This paper generalizes many-sorted algebra (hereafter, MSA) to order-sorted algebra (hereafter, 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.

This thesis proposes a general framework for equational logic programming, called categorybased equational logic by placing the general principles underlying the design of the programming language Eqlog and formulated by Goguen and Meseguer into an abstract form. This framework generalises equational deduction to an arbitrary category satisfying certain natural conditions; completeness is proved under a hypothesis of quantifier projectivity, using a semantic treatment that regards quantifiers as models rather than variables, and regards valuations as model morphisms rather than functions. This is used as a basis for a model theoretic category-based approach to a paramodulation-based operational semantics for equational logic programming languages. Category-based equational logic in conjunction with the theory of institutions is used to give mathematical foundations for modularisation in equational logic programming. We study the soundness and completeness problem for module imports i...

Abstract. As an example of the categorical apparatus of pseudo algebras over 2-theories, we show that pseudo algebras over the 2-theory of categories can be viewed as pseudo double categories with folding or as appropriate 2-functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2-cells of the horizontal 2-category. As a special case, strict 2-algebras with one object and everything invertible are crossed modules under a group.

Current methodology for compiler construction evolved from the need to release programmers form the burden of writing machine-language programs. This methodology does not assume a formal concept of a programming language and is not based on mathematical algorithms that model the behavior of a compiler. The side effect is that compiler implementation is a difficult task and the correctness of a compiler usually is not proven mathematically. Moreover, a compiler may be based on assumptions about its source and target languages that are not necessarily acceptable for another compiler that has the same source and target languages. The consequence is that programs are not portable between platforms of machines and between generations of languages. In addition, while a conventional compiler freezes the notation that programmers can use to develop their programs the problem domain evolves and requires extensions that are not supported by the compiler. These problems are addressed by two direc...

Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could be understood as an universal algebraic geometry. This geometry is parallel to universal algebra. In the monograph [51] algebraic logic was used for building up a model of a database. Later on, the structures arising there turned out to be useful for solving several problems from algebra. This is the position which the present paper is written from.

Context-free grammars are used in several algebraic specification formalisms instead of first-order signatures for the definition of the structure of algebras, because grammars provide better notation than signatures. The rigidity of these first-order structures enforces a choice between strongly typed structures with little genericity or generic operations over untyped structures. In two-level signatures level 1 defines the algebra of types used at level 0 providing the possibility to define polymorphic abstract data types. Two-level grammars are the grammatical counterpart of two-level signatures. This paper discusses the correspondence between context-free grammars and first-order signatures, the extension of this correspondence to two-level grammars and signatures, examples of the usage of two-level grammars for polymorphic syntax definition, a restriction of the class of two-level grammars for which the parsing problem is decidable, a parsing algorithm that yields a minimal and ...

As an example of the categorical apparatus of pseudo algebras over 2-theories, we show that pseudo algebras over the 2-theory of categories can be viewed as pseudo double categories with folding or as appropriate 2-functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2-cells of the horizontal 2-category. As a special case, strict 2-algebras with one object and everything invertible are crossed modules under a group. 1.

This thesis is primarily concerned with the algebraic semantics of non-terminating term rewriting systems. The usual semantics for rewrite system is based in interpreting rewrite rules as equations and rewriting as a particular case of equational reasoning. The termination of a rewrite system ensures that every term has a value (normal form). But, in general we cannot guarantee this. The research that has been done on non-terminating rewrite systems is centered on seeking semantics for these systems where the usual properties of confluent systems (like uniqueness of normal forms) still hold. These approaches extend the original set of terms (with infinite terms) in such a way that every term has a value. We propose a new semantics for rewrite systems based on interpreting rewrite rules as inequations between terms in an ordered algebra. We show that a variant of equational logic -- inequational logic -- is an institution and we further prove that rewriting is a sound and complete proof...

Chris PrestonThis study presents a systematic approach to specifying data objects with the help of initial algebras. The primary aim is to describe the set-up to be found in modern functional programming languages such as Haskell and ML, although it can also be applied to more general situations. The ‘initial algebra semantics ’ philosophy has been propagated by the ADJ group consisting of J.A. Goguen, J.W. Thatcher, E.G. Wagner and J.B. Wright, for example in [6], and is now well-established. The approach presented here can be seen as pushing this philosophy a stage further and consists of taking the following four steps. (1) Data types are specified by a signature and the ‘fully-defined ’ data objects are then described by the carrier sets in an initial algebra. (2) The initial algebra is extended to include ‘undefined ’ and ‘partially defined’ data objects, leading to what is known as a bottomed algebra. The correct bottomed algebra depends on the language being considered. However, it can always be defined to be an initial object in a class of bottomed algebras