Many-sorted Algebras Research Articles

Primitive recursive selection functions for existential assertions over abstract algebras

We generalize to abstract many-sorted algebras the classical proof-theoretic result due to Parsons, Mints and Takeuti that an assertion ∀ x ∃ y P ( x , y ) (where P is Σ 1 0 ), provable in Peano arithmetic with Σ 1 0 induction, has a primitive recursive selection function. This involves a corresponding generalization to such algebras of the notion of primitive recursiveness. The main difficulty encountered in carrying out this generalization turns out to be the fact that equality over these algebras may not be computable, and hence atomic formulas in their signatures may not be decidable. The solution given here is to develop an appropriate concept of realizability of existential assertions over such algebras, generalized to realizability of sequents of existential assertions. In this way, the results can be seen to hold for classical proof systems. This investigation may give some insight into the relationship between specifiability and computability for data types such as the reals, where the atomic formulas, i.e., equations between terms of type real, are not computable.

Piecewise initial algebra semantics

We investigate a free functor construction on arbitrary many-sorted algebras, and the extent to which it solves the problem of finding an initial algebra interpretation for the module algebra of Bergstra, Heering and Klint.

Universality and semicomputability for nondeterministic programming languages over abstract algebras

The Universal Function Theorem (UFT) originated in 1930s with the work of Alan Turing, who proved the existence of a universal Turing machine for computations on strings over a finite alphabet. This stimulated the development of stored-program computers. Classical computability theory, including the UFT and the theory of semicomputable sets, has been extended by Tucker and Zucker to abstract many-sorted algebras, with algorithms formalized as deterministic While programs. This paper investigates the extension of this work to the nondeterministic programming languages While RA consisting of While programs extended by random assignments, as well as sublanguages of While RA formed by restricting the random assignments to booleans or naturals only. It also investigates the nondeterministic language GC of guarded commands. There are two topics of investigation: (1) the extent to which the UFT holds over abstract algebras in these languages; (2) concepts of semicomputability for these languages, and the extent to which they coincide with semicomputability for the deterministic While language.

Partial algebras, meaning categories and algebraization

Many approaches to natural language semantics are essentially model-theoretic, typically cast in type theoretic terms. Many linguists have adopted type theory or many-sorted algebras [see H. Hendriks, Compositionality and model-theoretic interpretation, J. Logic, Language Inform. 10 (2001) 29–48 and references therein]. However, recently Hodges [Formal features of compositionality, J. Logic, Language Inform. 10 (2001) 7–28] has offered an approach to compositionality using just partial algebras. An approach in terms of partial algebras seems at the outset more justified, since the typing is often just artificially superimposed on language (and makes many words massively polymorphic). On the other hand, many-sorted algebras are easier to handle than partial algebras, and are therefore generally preferred. This paper investigates the dialectics between partial algebras and many-sorted algebras and tries to set the background for an approach in the spirit of Hodges [Formal features of compositionality, J. Logic, Language Inform. 10 (2001) 7–28], which also incorporates insights from algebraic logic, in particular from Blok and Pigozzi [Algebraizable logics, Mem. Amer. Math. Soc. 77 (396) (1990)]. The analytic methods that we shall develop here shall also be applied to combinatory algebras and algebraizations of predicate logic.

On the Compositional Extension Problem

A semantics may be compositional and yet partial, in the sense that not all well-formed expressions are assigned meanings by it. Examples come from both natural and formal languages. When can such a semantics be extended to a total one, preserving compositionality? This sort of extension problem was formulated by Hodges, and solved there in a particular case, in which the total extension respects a precise version of the fregean dictum that the meaning of an expression is the contribution it makes to the meanings of complex phrases of which it is a part. Hodges' result presupposes the so-called Husserl property, which says roughly that synonymous expressions must have the same category. Here I solve a different version of the compositional extension problem, corresponding to another type of linguistic situation in which we only have a partial semantics, and without assuming the Husserl property. I also briefly compare Hodges' framework for grammars in terms of partial algebras with more familiar ones, going back to Montague, which use many-sorted algebras instead.

Computable total functions on metric algebras, universal algebraic specifications and dynamical systems

Data such as real and complex numbers, discrete and continuous time data streams, waveforms, scalar and vector fields, and many other functions, are fundamental for many kinds of computation. In the theory of data, such data types are modelled using topological, or metric, many-sorted algebras and continuous homomorphisms. A theory of such topological data types is needed to answer the general questions: 1. What are the computable functions on topological algebras? 2. What methods exist to axiomatically specify functions on topological algebras? 3. Can all computable functions be specified? Such a theory seems to be in its infancy: there are many approaches to computability theory on general and specific spaces, and few approaches to specification theory. In some earlier papers, we have studied the questions 1 and 2 with the needs of data type theory in mind, and built a bridge between computations and specifications to try to answer 3. In this paper, we extend and combine several of our results, to prove new theorems that (i) show the equivalence of some six deterministic or non-deterministic models of computation on various metric algebras and, in particular, on spaces R n of real numbers; (ii) provide finite universal algebraic specifications for all the functions that can be computably approximated on metric algebras and, in particular, on Euclidean n-space R n ; (iii) show the existence of finite universal algebraic specifications of computably approximable finite dimensional deterministic dynamical systems. A technical issue is the localisation of uniform continuity using exhaustions of open sets. We use specifications composed of conditional equations, inequalities and, for convenience, new exhaustion primitives, that define functions uniquely up to isomorphism.

Mappings between probabilistic Boolean networks

Probabilistic Boolean Networks (PBNs) comprise a graphical model based on uncertain rule-based dependencies between nodes and have been proposed as a model for genetic regulatory networks. As with any algebraic structure, the characterization of important mappings between PBNs is critical for both theory and application. This paper treats the construction of mappings to alter PBN structure while at the same time maintaining consistency with the original probability structure. It considers projections onto sub-networks, adjunctions of new nodes, resolution reduction mappings formed by merging nodes, and morphological mappings on the graph structure of the PBN. It places PBNs in the framework of many-sorted algebras and in that context defines homomorphisms between PBNs.

Specification of Logic Programming Languages from Reusable Semantic Building Blocks

We present a Language Prototyping System that facilitates the modular development of interpreters from independent semantic building blocks. The abstract syntax is modelled as the fixpoint of a pattern functor which can be obtained as the sum of functors. For each functor we define an algebra whose carrier is the computational structure. This structure is obtained as the composition of several monad transformers applied to a base monad, where each monad transformer adds a new notion of computation. When the abstract syntax is composed from mutually recursive categories, we use many-sorted algebras. With this approach, the prototype interpreters are automatically obtained as a catamorphism over the defined algebras.As an example, in this paper, we independently specify an arithmetic evaluator and a simple logic programming language and combine both specifications to obtain a logic programming language with arithmetic capabilities.

Abstract computability and algebraic specification

Abstract computable functions are defined by abstract finite deterministic algorithms on many-sorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable functions on any many-sorted algebra; (ii) all functions effectively approximable by abstract computable functions on any metric algebra. We show that there exist universal algebraic specifications for all the classically computable functions on the set ℝ of real numbers. The algebraic specifications used are mainly bounded universal equations and conditional equations. We investigate the initial algebra semantics of these specifications, and derive situations where algebraic specifications precisely define the computable functions.

Algebraic methods for specification and formal development of programs

and development including applications to the formal development of reliable complex software systems. We regard this area of research as straddling the borderline between theory and practice. It has connections with work on the design and semantics of software systems and programming languages, on formal methods for system verication (various program logics in particular), and relies heavily on some basic concepts of mathematical logic, universal algebra and category theory, while being directly inspired by and potentially applicable in practice. In this note we sum up our experiences so far, hint at the need for further work in certain areas, and speculate a bit about the directions in which we expect to go in the future. The roots of our work in this area are in the theory of algebraic specication. The most fundamental assumption underlying this theory is that programs are modelled as many-sorted algebras consisting of a collection of sets of data values together with functions over those sets. This level of abstraction is commensurate with the view that the correctness of the input/output behaviour of a program takes precedence over all its other properties. Another common element is that

Computation by ‘While’ programs on topological partial algebras

The language of while programs is a fundamental model for imperative programming on any data type. It leads to a generalisation of the theory of computable functions on the natural numbers to the theory of computable functions on any many-sorted algebra. The language is used to express many algorithms in scientific computing where while programs are applied to continuous data. In the theory of data, continuous data types are modelled by topological many-sorted algebras. We study both exact and approximate computations by while programs, and while programs with arrays, over topological many-sorted algebras with partial operations. First, we establish that partial operations are necessary in order to compute a wide range of continuous functions. We prove basic continuity properties of our abstract computability: Any partial function computable over a partial topological algebra by a while- array program is continuous. Any set semicomputable, or computable, over a partial topological algebra by a while- array program is open, or clopen, respectively. Secondly, we contrast exact and approximate computations. The class of functions that can be computed exactly can be quite limited. We show that on connected total algebras, the while and while-array computable functions are precisely those that are explicitly definable by terms. We show that for certain general classes of topological algebras, the total functions that can be approximated by while programs are precisely those that can be effectively approximated by terms. This property we call generalised Weierstrass approximation. An application of this result is that a function on the set R of reals is computable in the sense of computable analysis if, and only if, it is while program approximable on a simple algebra based on R .

Order-Sorted Inductive Types

System F ω ⩽ is an extension of system F ω with subtyping and bounded quantification. Order-sorted algebra is an extension of many-sorted algebra with overloading and subtyping. We combine both formalisms to obtain IF ω ⩽ , a higher-order typed λ -calculus with subtyping, bounded quantification, and order-sorted inductive types, i.e., data types with built-in subtyping and overloading. Moreover we show that IF ω ⩽ enjoys important meta-theoretic properties, including confluence, strong normalization, subject reduction, and decidability of type checking.

Nonlinear science — The impact of biology

Nonlinear science has primarily developed from applications of mathematics to physics. The biological sciences are emerging as the dominant growth points of science and technology, and biological systems are characterised by being information dense, spatially extended, organised in interacting hierarchies, and rich in diversity. These characteristics, linked with an increase in available computing power and accessible memory, may lead to a nonlinear science of complicated interacting systems that will link different types of mathematical object within a framework of many-sorted algebras. Examples, drawn from current work on intracellular, cellular, tissue, organ and integrative physiology of an individual, are outlined within the theory of synchronous concurrent algorithms. Possible directions in population dynamics and applications to ecosystem management are outlined.

Many-sorted algebras in congruence modular varieties

We present several basic results on many-sorted algebras, most of them only valid in congruence modular varieties. We describe a connection between the properties of many-sorted varieties and those of varieties of one sort and give some results on functional completeness, the commutator and Abelian algebras.

A new approach to ADT specification support based on reuse of similar ADT by the application of Case-Based Reasoning

Abstract Data Type (ADT) is a powerful tool having a very good mathematic basis called many-sorted algebra and is also combined into several Formal Description Techniques (FDTs). By using ADT, a protocol or a software system can be specified in a very abstract way. It also facilitates the verification and validation of the specified system. However, in practice ADT is rarely used because it is difficult for a non-ADT expert to specify. As a consequence of this, in order to promote the widespread use of ADT and FDT, a support method for ADT specification is necessary. Unfortunately, there is almost no such a support method so far. We propose an ADT specification support method in terms of support system based on Case-Based Reasoning (CBR) which is a kind of problem solving by reusing solutions of the existing similar problem with some appropriate modifications. CBR has been used in some researches. But there is no research which uses this technique for ADT specification support so far. The support system can ease the specification of ADT and prevent the user from spending too much time and effort in specifying the whole ADT from the beginning. Our support system consists of four parts: (1) Case base of ADTs; (2) Requirement Acquisition Template; (3) Search mechanism for finding the most similar case in (1); (4) Modifier for modifying the retrieved case to meet the requirement of user. Among these parts, the core of the support system is the search mechanism which is based on the definition of similarity of ADTs. In this paper, we first propose an ADT specification model which defines a data carrier part of an ADT as a combination of constructors of four patterns. Based on the specification model, a similarity definition is given. Furthermore, we have designed the support system. Finally, in order to evaluate the effectiveness of the system, we experimented with data types of service and protocol in the data communication field. From the evaluation, the effectiveness of the system is ensured.

WHEN IS A CATEGORY OF MANY-SORTED PARTIAL ALGEBRAS CARTESIAN-CLOSED?

In this paper we study the cartesian closedness of the five most natural categories with objects all partial many-sorted algebras of a given signature. In particular, we prove that, from these categories, only the usual one and the one having as morphisms the closed homomorphisms can be cartesian closed. In the first case, it is cartesian closed exactly when the signature contains no operation symbol, in which case such a category is a slice category of sets. In the second case, it is cartesian closed if and only if all operations are unary. In this case, we identify it as a functor category and we show some relevant constructions in it, such as its subobjects classifier or the exponentials.

A process algebraic view of input/output automata

Input/output automata are a widely used formalism for the specification and verification of concurrent algorithms. Unfortunately, they lack an algebraic characterization, a formalization which has been fundamental for the success of theories like CSP, CCS and ACP. We present a many-sorted algebra for I/O automata that takes into account notions such as interface, input enabling, and local control. It is sufficiently expressive for representing all finitely branching transition systems; hence, all I/O automata with a finitely branching transition relation. Our presentation includes a complete axiomatization of the external trace preorder relation over recursion-free processes with input and output.

Order-Sorted Algebra Solves the Constructor-Selector, Multiple Representation, and Coercion Problems

Structured data are generally composed from constituent parts by constructors and decomposed by selectors . We show that the usual many-sorted algebra approach to abstract data types cannot capture this simple intuition in a satisfactory way. We also show that order-sorted algebra does solve this problem, and many others concerning partially defined, ill-defined, and erroneous expressions, in a simple and natural way. In particular, we show how order-sorted algebra supports and elegant solution to the problems of multiple representations and coercions. The essence of order-sorted algebra is that sorts have subsorts , whose semantic interpretation is the subset relation on the carriers of algebras.

Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations

This paper generalizes many-sorted algebra (MSA) to order-sorted algebra (OSA) by allowing a partial ordering relation on the set of sorts. This supports abstract data types with multiple inheritance (in roughly the sense of object-oriented programming), several forms of polymorphism and overloading, partial operations (as total on equationally defined subsorts), exception handling, and an operational semantics based on term rewriting. We give the basic algebraic constructions for OSA, including quotient, image, product and term algebra, and we prove their basic properties, including quotient, homomorphism, and initiality theorems. The paper's major mathematical results include a notion of OSA deduction, a completeness theorem for it, and an OSA Birkhoff variety theorem. We also develop conditional OSA, including initiality, completeness, and McKinsey-Malcev quasivariety theorems, and we reduce OSA to (conditional) MSA, which allows lifting many known MSA results to OSA. Retracts, which intuitively are left inverses to subsort inclusions, provide relatively inexpensive run-time error handling. We show that it is safe to add retracts to any OSA signature, in the sense that it gives rise to a conservative extension. A final section compares and contrasts many different approaches to OSA. This paper also includes several examples demonstrating the flexibility and applicability of OSA, including some standard benchmarks like stack and list, as well as a much more substantial example, the number hierarchy from the naturals up to the quaternions.

Rule-based optimization and query processing in an extensible geometric database system

Gral is an extensible database system, based on the formal concept of a many-sorted relational algebra. Many-sorted algebra is used to define any application's query language, its query execution language, and its optimiztion rules. In this paper we describe Gral's optimization component. It provides (1) a sophisticated rule language—rules are transformations of abstract algebra expressions, (2) a general optimization framework under which more specific optimization algorithms can be implemented, and (3) several control mechanisms for the application of rules. An optimization algorithm can be specified as a series of steps. Each step is defined by its own collection of rules together with a selected control strategy. The general facilities are illustrated by the complete design of an example optimizer—in the form of a rule file—for a small nonstandard query language and an associated execution language. The query language includes selection, join, ordering, embedding derived values, aggregate functions, and several geometric operations. The example shows in particular how the special processing techniques of a geometric database systems, such as spatial join methods and geometric index structures, can be integrated into query processing and optimization of a relational database system. A similar, though larger, optimizer is fully functional within the geometric database system implemented as a Gral prototype.