Transition systems with algebraic structure as models of computations

Probabilistic models of computation have been studied for over three decades now, from foundational, logical, coalgebraic, categorical, as well as more practical verification-motivated point of view. In my work, and in this talk, we focus on the foundational, semantics side of probabilistic automata and transition systems, and in particular the relevant monads, and their algebras. The interplay of probability and non-determinism has been particularly challenging from a semantics point of view for some decades, as it does not just yield a monad. There are by now well-studied solutions to this and several monads for probability and non-determinism: some enriching the structure with convexity to obtain a monad, albeit a non-commutative one, others deliberately simplifying it, by imposing commutativity. It is well known that monads have two faces: a computational one - we think here of the powerset monad modelling non-determinism, or the probability distribution monad modelling probabilistic choices, and a universal-algebraic one - where we think of the algebraic presentation of the monads, like semilattices for the powerset monad and (variants of) convex algebras for (variants of) the probability distribution monad. Combining non-determinism and probability yields other combined monads, among which probably the most studied is the convex-subsets-of-distributions monad. This monad is presented by so-called convex semilattices, algebraic structures that are both a semilattice and a convex algebra, with suitable distributivity connecting the operations. From a semantics point of view, the algebraic presentations give us a nice way to define (and sometimes compute) language (aka trace) equivalence of the corresponding automata. Moreover, the presentations are useful for axiomatizations of language equivalence. From an algebraic point of view, these algebras are interesting and many questions about them are still open. We will discuss language semantics, its axiomatization, as well as some obtained and open algebraic problems for convex algebras and convex semilattices - and their computational consequences. In particular, we have a full characterization of congruences of (variants of) convex algebras, which for example yields decidability of distribution semantics; other results on congruences, e.g. a result on a congruence being finitely generated as a subalgebra, surprisingly accellerated the proof of completeness of infinite trace semanitcs, etc. We will review some existing results: all congruences are described and fp = fg for convex algebras, termination is a black hole, useful functors are proper on convex algebras, cancellativity of convex semi-lattices; as well as mention ongoing work on: mid-point-cancellativity, subalgebras, and homomorphisms for convex semi-lattices, as well as topological convex semilattices. This talk is based on previous joint works with Filippo Bonchi, Alexandra Silva, Valeria Vignudelli, and Harald Woracek, as well as on ongoing work and discussions with Matteo Mio, Alex Simpson, and Harald Woracek.

Investigations of adequate mathematical model of digital computer have lasted for a few years. But when a digital computer was well represented by some suggested model, this model had extremely complicated algebraic structure and it couldnrt be studied by modern algebraic methods. On the other hand if investigated model has a simple algebraic structure (e.g. a unary algebra) then lot of notions associated with digital computers had to be excluded from such a model (e.g. programs). The starting-point of this pa~er is the notion of stored program computer presented by PAWIAK,Z. in ~4J. The stored program computers have a very rich algebraic structure and because of difficulties mentioned above to each stored program computer a new kind of object is associated, more useful in algebraic studies; it's called a M-groupoid. Such associations were done very often in mathematics. To investigate properties of some objects we associate a new kind of objects. And we find the properties of our investigated objects from studies of objects associated with them (e.g. associating to first order predicate calculus the Tarski-Lindenbaum algebra (see ~J), to given Boolean algebra the Stone space (see E6J), to given topological space the fundamental group (see F7J ), to given finite automaton the input monoid (see [1~)). Such an approach often makes possible simple proofs of complex results. On the other hand it#s good to be aware of differences between an associated object and the investigated one (it means how much information about the invastigated object is contained in the associated object). Theory of categories will be useful for it (it will be shown, that the assotiation between stored program computers and M-groupoids is a functor from the category of stored program computers to the category of M-groupoids).It will be shown that stored program computers and M-groupoids are closely related, namely theirs categories are equiva-

It is well-known that sums and products of holomorphic functions are holomorphic, and the holomorphic functions on a complex manifold form a commutative algebra over C . The study of complex manifolds using algebras of holomorphic functions upon them is called complex algebraic geometry. The purpose of this paper is to develop an analogue of complex algebraic geometry, in which the complex numbers C are replaced by the quaternions H . The natural quaternionic analogue of a complex manifold is called a hypercomplex manifold. A class of H -valued functions on hypercomplex manifolds will be defined, called q-holomorphic functions, that are analogues of holomorphic functions on complex manifolds. Now, the set of holomorphic functions on a complex manifold is a commutative algebra over C . Therefore one asks: does the set of q-holomorphic functions on a hypercomplex manifold have an analogous algebraic structure, and if so, what is it? We shall show that the q-holomorphic functions on a (noncompact) hypercomplex manifold do indeed possess a rich algebraic structure. To describe it, we shall introduce a theory of quaternionic algebra, which is a quaternionic analogue of real linear algebra. This theory is built on three building blocks: AH-modules, the analogues of vector spaces, AH-morphisms, the analogues of linear maps, and the quaternionic tensor product, the analogue of tensor product of real vector spaces. As far as the author can tell, these ideas seem to be new. They enable us to construct algebraic structures over H as though H were a commutative field. Quaternionic algebra describes the algebraic structure of hypercomplex manifolds in a remarkable way, and it seems to be the natural language of hypercomplex algebraic geometry, the algebraic geometry of hypercomplex manifolds. We believe that quaternionic algebra is worth studying for its own sake. It has many similarities with linear algebra over R or C , which is why the analogies between complex and quaternionic theories work so well, but there are also deep differences, which give quaternionic algebra a flavour all of its own. Quillen [12] has given a sheaf-theoretic interpretation of the ideas of quaternionic algebra, based on a previous version of this paper. He finds a contravariant equivalence between a class of AH-modules and regular sheaves on a real form of CP. Regular sheaves on CP are equivalent to representations of the Kronecker quiver, and out of such a representation Quillen constructs an AH-module. Under Quillen’s equivalence stable AH-modules correspond to regular vector bundles over the real form of CP.

We consider a uniform model of computation over algebraic structures resulting from a generalization of the Turing machine and the BSS model of computation. This model allows us to gain more insight into the reasons for unsolvability of algorithmic decision problems from different perspectives. For example, classes of undecidable problems can be introduced in several ways by analogy with the classical arithmetical hierarchy and, for many structures, the different definitions lead to different hierarchies of undecidable problems. Here, we will investigate some classes of a hierarchy that is defined semantically by our deterministic oracle machines and that can be syntactically characterized by formulas whose quantifiers range only over an enumerable set. Starting from machines over algebraic structures endowed with some relations and containing an infinite recursively enumerable sequence of individuals, we will also consider this hierarchy for BSS RAM's over the reals and some undecidable problems defined by algebraic properties of the real numbers.

Based on the boson realization, the ground state and phase structure are investigated in a many-quark model with algebraic structure developed in our previous paper, in which the two-body pairing and particle-hole type interactions are active. The physical interpretation of the exact eigenstates obtained in the boson realization in our previous paper is given in a many-quark model with su(4) algebraic structure.

Concurrent histories: A basis for observing distributed systems

To the best of our knowledge, we characterize for the first time the importance of resources (counters, channels, alphabets) when measuring expressiveness of WSTS. We establish, for usual classes of wpos, the equivalence between the existence of order reflections (nonmonotonic order embeddings) and the simulations with respect to coverability languages. We show that the non-existence of order reflections can be proved by the computation of order types. This allows us to solve some open problems and to unify the existing proofs of the WSTS classification.

Reaction systems originated as a formal model for processes inspired by the functioning of the living cell. The underlying idea of this model is that the functioning of the living cell is determined by the interactions of biochemical reactions and these interactions are based on the mechanisms of facilitation and inhibition. Since their inception, reaction systems became a well-investigated novel model of computation. Following this line of research, in this paper we discuss a systematic framework for investigating a whole range of equivalence notions for reaction systems. Some of the equivalences are defined directly on reaction systems while some are defined through transition systems associated with reaction systems. In this way we establish a new bridge between reaction systems and transition systems. In order to define equivalences which capture various ways of interacting with an environment, we also introduce models of the environment which evolve in a finite-state fashion.

We introduce geometric consideration into the theory of formal languages. We aim to shed light on our understanding of global patterns that occur on infinite strings. We utilise methods of geometric group theory. Our emphasis is on large scale geometries. Two infinite strings have the same large scale geometry if there are colour preserving bi-Lipschitz maps with distortions between the strings. Call these maps quasi-isometries. Introduction of large scale geometries poses several questions. The first question asks to study the partial order induced by quasi-isometries. This partial order compares large scale geometries; as such it presents an algebraic tool for classification of global patterns. We prove there is a greatest large scale geometry and infinitely many minimal large scale geometries. The second question is related to understanding the quasi-isometric maps on various classes of strings. The third question investigates the sets of large scale geometries of strings accepted by computational models, e.g. Buchi automata. We provide an algorithm that describes large scale geometries of strings accepted by Buchi automata. This links large scale geometries with automata theory. The fourth question studies the complexity of the quasi-isometry problem. We show the problem is Σ 3 0-complete thus providing a bridge with computability theory. Finally, the fifth question asks to build algebraic structures that are invariants of large scale geometries. We invoke asymptotic cones, a key concept in geometric group theory, defined via model-theoretic notion of ultra-product. Partly, we study asymptotic cones of algorithmically random strings thus connecting the topic with algorithmic randomness.

Using as example an incomplete information system with support a set of objects X, we discuss a possible algebraization of the concrete algebra of the power set of X through quasi BZ lattices. This structure enables us to define two rough approximations based on a similarity and on a preclusive relation, with the second one always better that the former. Then, we turn our attention to Pawlak rough sets and consider some of their possible algebraic structures. Finally, we will see that also Fuzzy Sets are a model of the same algebras. Particular attention is given to HW algebra which is a strong and rich structure able to characterize both rough sets and fuzzy sets.

An abstract definition of bisimulation is presented. It enables a uniform definition of bisimulation across a range of different models for parallel computation presented as categories. As examples, transition systems, synchronisation trees, transition systems with independence (an abstraction from Petri nets) and labelled event structures are considered. On transition systems the abstract definition readily specialises to Milner's strong bisimulation. On event structures it explains and leads to a revision of history-preserving bisimulation of Rabinovitch and Traktenbrot, Goltz and van Glabeek. A tie-up with open maps in a (pre)topos, as they appear in the work of Joyal and Moerdijk, brings to light a promising new model, presheaves on categories of pomsets, into which the usual category of labelled event structures embeds fully and faithfully. As an indication of its promise, this new presheaf model has ``refinement'' operators, though further work is required to justify their appropriateness and understand their relation to previous attempts. The general approach yields a logic, generalising Hennessy-Milner logic, which is characteristic for the generalised notion of bisimulation.

The first theoretical model of DNA computing, called a splicing system, for the study of the generative power of deoxyribonucleic acid (DNA) in the presence of restriction enzymes and ligases was introduced by Head in 1987. Splicing systems model the recombinant behavior of double-stranded DNA (dsDNA) and the enzymes which perform operation of cutting and pasting on dsDNA. Splicing systems with finite sets of axioms and rules generate only regular languages when no additional control is assumed. With several restrictions to splicing rules, the generative power increase up to recursively enumerable languages. Algebraic structures can also be used in order to control the splicing systems. In the literature, splicing systems with additive and multiplicative valences have been investigated, and it has been shown that the family of languages generated by valence splicing systems is strictly included in the family of context-sensitive languages. This motivates the study of splicing systems over permutation groups. In this paper, we define splicing systems over permutation groups and investigate the generative power of the languages produced.

In this study, we introduce the generalized dual Edouard numbers, a novel class of numerical sequences that extend classical recurrence relations within a broader mathematical framework. Several notable special cases particularly the dual Edouard numbers and dual Edouard-Lucas numbers are examined in depth, each exhibiting rich combinatorial and algebraic structures. We derive explicit formulations for these sequences, including Binet-type expressions, generating functions, and summation identities, which provide analytical insight into their intrinsic behavior and structural characteristics. Furthermore, we investigate their matrix representations, offering a refined algebraic approach for theoretical exploration and potential applications in related fields.

Considering resource usage is a powerful insight in the analysis of many phenomena in the sciences. Much of the current research on these resource theories focuses on the analysis of specific resources such quantum entanglement, purity, randomness or asymmetry. However, the mathematical foundations of resource theories are at a much earlier stage, and there has been no satisfactory account of quantitative aspects such as costs, rates or probabilities. We present a categorical foundation for quantitative resource theories, derived from enriched category theory. Our approach is compositional, with rich algebraic structure facilitating calculations. The resulting theory is parameterized, both in the quantities under consideration, for example costs or probabilities, and in the structural features of the resources such as whether they can be freely copied or deleted. We also achieve a clear separation of concerns between the resource conversions that are freely available, and the costly resources that are typically the object of study. By using an abstract categorical approach, our framework is naturally open to extension. We provide many examples throughout, emphasising the resource theoretic intuitions for each of the mathematical objects under consideration.

We examine quantum field theory in spacetimes that are time nonorientable but have no other causal pathology. These are Lorentzian universes-from-nothing, spacetimes with a single spacelike boundary that nevertheless have a smooth Lorentzian metric. Classically, such spacetimes are locally indistinguishable from their globally hyperbolic covering spaces. However, the construction of a quantum field theory (QFT) is more problematic. One can define a family of local algebras on an atlas of globally hyperbolic subspacetimes. But one cannot extend a generic positive linear function from a single algebra to the collection of all local algebras without violating positivity, while satisfying the physically appropriate overlap conditions. This difficulty can be overcome by restricting the size of neighborhoods so that the union of any pair is time- orientable. The structure of local algebras and states is then locally indistinguishable from that of QFT on a globally hyperbolic spacetime. But this size restriction on neighborhoods makes the structure unsatisfactory as a global field theory. The theory allows less information than QFT in a globally hyperbolic spacetime, because correlations between field operators at a pair of points are defined only if a curve joining the points lies in a single neigh- borhood. Moreover, to extend a local state to a collection of states, we use an antipodally symmetric state on the covering space, a state that would not yield a sensible state on the spacetime if all correlations could be measured.