A coalgebraic approach to non-determinism: Applications to multilattices

Multilattices are a suitable generalization of lattices which enables to accommodate the formalization of non-deterministic computation; specifically, the algebraic characterization for multilattices provides a formal framework to develop tools in several fields of computer science. On the other hand, the usefulness of coalgebra theory has been increasing in the recent years, and its importance is undeniable. In this paper, somehow mimicking the use of universal algebra, we define a new kind of coalgebras (the ND-coalgebras) that allows to formalize non-determinism, and show that several concepts, widely used in computer science are, indeed, ND-coalgebras. Within this formal context, we study a minimal set of properties which provides a coalgebraic definition of multilattices.     

A coalgebraic equational approach to specifying observational structures

A coalgebraic, equational approach to the specification of observational structures allowing for a choice in the result type of observations is presented. Observers whose result type is structured as a coproduct of basic types are considered, and notions of covariable, coterm and coequation, dual to the algebraic notions of variable, term and equation are used to specify the associated structures. A sound and complete deduction calculus for reasoning about observational structures is then formulated. Finally, the approach is extended in order to account for the availability of a fixed data universe in the specification of such structures.     

Towards coalgebraic behaviourism

In this paper we show that it is possible to model observable behaviour of coalgebras independently from their internal dynamics, but within the general framework of representing behaviour by a map into a “final” coalgebra.In the first part of the paper we characterise Set-endofunctors F with the property that bisimilarity of elements of F-coalgebras coincides with having the same observable behaviour. We show that such functors have the final coalgebra of a rather simple nature, and preserve some weak pullbacks. We also show that this is the case if and only if F-bisimilarity corresponds to logical equivalence in the finitary fragment of the coalgebraic logic.In the second part of the paper, we present a construction of a “final” coalgebra that captures the observable behaviour of F-coalgebras. We keep the word “final” quoted since the object we are going to construct need not belong to the original category. The construction is carried out for arbitrary Set-endofunctor F, throughout the construction we remain in Set, but the price to pay is the introduction of new morphisms. The paper concludes with a hint to a possible application to modelling weak bisimilarity for coalgebras.     

A finite model construction for coalgebraic modal logic

In recent years, a tight connection has emerged between modal logic on the one hand and coalgebras, understood as generic transition systems, on the other hand. Here, we prove that (finitary) coalgebraic modal logic has the finite model property. This fact not only reproves known completeness results for coalgebraic modal logic, which we push further by establishing that every coalgebraic modal logic admits a complete axiomatisation in rank 1; it also enables us to establish a generic decidability result and a first complexity bound. Examples covered by these general results include, besides standard Hennessy–Milner logic, graded modal logic and probabilistic modal logic.     

Automata and fixed point logic: A coalgebraic perspective

This paper generalizes existing connections between automata and logic to a coalgebraic abstraction level. Let F: Set to Set be a standard functor that preserves weak pullbacks. We introduce various notions of F-automata, devices that operate on pointed F-coalgebras. The criterion under which such an automaton accepts or rejects a pointed coalgebra is formulated in terms of an infinite two-player graph game. We also introduce a language of coalgebraic fixed point logic for F-coalgebras, and we provide a game semantics for this language. Finally, we show that the two approaches are equivalent in expressive power. We prove that any coalgebraic fixed point formula can be transformed into an F-automaton that accepts precisely those pointed F-coalgebras in which the formula holds. And conversely, we prove that any F-automaton can be converted into an equivalent fixed point formula that characterizes the pointed F-coalgebras accepted by the automaton.     

Probabilistic anonymity via coalgebraic simulations

There is a growing concern about anonymity and privacy on the Internet, resulting in lots of work on formalization and verification of anonymity. In particular, the importance of probabilistic aspects of anonymity has recently been highlighted by many authors. Several different notions of “probabilistic anonymity” have been studied so far, but proof methods for such probabilistic notions have not yet been elaborated. In this paper we introduce a simulation-based proof method for one notion of probabilistic anonymity introduced by Bhargava and Palamidessi, called strong probabilistic anonymity. The method is a probabilistic adaptation of the one by Kawabe, Sakurada et al. for non-deterministic anonymity; anonymity of a protocol is proved by finding a forward/backward simulation between certain automata. For the jump from non-determinism to probability we exploit a generic, coalgebraic theory of traces and simulations developed by Hasuo, Jacobs and Sokolova. In particular, an appropriate notion of probabilistic simulation is obtained as an instantiation of the generic definition, for which soundness theorem comes for free. Additionally, we show how we can use a similar idea to verify a weaker notion of probabilistic anonymity called probable innocence.     

On institutions for modular coalgebraic specifications

We present an algebraic extension of standard coalgebraic specification techniques for state-based systems which allows us to integrate constants and n-ary operations in a smooth way and which leads to institutions enabling the use of modular specification techniques. A sound and complete proof system for first-order observational properties of modular specifications is given. The framework of (Ω,Ξ)-structures that we present can be considered as the result of a transformation of concepts of observational logic as in Hennicker and Bidoit (in: A. Haeberer (Ed.), Algebraic Methodology and Software Technology (AMAST’98), Lecture Notes in Computer Science, vol. 1548, Springer, Berlin, 1999) into the coalgebraic world. Moreover, it is shown that the features of (Ω,Ξ)-structures that make them suitable models for an observational approach to specifications can be categorically expressed by the fact that the operation mapping an (Ω,Ξ)-structure to its behaviour is a fibred idempotent monad.     

Subsequential transducers: a coalgebraic perspective

Subsequential transducers combine (input) language recognition with transduction and thereby generalise classic deterministic automata as well as Mealy and Moore type state machines. These well known subclasses all have a natural coalgebraic characterisation, and the question arises whether their coalgebraic modelling can be extended to subsequential transducers and their underlying structures. In this paper, we show that although subsequential structures cannot generally be regarded as coalgebras, the subclass of normalised structures do form a subcategory of coalgebras. Moreover, normalised structures are reflective in the category of all subsequential structures, and a final normalised structure exists. The existence and properties of the minimal subsequential transducer can be derived from this result. We also show that for the class of subsequential structures in which all states are accepting, an alternative coalgebraic representation is obtained by taking differentials. This differential representation gives rise to a new method of deciding equivalence and computing minimal representations which does not involve normalisation. Both normalisation and taking differentials can be formalised as functors into reflective subcategories of coalgebras, and we can therefore see these constructions as coalgebraisation.     

Maximal traces and path-based coalgebraic temporal logics

This paper gives a general coalgebraic account of temporal logics whose semantics involves a notion of computation path. Examples of such logics include the logic CTL* for transition systems and the logic PCTL for probabilistic transition systems. Our path-based temporal logics are interpreted over coalgebras of endofunctors obtained as the composition of a computation type (e.g. non-deterministic or stochastic) with a general transition type. The semantics of such logics relies on the existence of execution maps similar to the trace maps introduced by Jacobs and co-authors as part of the coalgebraic theory of finite traces (Hasuo et al., 2007 [1]). We consider finite execution maps derived from the theory of finite traces, and a new notion of maximal execution map that accounts for maximal, possibly infinite executions. The latter is needed to recover the logics CTL* and PCTL as specific path-based logics.     

A quantitative solution to Constraint Satisfaction Problem (CSP)

Constraint Satisfaction Problem (CSP) is an important problem in artificial intelligence and operations research. Many practical problems can be formulated as CSP, i.e., finding a consistent value assignment to variables subject to a set of constraints. In this paper, we give a quantitative approach to solve the CSPs which deals uniformly with binary constraints as well as high order,k-ary (k ≥ 2) constraints. In this quantitative approach, using variable transformation and constraint transformation, a CSP is transformed into a satisfiability (SAT) problem. The SAT problem is then solved within a continuous search space. We will evaluate the performance of this method based on randomly generated SAT problem instances and regularly generatedk-ary (k ≥ 2) CSP problem instances.     

A Continuous Algebraic Semantics of CSP

This is an attempt to use continuous algebras to describe the semantics of CSP-continuity being used to solve recursive definitions of processes as infinite objects.By so doing,we combine the algebraic specifications of abstract data types with CSP 50 make ups 2 new language,which is recommended as a promising candidate of specification language for designing and developing communicating systems.     

A quick introduction to membrane computing

Membrane computing is a branch of natural computing inspired from the architecture and the functioning of biological cells. The obtained computing models are distributed parallel devices, called P systems, processing multisets of objects in the compartments defined by hierarchical or more general arrangements of membranes. Many classes of P systems were investigated – mainly from the point of view of computing power and computing efficiency; also, a series of applications (especially in modeling biological processes) were reported. This note is a short and informal introduction to this research area, introducing a few basic notions, research topics, types of results, and pointing out to some relevant references.     

A specifier's introduction to formal methods

Formal methods used in developing computer systems (i.e. mathematically based techniques for describing system properties) are defined, and their role is delineated. Formal specification languages, which provide the formal method's mathematical basis, are examined. Certain pragmatic concerns about formal methods and their users, uses, and characteristics are discussed. Six well-known or commonly used formal methods are illustrated by simple examples. They are Z, VDM, Larch, temporal logic, CSP, and transition axioms     

A brief introduction to agent mining

Agent mining is an emerging interdisciplinary area that integrates multiagent systems, data mining and knowledge discovery, machine learning and other relevant areas. It brings new opportunities to tackling issues in relevant fields more efficiently by engaging together the individual technologies. It will also bring about symbiosis and symbionts that combine advantages from the corresponding constituent systems. In this editorial, we briefly introduce the concept of agent mining, the main areas of research, and challenges and opportunities in agent mining. Finally, we give an overview of the papers in this special issue.     

A brief introduction to expert systems

Recent interest in the topic of expert systems has been enormous, and continues to grow. This allegedly new tool has been proposed for implementation in an incredibly diverse array of problems, including a host of problem types that are (or should be) of interest to the operations researcher. In this tutorial, we present a very brief overview of expert systems. In particular, we examine its relationship and usefulness to the operations research profession.