Publications (18)4.23 Total impact
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ABSTRACT: Following Lockwood Morris, a method for algebraically structuring a compiler and proving it correct is described. An example language with block structure and sideeffects is presented. This determines an initial manysorted 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 flowcharts 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. 
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ABSTRACT: The purpose of this paper is twofold: first to show how a natural mathematical formulation of the “solution” of a system of recursion equations is formally almost identical with wellknown 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.  [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 ‘Pcategories’. A Pcategory C is a category such that each ‘Homset’ 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 Pcategories. 
Article: Finitary quasivarieties
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ABSTRACT: A pair (,U) consisting of a category with coequalizers and a functor U: → Set is a weak quasivariety if U has a left adjoint and U preserves and reflects regular epis. It is known that every weak quasivariety is equivalent to a concrete quasivariety, i.e. a category of Σalgebras 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 quasivariety of Σalgebras in which Σ contains no function symbols of infinite rank; and if U preserves all direct limits, is equivalent to a concrete quasivariety of Σalgebras definable by a set of implications of the form where ti and si are Σterms and m is a nonnegative integer. This paper concerns several definitions of ‘finiteness’ in a category theoretic setting and some theorems on weak quasivarieties. Two main theorems characterize those weak quasivarieties (, U) such that U preserves all direct limits.  [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., setsof(), stacksof(), 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 finitesetsofd 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. 

Conference Paper: Parameter Passing in Algebraic Specification Languages.

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Conference Paper: More on advice on structuring compilers and proving them correct.
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Conference Paper: Programming Languages as Mathematical Objects.
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Conference Paper: A Uniform Approach to Inductive Posets and Inductive Closure.


Conference Paper: Some Fundamentals of OrderAlgebraic Semantics.
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Publication Stats
447  Citations  
4.23  Total Impact Points  
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Institutions

1983

IBM
 Thomas J. Watson Research Center
Armonk, New York, United States
