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Equivalence in Functional Languages with Effects
, 1991
"... Traditionally the view has been that direct expression of control and store mechanisms and clear mathematical semantics are incompatible requirements. This paper shows that adding objects with memory to the callbyvalue lambda calculus results in a language with a rich equational theory, satisfying ..."
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Cited by 111 (13 self)
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Traditionally the view has been that direct expression of control and store mechanisms and clear mathematical semantics are incompatible requirements. This paper shows that adding objects with memory to the callbyvalue lambda calculus results in a language with a rich equational theory, satisfying many of the usual laws. Combined with other recent work this provides evidence that expressive, mathematically clean programming languages are indeed possible. 1. Overview Real programs have effectscreating new structures, examining and modifying existing structures, altering flow of control, etc. Such facilities are important not only for optimization, but also for communication, clarity, and simplicity in programming. Thus it is important to be able to reason both informally and formally about programs with effects, and not to sweep effects either to the side or under the store parameter rug. Recent work of Talcott, Mason, Felleisen, and Moggi establishes a mathematical foundation for...
A Variable Typed Logic of Effects
 Information and Computation
, 1993
"... In this paper we introduce a variable typed logic of effects inspired by the variable type systems of Feferman for purely functional languages. VTLoE (Variable Typed Logic of Effects) is introduced in two stages. The first stage is the firstorder theory of individuals built on assertions of equalit ..."
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Cited by 48 (12 self)
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In this paper we introduce a variable typed logic of effects inspired by the variable type systems of Feferman for purely functional languages. VTLoE (Variable Typed Logic of Effects) is introduced in two stages. The first stage is the firstorder theory of individuals built on assertions of equality (operational equivalence `a la Plotkin), and contextual assertions. The second stage extends the logic to include classes and class membership. The logic we present provides an expressive language for defining and studying properties of programs including program equivalences, in a uniform framework. The logic combines the features and benefits of equational calculi as well as program and specification logics. In addition to the usual firstorder formula constructions, we add contextual assertions. Contextual assertions generalize Hoare's triples in that they can be nested, used as assumptions, and their free variables may be quantified. They are similar in spirit to program modalities in ...
References, Local Variables and Operational Reasoning
 In Seventh Annual Symposium on Logic in Computer Science
, 1992
"... this paper we regard the following as synonyms: references, program variables, pointers, locations, and unary cells) to a programming language complicates life. Adding them to the simply typed lambda calculus causes the failure of most of the nice mathematical properties and some of the more basic r ..."
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Cited by 30 (4 self)
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this paper we regard the following as synonyms: references, program variables, pointers, locations, and unary cells) to a programming language complicates life. Adding them to the simply typed lambda calculus causes the failure of most of the nice mathematical properties and some of the more basic rules (such as j). For example strong normalization fails since it is possible, for each provably nonempty function type, to construct a Y combinator for that type. References also interact unpleasantly with polymorphism [34, 35]. They are also troublesome from a denotational point of view as illustrated by the lack of fully abstract models. For example, in [22] Meyer and Sieber give a series of examples of programs that are operationally equivalent (according to the intended semantics of blockstructured Algollike programs) but which are not given equivalent denotations in traditional denotational semantics. They propose various modifications to the denotational semantics which solve some of these discrepancies, but not all. In [27, 26] a denotational semantics that overcomes some of these problems is presented. However variations on the seventh example remain problematic. Since numerous proof systems for Algol are sound for the denotational models in question, [8, 7, 32, 28, 16, 27, 26], these equivalences, if expressible, must be independent of these systems. The problem which motivated Meyer and Sieber's paper, [22], was to provide mathematical justification for the informal but convincing proofs of the operational equivalence of their examples. In this paper we approach the same problem, but from an operational rather than denotational perspective. This paper accomplishes two goals. Firstly, we present the firstorder part of a new logic for reasoning about programs....
Theories With SelfApplication and Computational Complexity
 Information and Computation
, 2002
"... Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the early seventies. In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: selfapplication is meaningful, but n ..."
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Cited by 12 (9 self)
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Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the early seventies. In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: selfapplication is meaningful, but not necessarily total. It has turned out that theories with selfapplication provide a natural setting for studying notions of abstract computability, especially from a prooftheoretic perspective.
Using Reflection to Explain and Enhance Type Theory
 Proof and Computation, volume 139 of NATO Advanced Study Institute, International Summer School held in Marktoberdorf, Germany, July 20August 1, NATO Series F
, 1994
"... The five lectures at Marktoberdorf on which these notes are based were about the architecture of problem solving environments which use theorem provers. Experience with these systems over the past two decades has shown that the prover must be extensible, yet it must be kept safe. We examine a way to ..."
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Cited by 11 (5 self)
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The five lectures at Marktoberdorf on which these notes are based were about the architecture of problem solving environments which use theorem provers. Experience with these systems over the past two decades has shown that the prover must be extensible, yet it must be kept safe. We examine a way to safely add new decision procedures to the Nuprl prover. It relies on a reflection mechanism and is applicable to any tacticoriented prover with sufficient reflection. The lectures explain reflection in the setting of constructive type theory, the core logic of Nuprl.
Reasoning about Object Systems in VTLoE
, 1994
"... VTLoE (Variable Type Logic of Effects) is a logic for reasoning about imperative functional programs inspired by the variable type systems of Feferman. The underlying programming language, mk , extends the callbyvalue lambda calculus with primitives for arithmetic, pairing, branching, and refere ..."
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Cited by 11 (3 self)
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VTLoE (Variable Type Logic of Effects) is a logic for reasoning about imperative functional programs inspired by the variable type systems of Feferman. The underlying programming language, mk , extends the callbyvalue lambda calculus with primitives for arithmetic, pairing, branching, and reference cells (mutable data). In VTLoE one can reason about program equivalence and termination, input/output relations, program contexts, and inductively (and coinductively) define data structures. In this paper we present a refinement of VTLoE. We then introduce a notion of object specification and establish formal principles for reasoning about object systems within VTLoE. Objects are selfcontained entities with local state. The local state of an object can only be changed by action of that object in response to a message. In mk objects are represented as closures with mutable data bound to local variables. A semantic principle called simulation induction was introduced in our earlier wor...
A Theory of Classes for a Functional Language with Effects
 In Proceedings of CSL92, volume 702 of Lecture Notes in Computer Science
, 1993
"... this paper we introduce a variable typed logic of effects (i.e. a logic of effects where classes can be defined and quantified over) inspired by the variable type systems of Feferman [3, 4] for purely functional languages. A similar extension incorporating nonlocal control operations was introduced ..."
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Cited by 7 (6 self)
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this paper we introduce a variable typed logic of effects (i.e. a logic of effects where classes can be defined and quantified over) inspired by the variable type systems of Feferman [3, 4] for purely functional languages. A similar extension incorporating nonlocal control operations was introduced in [27]. The logic we present provides an expressive language for defining specifications and constraints and for studying properties and program equivalences, in a uniform framework. Thus it has an advantage over a plethora of systems in the literature that aim to capture solitary aspects of computation. The theory also allows for the construction of inductively defined sets and derivation of the corresponding induction principles. Classes can be used to express, inter alia, the nonexpansiveness of terms [29]. Other effects can also be represented within the system. These include read/write effects and various forms of interference [24]. The first order fragment is described in [16] where it is used to resolve the denotationally problematic examples of [17]. In our language atoms, references and lambda abstractions are all first class values and as such are storable. This has several consequences. Firstly, mutation and variable binding are separate and so we avoid the problems that typically arise (e.g. in Hoare's and dynamic logic) from the conflation of program variables and logical variables. Secondly, the equality and sharing of references (aliasing) is easily expressed and reasoned about. Thirdly, the combination of mutable references and lambda abstractions allows us to study object based programming within our framework. Our atomic formulas express the (operational or observational) equivalence of programs `a la Plotkin [23]. Neither Hoare's logic nor Dynamic logi...
Program Transformation via Contextual Assertions
 In Logic, Language and Computation. Festschrift in Honor of Satoru Takasu
, 1994
"... . In this paper we describe progress towards a theory of tranformational program development. The transformation rules are based on a theory of contextual equivalence for functional languages with imperative features. Such notions of equivalence are fundamental for the process of program specificati ..."
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Cited by 7 (4 self)
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. In this paper we describe progress towards a theory of tranformational program development. The transformation rules are based on a theory of contextual equivalence for functional languages with imperative features. Such notions of equivalence are fundamental for the process of program specification, derivation, transformation, refinement and other forms of code generation and optimization. This paper is dedicated to Professor Satoru Takasu. 1 Introduction This paper describes progress towards a theory of program development by systematic refinement beginning with a clean simple program thought of as a specification. Transformations include reuse of storage, and rerepresentation of abstract data. The transformation rules are based on a theory of constrained equivalence for functional languages with imperative features (i.e. Lisp, Scheme or ML). Such notions of equivalence are fundamental for the process of program specification, derivation, transformation, refinement, and other for...
First Steps Into Metapredicativity in Explicit Mathematics
, 1999
"... The system EMU of explicit mathematics incorporates the uniform construction of universes. In this paper we give a prooftheoretic treatment of EMU and show that it corresponds to transfinite hierarchies of fixed points of positive arithmetic operators, where the length of these fixed point hierarc ..."
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Cited by 5 (2 self)
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The system EMU of explicit mathematics incorporates the uniform construction of universes. In this paper we give a prooftheoretic treatment of EMU and show that it corresponds to transfinite hierarchies of fixed points of positive arithmetic operators, where the length of these fixed point hierarchies is bounded by # 0 . 1 Introduction Metapredicativity is a new general term in proof theory which describes the analysis and study of formal systems whose prooftheoretic strength is beyond the FefermanSchutte ordinal # 0 but which are nevertheless amenable to purely predicative methods. Typical examples of formal systems which are apt for scaling the initial part of metapredicativity are the transfinitely iterated fixed point theories # ID # whose detailed prooftheoretic analysis is given by Jager, Kahle, Setzer and Strahm in [18]. In this paper we assume familiarity with [18]. For natural extensions of Friedman's ATR that can be measured against transfinitely iterated fixed point ...
Themes in Final Semantics
 Dipartimento di Informatica, Università di
, 1998
"... C'era una volta un re seduto in canap`e, che disse alla regina raccontami una storia. La regina cominci`o: "C'era una volta un re seduto in canap`e ..."
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Cited by 5 (2 self)
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C'era una volta un re seduto in canap`e, che disse alla regina raccontami una storia. La regina cominci`o: &quot;C'era una volta un re seduto in canap`e