Stone duality, topological algebra, and recognition

This is a paper on Stone duality in computer science with special focus on topics with applications in formal language theory. In Section 2 we give a general overview of Stone duality in its various forms: for Boolean algebras, distributive lattices, and frames. For distributive lattices, we discuss both Stone and Priestley duality. We identify how to move between the different dualities and which dual spaces carry the Scott topology. We then focus on three themes. The first theme is additional operations on distributive lattices and Boolean algebras. Additional operations arise in denotational semantics in the form of predicate transformers. In verification they occur in the form of modal operators. They play an essential role in Eilenberg's variety theorem in the form of quotient operations. Quotient operations are unary instantiations of residual operators which are dual to the operations in the profinite algebras of algebraic language theory. We discuss additional operations in Section 3. The second theme is that of hyperspaces, that is, spaces of subsets of an underlying space. Some classes of algebras may be seen as the class of algebras for a functor. In the case of predicate transformers the dual functors are hyperspace constructions such as the Plotkin, Smyth, and Hoare powerdomain constructions. The algebras-for-a-functor point of view is central to the coalgebraic study of modal logic and to the solution of domain In the algebraic theory of formal languages various hyperspace-related product constructions, such as block and Schutzenberger products, are used to study complexity hierarchies. We describe a construction, similar to the Schutzenberger product, which is dual to adding a layer of quantification to formulas describing formal languages. We discuss hyperspaces in Section 4. The final theme is that of equations. These are pairs of elements of dual spaces. They arise via the duality between subalgebras and quotient spaces and have provided one of the most successful tools for obtaining decidability results for classes of regular languages. The perspective provided by duality allows us to obtain a notion of equations for the study of arbitrary formal languages. Equations in language theory is the topic of Section 5.

Introduction.Boolean equalities occur naturally in Logic and especially in the theory of polyadic algebras.For instance, they have been used in [2] under the name of Champ Logique.The purpose of this paper is to develop a duality theory for Boolean equalities and by doing so establish a representation theory for them.0. Notation.The basic notation for Boolean algebras is the standard one (for instance, see [l]).Throughout this paper, the letter O will denote the two-element Boolean algebra; whenever a topology is assumed to be defined on 0, then this topology is the discrete one.If 73 is a Boolean algebra, then a valuation of B is a homomorphism from 73 onto 0. We make precise the terminology we shall be using in connection with the Stone duality theory for Boolean algebras.A Boolean space is a compact Hausdorff space whose topology is generated by the clopen sets.If X is a Boolean space, then the dual algebra of X is the Boolean algebra of clopen sets of X.If 73 is a Boolean algebra, then the dual space of 73 is the Boolean space X of all valuations of 73 (the topology on X is the topology generated by all sets of the form {v: vp = 1} where p belongs to B).

We extend the theory of profinite algebras, as it is used in the algebraic theory of regular languages, to the more general setting of Stone topological algebras with the ultimate goal of going beyond classes of regular languages. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed topological signature that are closed under taking Stone quotients, closed subalgebras and finite direct products. Looking at the dual Stone spaces of Boolean algebras of subsets of a given topological algebra, we find a simple characterization of when the dual space admits a natural structure of topological algebra; the characterization is given in terms of how the Boolean algebra behaves under the inverses of the operation evaluation mappings of each arity. This provides an alternative, which is presented in the form of an equivalence of categories, to the duality theory of M. Gehrke. As an application, a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is shown to be also residually in it, thereby extending the corresponding result of M. Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures, which was recently extended to topological signatures jointly by the authors and H. Goulet-Ouellet. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety, and these are precisely the Stone analogues of varieties. A Birkhoff type theorem for Stone varieties is also established and it is shown how Reiterman’s theorem can be derived from it.

In this paper we show that the (co)chain complex associated with a decomposition of the computational domain, commonly called a mesh in computational science and engineering, can be represented by a block-bidiagonal matrix that we call the Hasse matrix. Moreover, we show that topology-preserving mesh refinements, produced by the action of (the simplest) Euler operators, can be reduced to multi-linear transformations of the Hasse matrix representing the complex.Our main result is a new representation of the (co)chain complex underlying field computations, a representation that provides new insights into the transformations induced by local mesh refinements. This paper is a further contribution towards bridging the subject of computer representations for solid and physical modeling---which flourished border-line between computer graphics, engineering mechanics and computer science with its own methods and data structures---under the general cover of linear algebra and algebraic topology. The main advantage of such an approach is that topology, geometry and physics may coexist in one and the same formalized framework, concurring together to define, represent and simulate the behavior of the model.Our approach is based on first principles and is general in that it applies to most representational domains that can be characterized as cell complexes, without any restrictions on their type, dimension, codimension, orientability, manifoldness, connectedness. Contrary to what might appear at first sight, the theoretical complexity of the present approach is not greater than that of current methods, provided that sparse-matrix techniques with double element access (by rows and by columns) are employed. Last but not least, our tensorbased approach is a significant step forward in achieving close integration of geometrical representations and physics-based simulations, i.e., in the concurrent modeling of shape and behavior.

Preface: Volume 81

Preface: Volume 52, Issue 2

Contents: Introduction. §1: Computable rings and modules. §2: Ideal membership relation. §3: Effective structured domains. §4: Completion of a local ring as a domain. §5: The recursive completion. Epilogue. References.Introduction. Completion is an important general mathematical device. Often, but not always, a completion takes the following form. Let A be a topological algebraic structure whose topology is derived from a metric. For A, a topological algebra  and an embedding i: A →  are constructed such that  is a complete metric space in which A is densely embedded by i. The long list of structures for which such completions exist begins with Cantor's construction of the real number field and includes objects like the p-adic integers, Baire space, and Boolean algebras. In Bourbaki [6] a careful and thorough account of completions for arbitrary topological groups and fields is given, for which it is important to note that the topological structures need not be metrizable, but must possess a uniformity.The effectiveness of the completion process of a computable structure A cannot be readily studied using the tools of computable algebra, simply because the resulting structure  is almost invariably uncountable. However, in particular cases, it has been possible to define and study the substructure Ak of computable elements of Â; this has been done for the structures mentioned above, starting with the field of recursive real numbers.In this paper we analyse the effectivity of the completion of a local ring R. We do this using structured Scott-Ershov domains. Our study may be considered as a prototype containing methods applicable to a broad class of completions, including all the examples mentioned above, except for the real number field, which needs a generalisation of the domain concept.A Scott-Ershov domain D formalises how a set Dt of possibly “infinite” elements, called total elements, is constructed from a set Dc of “finite” elements, called compact elements. This is achieved by means of an approximation ordering which determines a topology on D and, in particular, on Dt. Our methodology is to associate to a given topological algebra A a structured domain D(A) such that the total elements D(A)t form a topological algebra topologically isomorphic to A. In such circumstances A is said to be domain definable by D(A). The theory of computability for domains is now applied to study the effectivity of the topological algebra A.

Stone duality relates logic, in the form of Boolean algebra, to spaces. Stone-type dualities abound in computer science and have been of great use in understanding the relationship between computational models and the languages used to reason about them. Recent work on probabilistic processes has established a Stone-type duality for a restricted class of Markov processes. The dual category was a new notion---Aumann algebras---which are Boolean algebras equipped with countable family of modalities indexed by rational probabilities. In this article we consider an alternative definition of Aumann algebra that leads to dual adjunction for Markov processes that is a duality for many measurable spaces occurring in practice. This extends a duality for measurable spaces due to Sikorski. In particular, we do not require that the probabilistic modalities preserve a distinguished base of clopen sets, nor that morphisms of Markov processes do so. The extra generality allows us to give a perspicuous definition of event bisimulation on Aumann algebras.

This papers reviews the classical theory of deterministic automata and regular languages from a categorical perspective. The basis is formed by Rutten’s description of the Brzozowski automaton structure in a coalgebraic framework. We enlarge the framework to a so-called bialgebraic one, by including algebras together with suitable distributive laws connecting the algebraic and coalgebraic structure of regular expressions and languages. This culminates in a reformulated proof via finality of Kozen’s completeness result. It yields a complete axiomatisation of observational equivalence (bisimilarity) on regular expressions. We suggest that this situation is paradigmatic for (theoretical) computer science as the study of “generated behaviour”.

For any Boolean space X and a discrete almost distributive lattice D, it is proved that the set C(X, D) of all continuous mappings of X into D, when D is equipped with the discrete topology, is an almost Boolean algebra under pointwise operations. Conversely, it is proved that any almost Boolean algebra is a homomorphic image of C(X,D) for a suitable Boolean space X and a discrete almost distributive lattice D.

The main purpose of this work is to study some infinite combinatorics applications in spaces of continuous functions, defined in Hausdorff compact spaces. Using infinite combinatorics in Boolean algebras, through Stone duality, we obtain a compact Hausdorff space. With certain properties in Boolean algebras it is possible to analyze analytic properties in the space of continuous functions defined in such space. Specifically, we analyze the Grothendieck property.

We study finite automata with both nondeterministic and random states (npfa's). We restrict our attention to those npfa's that accept their languages with a small probability of error and run in polynomial expected time. Equivalently, we study Arthur--Merlin games where Arthur is limited to polynomial time and constant space. Dwork and Stockmeyer [SIAM J. Comput., 19 (1990), pp. 1011--1023] asked whether these npfa's accept only the regular languages (this was known if the automaton has only randomness or only nondeterminism). We show that the answer is yes in the case of npfa's with a 1-way input head. We also show that if L is a nonregular language, then either L or $\bar{L}$ is not accepted by any npfa with a 2-way input head. Toward this end, we define a new measure of the complexity of a language L, called its 1-tiling complexity. For each n, this is the number of tiles needed to cover the 1's in the "characteristic matrix" of L, namely, the binary matrix with a row and column for each string of length $\le n$, where entry [x,y]=1 if and only if the string $xy \in L$. We show that a language has constant 1-tiling complexity if and only if it is regular, from which the result on 1-way input follows. Our main result regarding the general 2-way input tape follows by contrasting two bounds: an upper bound of polylog(n) on the 1-tiling complexity of every language computed by our model and a lower bound stating that the 1-tiling complexity of a nonregular language or its complement exceeds a function in $2^{\Omega (\sqrt{\log n})}$ infinitely often. The last lower bound follows by proving that the characteristic matrix of every nonregular language has rank n for infinitely many n. This is our main technical result, and its proof extends techniques of Frobenius and Iohvidov developed for Hankel matrices [Sitzungsber. der Königl. Preuss. Akad. der Wiss., 1894, pp. 407--431], [Hankel and Toeplitz Matrices and Forms: Algebraic Theory, Birkhauser, Boston, 1982].

A monadic (Boolean) algebra is a Boolean algebra A together with an operator 3 on A (called an existential quantifier, or, simply, a quantifier) such that 30=0, pfk 3p, and 3(^A 3q) = 3p* 3g whenever p and q are in A. Most of this note uses nothing more profound about monadic algebras than the definition. The reader interested in the motivation for and the basic facts in the theory of monadic algebras may, however, wish to consult [2]. Every Boolean algebra can be converted into a monadic algebra, usually in several ways. (One way is to write 3p = p for all p; another is to write Bp =0 or 1 according p =0 or py^O. These special operators are known the discrete and the simple quantifier, respectively.) It follows, a fortiori, that every Boolean algebra can be embedded into a monadic algebra, and it is clear, on grounds of universal algebra, that among the monadic extensions of a Boolean algebra there is one that is as free possible. To be more precise, let us say that a monadic algebra A is a free monadic extension of a Boolean algebra B if (i) B is a Boolean subalgebra of A, (ii) A is (monadically) generated by B, (iii) every Boolean homomorphism g that maps B into an arbitrary monadic algebra C has a (necessarily unique) extension to a monadic homomorphism f that maps A into C. The statement that a monadic extension as free possible always exists means that every Boolean algebra B has a free monadic extension A; the algebra A is uniquely determined to within a monadic isomorphism that is equal to the identity on B. The purpose of what follows is to give a constructive proof of this fact, i.e., a proof that exhibits A by means of certain set-theoretic constructions based on B (instead of a structurally not very informative proof via equivalence classes of strings of symbols). A by-product of the proof is a rather satisfying insight into the structure of the (Stone) dual space of the free monadic extension. The theorem can (and will) be formulated in such a way to subsume the results of Hyman Bass [l ] on the cardinal number of finitely generated free monadic algebras. The idea of the construction is this. Step 1: form a free Boolean algebra generated by a copy of B. Step 2: adjoin the elements of that

It is well known that the laws of logic governing the sentence connectives—“and”, “or”, “not”, etc.—can be expressed by means of equations in the theory of Boolean algebras. The task of providing a similar algebraic setting for the full first-order predicate logic is the primary concern of algebraic logicians. The best-known efforts in this direction are the polyadic algebras of Halmos (cf. [2]) and the cylindric algebras of Tarski (cf. [3]), both of which may be described as Boolean algebras with infinitely many additional operations. In particular, there is a primitive operator, cκ, corresponding to each quantification, ∃υκ. In this paper we explore a version of algebraic logic conceived by A. H. Copeland, Sr., and described in [1], which has this advantage: All operators are generated from a finite set of primitive operations.Following the theory of cylindric algebras, we introduce, in the natural way, the classes of Copeland set algebras (SCpA), representable Copeland algebras (RCpA), and Copeland algebras of formulas. Playing a central role in the discussion is the set, Γ, of all equations holding in every set algebra. The reason for this is that the operations in a set algebra reflect the notion of satisfaction of a formula in a model, and hence an equation expresses the fact that two formulas are satisfied by the same sequences of objects in the model. Thus to say that an equation holds in every set algebra is to assert that a certain pair of formulas are logically equivalent.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.