph modulo that formalizes the interaction between the rules of an algebraic rewriting system and the inherent algebraic axioms, and we show a critical branching lemma for algebraic polygraphs. We deduce a critical branching lemma for rewriting systems on algebraic models whose axioms are specified by convergent modulo rewriting systems. We illustrate our constructions for string, linear, and group rewriting systems.

We present Geometry of Interaction (GoI) models for Multiplicative Polarized Linear Logic, MLLP, which is the multiplicative fragment of Olivier Laurent's Polarized Linear Logic. This is done by uniformly adding multi-points to various categorical models of GoI. Multi-points are shown to play an essential role in semantically characterizing the dynamics of proof networks in polarized proof theory. For example, they permit us to characterize the key feature of polarization, focusing, as well as being fundamental to our construction of concrete polarized GoI models.

Our approach to polarized GoI involves following two independent studies, based on different categorical perspectives of GoI:

1. (i) Inspired by the work of Abramsky, Haghverdi and Scott, a polarized GoI situation is defined in which multi-points are added to a traced monoidal category equipped with a reflexive object U. Using this framework, categorical versions of Girard's execution formula are defined, as well as the GoI interpretation of MLLP proofs. Running the execution formula is shown to characterize the focusing property (and thus polarities) as well as the dynamics of cut elimination.

2. (ii) The Int construction of Joyal–Street–Verity is another fundamental categorical structure for modelling GoI. Here, we investigate it in a multi-pointed setting. Our presentation yields a compact version of Hamano–Scott's polarized categories, and thus denotational models of MLLP. These arise from a contravariant duality between monoidal categories of positive and negative objects, along with an appropriate bimodule structure (representing ‘non-focused proofs’) between them.

Finally, as a special case of (ii) above, a compact model of MLLP is also presented based on Rel (the category of sets and relations) equipped with multi-points.

In the article below, Theorem 3.4 requires the additional assumption that A is Kan as well. Indeed, the inductive proof as given only shows that if W(f)<α is a Kan complex, then W(f)<α+1 → A is a Kan fibration.

It has been known that categorical interpretations of dependent type theory with Σ- and Id-types induce weak factorization systems. When one has a weak factorization system $({\cal L},{\cal R})$ on a category $\mathbb{C}$ in hand, it is then natural to ask whether or not $({\cal L},{\cal R})$ harbors an interpretation of dependent type theory with Σ- and Id- (and possibly Π-) types. Using the framework of display map categories to phrase this question more precisely, one would ask whether or not there exists a class ${\cal D}$ of morphisms of $\mathbb{C}$ such that the retract closure of ${\cal D}$ is the class ${\cal R}$ and the pair $(\mathbb{C},{\cal D})$ forms a display map category modeling Σ- and Id- (and possibly Π-) types. In this paper, we show, with the hypothesis that $\cal{C}$ is Cauchy complete, that there exists such a class $\cal{D}$ if and only if $(\mathbb{C},\cal{R})$itself forms a display map category modeling Σ- and Id- (and possibly Π-) types. Thus, we reduce the search space of our original question from a potentially proper class to a singleton.

We extend the simply-typed lambda-calculus with a mechanism for dynamic and incremental rebinding of code. Fragments of open code which can be dynamically rebound are values. Differently from standard static binding, which is done on a positional basis, rebinding is done on a nominal basis, that is, free variables in open code are associated with names which do not obey α-equivalence. Moreover, rebinding is incremental, that is, just a subset of names can be rebound, making possible code specialization, and rebinding can even introduce new names. Finally, rebindings, which are associations between names and terms, are first-class values, and can be manipulated by operators such as overriding and renaming. We define a type system in which the type for a rebinding, in addition to specify an association between names and types (similarly to record types), is also annotated. The annotation says whether or not the domain of the rebinding having this type may contain more names than the ones that are specified in the type. We show soundness of the type system.

The Coalgebraic Logic seminar took place between the 6th and 9th of December 2009 at the Leibniz Forschungszentrum Schloss Dagstuhl. The event was very well received, with more than 35 scientists from Europe, the United States, Canada and China attending, and with more than thirty presentations during the three days. Given the unforeseen enthusiasm of the meeting's reception, we thought that it would be a good idea for us to put together a special journal issue dedicated to this topic. We were also encouraged by the positive response we received from Guiseppe Longo, who indicated his interest in a special issue for Mathematical Structures in Computer Science, and here we are.

This paper presents a bisimulation-based method for establishing the soundness of equations between terms constructed using operations whose semantics are specified by rules in the GSOS format of Bloom, Istrail and Meyer. The method is inspired by de Simone's FH-bisimilarity and uses transition rules as schematic transitions in a bisimulation-like relation between open terms. The soundness of the method is proved and examples showing its applicability are provided. The proposed bisimulation-based proof method is incomplete, but we do offer some completeness results for restricted classes of GSOS specifications. An extension of the proof method to the setting of GSOS languages with predicates is also offered.

Ishihara's tricks have proven to be a highly useful tool in constructive mathematics, since they enable one to make decisions that seem, on first glance, impossible. They do, however, require that one deals with strongly extensional mappings on complete spaces. In this short note, we show how these assumptions can be weakened. Furthermore, we apply these generalizations to give a partial answer to the question, whether constructively we can rule out the existence of injections from Baire space into the natural numbers, to a version of Riemann's per mutation theorem and to a classification problem about cardinalities in constructive reverse mathematics.

Non-strict don't care functions, whose foremost representative is the ubiquitous if_then_else, play an essential role in computer science. As far as the semantics is concerned, they can be modelled by their totalizations with the appropriate use of elements representing undefinedness, as D. Scott has shown in his denotational approach. The situation is not so straightforward when we consider non-strict functions in the context of an algebraic framework; this point is discussed in the last section, where we explore the relationship between non-strict don't care and total algebras. The central part of this paper, after presenting the basic properties of the category of non-strict algebras, is an investigation of conditional algebraic specifications. It is shown that non-strict conditional specifications are equivalent to disjunctive specifications, and necessary and sufficient conditions for the existence of initial models are given. Since non-strict don't care specifications generalize both the total and the partial case, it is shown how the results for initiality can be obtained as specializations.

We introduce a strict version of the concept of a load/store instruction set architecture in the setting of Maurer machines. We take the view that transformations on the states of a Maurer machine are achieved by applying threads as considered in thread algebra to the Maurer machine. We study how the transformations on the states of the main memory of a strict load/store instruction set architecture that can be achieved by applying threads depend on the operating unit size, the cardinality of the instruction set and the maximal number of states of the threads.

We consider the problem of the unification modulo an equational theory associativity and commutativity (ACh), which consists of a function symbol h that is homomorphic over an associative–commutative operator +. Since the unification modulo ACh theory is undecidable, we define a variant of the problem called bounded ACh unification. In this bounded version of ACh unification, we essentially bound the number of times h can be applied to a term recursively and only allow solutions that satisfy this bound. There is no bound on the number of occurrences of h in a term, and the + symbol can be applied an unlimited number of times. We give inference rules for solving the bounded version of the problem and prove that the rules are sound, complete, and terminating. We have implemented the algorithm in Maude and give experimental results. We argue that this algorithm is useful in cryptographic protocol analysis.

It is well established that equational algebraic theories and the monads they generate can be used to encode computational effects. An important insight of Power and Shkaravska is that comodels of an algebraic theory $\mathbb{T}$ – i.e., models in the opposite category $\mathcal{S}\mathrm{et}^{\mathrm{op}}$ – provide a suitable environment for evaluating the computational effects encoded by $\mathbb{T}$ . As already noted by Power and Shkaravska, taking comodels yields a functor from accessible monads to accessible comonads on $\mathcal{S}\mathrm{et}$ . In this paper, we show that this functor is part of an adjunction – the “costructure–cosemantics adjunction” of the title – and undertake a thorough investigation of its properties. We show that, on the one hand, the cosemantics functor takes its image in what we term the presheaf comonads induced by small categories; and that, on the other, costructure takes its image in the presheaf monads induced by small categories. In particular, the cosemantics comonad of an accessible monad will be induced by an explicitly-described category called its behaviour category that encodes the static and dynamic properties of the comodels. Similarly, the costructure monad of an accessible comonad will be induced by a behaviour category encoding static and dynamic properties of the comonad coalgebras. We tie these results together by showing that the costructure–cosemantics adjunction is idempotent, with fixpoints to either side given precisely by the presheaf monads and comonads. Along the way, we illustrate the value of our results with numerous examples drawn from computation and mathematics.

We define a number of stochastic equivalences in the dtsPBC framework, which is a discrete time stochastic extension of finite Petri box calculus (PBC) enriched with iteration. These equivalences allow the identification of stochastic processes that have similar behaviour but are differentiated by the semantics of the calculus. We explain how the equivalences we propose can be used to reduce transition systems of expressions, and demonstrate how to apply the equivalences to compare the stationary behaviour. The equivalences guarantee a coincidence of performance indices for stochastic systems, and can be used for performance analysis simplification. We use a case study to outline a method of modelling, performance evaluation and behaviour preserving reduction of concurrent computing systems, and apply it to the dining philosophers system.

Confluent node-rewriting hypergraph grammars represent the most comprehensive known method for defining sets of hypergraphs in a recursive way. For a large natural subclass of these grammars, we show that the maximal rank of hyperedges indispensable for generating some set of hypergraphs equals the maximal rank of the hyperedges occurring in the hypergraphs of that set. Moreover, if such a grammar generates a set of graphs, one can construct from that grammar a C-edNCE graph grammar generating the same set of graphs.

Working constructively, we study continuous directed complete posets (dcpos) and the Scott topology. Our two primary novelties are a notion of intrinsic apartness and a notion of sharp elements. Being apart is a positive formulation of being unequal, similar to how inhabitedness is a positive formulation of nonemptiness. To exemplify sharpness, we note that a lower real is sharp if and only if it is located. Our first main result is that for a large class of continuous dcpos, the Bridges–Vîţǎ apartness topology and the Scott topology coincide. Although we cannot expect a tight or cotransitive apartness on nontrivial dcpos, we prove that the intrinsic apartness is both tight and cotransitive when restricted to the sharp elements of a continuous dcpo. These include the strongly maximal elements, as studied by Smyth and Heckmann. We develop the theory of strongly maximal elements highlighting its connection to sharpness and the Lawson topology. Finally, we illustrate the intrinsic apartness, sharpness, and strong maximality by considering several natural examples of continuous dcpos: the Cantor and Baire domains, the partial Dedekind reals, the lower reals and, finally, an embedding of Cantor space into an exponential of lifted sets.

Robustness is a property of system analyses, namely monotonic maps from the complete lattice of subsets of a (system’s state) space to the two-point lattice. The definition of robustness requires the space to be a metric space. Robust analyses cannot discriminate between a subset of the metric space and its closure; therefore, one can restrict to the complete lattice of closed subsets. When the metric space is compact, the complete lattice of closed subsets ordered by reverse inclusion is $\omega$-continuous, and robust analyses are exactly the Scott-continuous maps. Thus, one can also ask whether a robust analysis is computable (with respect to a countable base). The main result of this paper establishes a relation between robustness and Scott continuity when the metric space is not compact. The key idea is to replace the metric space with a compact Hausdorff space, and relate robustness and Scott continuity by an adjunction between the complete lattice of closed subsets of the metric space and the $\omega$-continuous lattice of closed subsets of the compact Hausdorff space. We demonstrate the applicability of this result with several examples involving Banach spaces.

We present algorithms and a C code to reveal quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al. [(2022). Journal of Physics A: Mathematical and Theoretical 55 475301], but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from 2 to 7. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension 2 and higher, (ii) non-existence of negative subspaces of dimension 3 and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for rank 4, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree. Finally, in the three-qubit polar space we correct and improve the contextuality degree of the full configuration and also describe finite geometric configurations formed by unsatisfiable/invalid constraints for both types of quadrics as well as for the geometry whose contexts are all 315 lines of the space.

Processes are often viewed as coalgebras, with the structure maps specifying the state transitions. In the simplest case, the state spaces are discrete, and the structure map simply takes each state to the next states. But the coalgebraic view is also quite effective for studying processes over structured state spaces, e.g. measurable, or continuous. In the present paper, we consider coalgebras over manifolds. This means that the captured processes evolve over state spaces that are not just continuous, but also locally homeomorphic to normed vector spaces, and thus carry a differential structure. Both dynamical systems and differential forms arise as coalgebras over such state spaces, for two different endofunctors over manifolds. A duality induced by these two endofunctors provides a formal underpinning for the informal geometric intuitions linking differential forms and dynamical systems in the various practical applications, e.g. in physics. This joint functorial reconstruction of tangent bundles and cotangent bundles uncovers the universal properties and a high-level view of these fundamental structures, which are implemented rather intricately in their standard form. The succinct coalgebraic presentation provides unexpected insights even about the situations as familiar as Newton's laws.

We show that the diamond lattice can be embedded into the Σ02 enumeration degrees preserving 0 and 1, with atoms one high and Π01, and the other one low.