[Show abstract][Hide abstract] ABSTRACT: Following Lockwood Morris, a method for algebraically structuring a compiler and proving it correct is described. An example language with block structure and side-effects is presented. This determines an initial many-sorted algebra L which is the ‘abstract syntax’ of the example language. Then the semantics of L is completely determined by describing a semantic algebra M ‘similar’ to L. In particular, initiality of L ensures that there is a unique homomorphism Lsem:L→>M. This is algebraically structuring the semantic definition of the language.A category of flow-charts over a stack machine is used as a target language for the purposes of compilation. The semantics of the flow charts (Tsem:T→S) is also algebraically determined given interpretations of the primitive operations on the stack and store. The homomorphism comp:L→ T is the compiler which is also uniquely determined by presenting an algebra T of flowcharts similar to L. This is algebraically structuring the compiler.Finally a function encode:M→S describes source meanings in terms of target meanings. The proof that the compiler is correct reduces to a proof that encode:M→S is a homomorphism; then both comp ∘ Tsem and Lsem ∘ encode are homomorphisms from L to S and they must be equal because there is only one homomorphism from L to S.
[Show abstract][Hide abstract] ABSTRACT: The purpose of this paper is two-fold: first to show how a natural mathematical formulation of the “solution” of a system of recursion equations is formally almost identical with well-known formulations of a solution of a system of “iteration equations.” The second aim is to present a construction which takes an algebraic theory T and yields another algebraic theory M(T) whose morphisms correspond to systems of recursion equations over T. This construction is highly uniform, i.e., the correspondence between T and M(T) is functorial.
Preview · Article · Oct 1983 · Journal of Computer and System Sciences
[Show abstract][Hide abstract] ABSTRACT: This paper is concerned mainly with classes (categories) of ordered algebras which in some signature are axiomatizable by a set of inequations between terms (‘varieties’ of ordered algebras) and also classes which are axiomatizable by implications between inequations (‘quasi varieties’ of ordered algebras). For example, if the signature contains a binary operation symbol (for the monoid operation) and a constant symbol (for the identity) the class of ordered monoids M can be axiomatized by a set of inequations (i.e. expressions of the form t≤t'. However, if the signature contains only the binary operation symbol, the same class M cannot be so axiomatized (since it is not now closed under subalgebras). Thus, there is a need to find structural, signature independent conditions on a class of ordered algebras which are necessary and sufficient to guarantee the existence of a signature in which the class is axiomatizable by a set of inequations (between terms in this signature). In this paper such conditions are found by utilizing the notion of ‘P-categories’. A P-category C is a category such that each ‘Hom-set’ C(a,b) is equipped with a distiguished partial order which is preserved by composition. Aside from proving the characterization theorem, it is also the purpose of the paper to begin the investigation of P-categories.
Preview · Article · Jul 1983 · Journal of Pure and Applied Algebra
[Show abstract][Hide abstract] ABSTRACT: A pair (,U) consisting of a category with coequalizers and a functor U: → Set is a weak quasi-variety if U has a left adjoint and U preserves and reflects regular epis. It is known that every weak quasi-variety is equivalent to a concrete quasi-variety, i.e. a category of Σ-algebras which has all free algebras and which is closed with respect to products and subalgebras. It is also known that if U preserves monic direct limits, is equivalent to a concrete quasi-variety of Σ-algebras in which Σ contains no function symbols of infinite rank; and if U preserves all direct limits, is equivalent to a concrete quasi-variety of Σ-algebras definable by a set of implications of the form where ti and si are Σ-terms and m is a nonnegative integer. This paper concerns several definitions of ‘finiteness’ in a category theoretic setting and some theorems on weak quasi-varieties. Two main theorems characterize those weak quasi-varieties (, U) such that U preserves all direct limits.
Preview · Article · Aug 1982 · Journal of Pure and Applied Algebra
[Show abstract][Hide abstract] ABSTRACT: This paper extends our earlier work on abstract data types by providing an algebraic treatment of parametrized data types (e.g., sets-of-(), stacks-of-(), etc.), as well as answering a number of questions on the power and limitations of algebraic specification techniques. In brief: we investigate the “hidden function” problem (the need to include operations in specifications which we want to be hidden from the user); we prove that conditional specifications are inherently more powerful than equational specifications; we show that parameterized specifications must contain “side conditions” (e.g., that finite-sets-of-d requires an equality predicate on d), and we compare the power of the algebraic approach taken here with the more categorical approach of Lehman and Smyth.