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.

The properties of a simple and natural notion of observational equivalence of algebras and the corresponding specification-building operation are studied. We begin with a defmition of observational equivalence which is adequate to handle reachable algebras only, and show how to extend it to cope with unreachable algebras and also how it may be generalised to make sense under an arbitrary institution. Behavioural equivalence is treated as an important special case of observational equivalence, and its central role in program development is shown by means of an example.

. We show the benefits of applying modular monadic semantics to compiler construction. Modular monadic semantics allows us to define a language with a rich set of features from reusable building blocks, and use program transformation and equational reasoning to improve code. Compared to denotational semantics, reasoning in monadic style offers the added benefits of highly modularized proofs and more widely applicable results. To demonstrate, we present an axiomatization of environments, and use it to prove the correctness of a well-known compilation technique. The monadic approach also facilitates generating code in various target languages with different sets of built-in features. 1 Introduction We propose a modular semantics which allows language designers to add (or remove) programming language features without causing global changes to the existing specification, derive a compilation scheme from semantic descriptions, prove the correctness of program transformation and compilation...

Modular monadic semantics is a high-level and modular form of denotational semantics. It is capable of capturing individual programming language features and their interactions. This thesis explores the theory and applications of modular monadic semantics, including: building blocks for individual programming features, equational reasoning with laws and axioms, modular proofs, program transformation, modular interpreters, and compiler construction. We will demonstrate that the modular monadic semantics framework makes programming languages easy to specify, reason about, and implement.

: Existing works on the construction of correct compilers have at least one of the following drawbacks: (i) correct compilers do not compile into machine code of existing processors. Instead they compile into programs of an abstract machine which ignores limitations and properties of real-life processors. (ii) the code generated by correct compilers is orders of magnitudes slower than the code generated by unverified compilers. (iii) the considered source language is much less complex than real-life programming languages. This paper focuses on the construction of correct compiler backends which generate machine-code for real-life processors from realistic intermediate languages. Our main results are the following: (i) We present a proof approach based on abstract state machines for bottom-up rewriting system specifications (BURS) for back-end generators. A significant part of this proof can be parametrized with the intermediate and machine language. (ii) The performance of the code con...

We have designed, implemented, and proved the correctness of a compiler generator that accepts action semantic descriptions of imperative programming languages. The generated compilers emit absolute code for an abstract RISC machine language that currently is assembled into code for the SPARC and the HP Precision Architecture. Our machine language needs no run-time type-checking and is thus more realistic than those considered in previous compiler proofs. We use solely algebraic specifications; proofs are given in the initial model. 1 Introduction The previous approaches to proving correctness of compilers for non-trivial languages all use target code with run-time type-checking. The following semantic rule is typical for these target languages: (FIRST : C; hv 1 ; v 2 i : S) ! (C; v 1 : S) The rule describes the semantics of an instruction that extracts the first component of the top-element of the stack, provided that the top-element is a pair. If not, then it is implicit that the...

We report about a joint project of the universities at Karlsruhe, Kiel and Ulm on how to get correct compilers for realistic programming languages. Arguing about compiler correctness must start from a compiling specification describing the correspondence of source and target language in formal terms. We have chosen to use abstract state machines to formalize this correspondence. This allows us to stay with traditional compiler architectures for subdividing the compiler task. A main achievement is the use of program checking for replacing large parts of compiler verification by the much simpler task of verifying program checkers.

this paper we study the constrained equivalence of programs with effects. In particular, we present a formal system for deriving such equivalences. Constrained equivalence is defined via a model theoretic characterization of operational, or observational, equivalence called strong isomorphism. Operational equivalence, as introduced by Morris [23] and Plotkin [27], treats programs as black boxes. Two expressions are operationally equivalent if they are indistinguishable in all program contexts. This equivalence is the basis for soundness results for program calculi and program transformation theories. Strong isomorphism, as introduced by Mason [14], also treats programs as black boxes. Two expressions are strongly isomorphic if in all memory states they return the same value, and have the same effect on memory (modulo the production of garbage). Strong isomorphism implies operational equivalence. The converse is true for first-order languages; it is false for full higher-order languages. However, even in the higher-order case, it remains an useful tool for establishing equivalence. Since strong isomorphism is defined by quantifying over memory states, rather than program contexts, it is a simple matter to restrict this equivalence to those memory states which satisfy a set of constraints. It is for this reason that strong isomorphism is a useful relation, even in the higher-order case. The formal system we present defines a single-conclusion consequence relation \Sigma ` \Phi where \Sigma is a finite set of constraints and \Phi is an assertion. The semantics of the formal system is given by a semantic consequence relation, \Sigma j= \Phi, defined in terms of a class of memory models for assertions and constraints. The assertions we consider are of the following two forms...

We describe computational science as an interdisciplinary approach to doing science on computers. Our purpose is to introduce computational science as a legitimate interest of computer scientists. We present a foundation for computational science based on the need to incorporate computation at the scientific level; i.e., computational aspects must be considered when a model is formulated. We next present some obstacles to computer scientists' participation in computational science, including a cultural bias in computer science that inhibits participation. Finally, we look at some areas of conventional computer science and indicate areas of mutual interest between computational science and computer science. Keywords: education, computational science. 1 What is Computational Science ? In December, 1991, the U. S. Congress passed the High Performance Computing and Communications Act, commonly known as the HPCC . This act focuses on several aspects of computing technology, but two have...

We describe the automatic generation of a provably correct compiler for a non-trivial subset of Ada. The compiler is generated from an action semantic description; it emits absolute code for an abstract RISC machine language that currently is assembled into code for the SPARC and the HP Precision Architecture. The generated code is an order of magnitude better than what is produced by compilers generated by the classical systems of Mosses, Paulson, and Wand. The use of action semantics makes the processable language specification easy to read and pleasant to work with. In Proc. ICCL'92, Fourth IEEE International Conference on Computer Languages, pages 117--126. 1 Introduction The purpose of a language designer's workbench, envisioned by Pleban, is to drastically improve the language design process. The major components in such a workbench are: ffl A specification language whose specifications are easily maintainable, and accessible without knowledge of the underlying theory; and f...