## A Variable Typed Logic of Effects (1993)

Venue: | Information and Computation |

Citations: | 49 - 13 self |

### BibTeX

@ARTICLE{Honsell93avariable,

author = {Furio Honsell and Ian A. Mason and Scott Smith and Carolyn Talcott},

title = {A Variable Typed Logic of Effects},

journal = {Information and Computation},

year = {1993},

volume = {119},

pages = {55--90}

}

### Years of Citing Articles

### OpenURL

### Abstract

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 first-order 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 first-order 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 ...

### Citations

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455 | Computational lambda-calculus and monads
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347 |
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Citation Context ... reduction system for this, and related, imperatives-calculi was discovered independently by three people in 1986-1987: Carolyn Talcott (Mason and Talcott, 1991a), Matthias Felleisen and Robert Hieb (=-=Felleisen and Hieb, 1992-=-). As well as being conceptually elegant, it has also provided the necessary tools for several key results and proofs. It provided the basis for an elegant revision of (Felleisen, 1987; Felleisen and ... |

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233 | Domain theory in logical form
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191 | The call-by-need lambda calculus
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Citation Context ...notable successful uses of the technique is the type soundness proof, via subject reduction, of the imperative ML type system (Felleisen and Wright, 1991). The analysis of parameter passing in Algol (=-=Crank and Felleisen, 1991-=-; Weeks and Felleisen, 1993). The analysis of reduction calculi for Scheme-like languages (Sabry and Felleisen, 1993; Fields and Sabry, 1993). In 1987 Ian Mason realized that it provided the ideal not... |

164 | Reasoning about programs in continuation-passing style
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Citation Context ...pe system (Felleisen and Wright, 1991). The analysis of parameter passing in Algol (Crank and Felleisen, 1991; Weeks and Felleisen, 1993). The analysis of reduction calculi for Scheme-like languages (=-=Sabry and Felleisen, 1993-=-; Fields and Sabry, 1993). In 1987 Ian Mason realized that it provided the ideal notion of a normal form and symbolic evaluation needed in the completeness result presented in (Mason and Talcott, 1992... |

134 |
Lambda Calculus Models of Programming Languages
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- 1968
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Citation Context ...es, reduction contexts, and memory contexts respectively. We define the operational equivalence relation and study its general properties. Our definition extends the equivalence relations defined in (=-=Morris, 1968-=-) and (Plotkin, 1975). In general it is very difficult to establish the operational equivalence of expressions. Thus it is desirable to have a simpler characterization of operational equivalence, one ... |

134 |
Syntactic Control of Interference
- Reynolds
- 1978
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Citation Context .... Other effects can also be represented within the system. These include read/write effects (Lucassen, 1987; Lucassen and Gifford, 1988; Jouvelot and Gifford, 1991) and various forms of interference (=-=Reynolds, 1978-=-; Reynolds, 1982). 3 The paper is divided into three parts. (I) The first part describes the syntax and operational semantics of our language, and establishes the basic results concerning the correspo... |

118 | Observable properties of higher order functions that dynamically create local names, or: What’s new
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- 1993
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Citation Context ...over memory structures. As shown by (Abramsky, 1990; Abramsky, 1991; Bloom, 1990; Egidi et al., 1992; Howe, 1989; Mason, 1986b; Mason and Talcott, 1991a; Jim and Meyer, 1991; Milner, 1977; Ong, 1988; =-=Pitts and Stark, 1993; Smith,-=- 1992; Talcott, 1985) operational equivalence and approximation can be characterized in various ways. Definition ( �� =): Two expressions are operationally equivalent, written e 0 �� = e 1 , i... |

117 |
Type inference for polymorphic references
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- 1990
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Citation Context ...ll z to be v. Then the desired functional is Y = f:letfz := mk(nil)gseq(set(z; x:f(get(z); x)); get(z)) In addition, references are problematic for polymorphic type systems (Damas, 1985; Tofte, 1988; =-=Tofte, 1990-=-; Leroy and Wies, 1990). Even when references are only allowed to contain numbers they are troublesome from a denotational point of view as illustrated by the absence of fully abstract models. For exa... |

113 | Equivalence in functional languages with effects
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Citation Context ...sults concerning the corresponding notion of operational equivalence. We also discuss some of the subtleties that arise due to the presence of mutable data. Most of the results in this part are from (=-=Mason and Talcott, 1991-=-a). (II) In part two we describe the first order aspects of our logic of effects, and provide examples of their use. These examples include proofs within VTLoE of the validity of the examples of Meyer... |

111 | Algebraic reconstruction of types and effects
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Citation Context ...press, inter alia, the non-expansiveness of terms (Tofte, 1990). Other effects can also be represented within the system. These include read/write effects (Lucassen, 1987; Lucassen and Gifford, 1988; =-=Jouvelot and Gifford, 1991-=-) and various forms of interference (Reynolds, 1978; Reynolds, 1982). 3 The paper is divided into three parts. (I) The first part describes the syntax and operational semantics of our language, and es... |

103 | Parametricity and local variables
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Citation Context ...enotations in traditional denotational semantics. They propose various modifications to the denotational semantics which solve some of these discrepancies, but not all. In (O'Hearn and Tennent, 1992; =-=O'Hearn and Tennent, 1993-=-b) 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 d... |

98 |
Type Assignment in Programming Languages
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- 1985
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Citation Context ...ets the contents of the cell z to be v. Then the desired functional is Y = f:letfz := mk(nil)gseq(set(z; x:f(get(z); x)); get(z)) In addition, references are problematic for polymorphic type systems (=-=Damas, 1985-=-; Tofte, 1988; Tofte, 1990; Leroy and Wies, 1990). Even when references are only allowed to contain numbers they are troublesome from a denotational point of view as illustrated by the absence of full... |

98 |
Towards a fully abstract semantics for local variables
- Meyer, Sieber
- 1988
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Citation Context ...d Wies, 1990). Even when references are only allowed to contain numbers they are troublesome from a denotational point of view as illustrated by the absence of fully abstract models. For example, in (=-=Meyer and Sieber, 1988-=-) they give a series of seven examples of programs that are operationally equivalent (according to the intended semantics of block-structured Algol-like programs) but which are not given equivalent de... |

94 | Operational Semantics and Polymorphic Type Inference
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- 1988
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Citation Context ...mma j= \Phi[oefX := Cg]) It is important to note that if \Gamma j= e 2 K [oe], then e evaluates (in the appropriate state) to a value without altering memory, the so-called non-expansive expressions (=-=Tofte, 1988-=-; Tofte, 1990). 6 5.3. Simple Examples We define (extensional) equality and subset relations on classes in the usual manner. K 0 ` K 1 , (8x)(x 2 K 0 ) x 2 K 1 ) K 0 j K 1 , K 0 ` K 1sK 1 ` K 0 As a c... |

91 |
A language and axioms for explicit mathematics
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- 1975
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Citation Context ...ic of effects inspired by the variable type systems of Feferman. These systems are two sorted theories of operations and classes initially developed for the formalization of constructive mathematics (=-=Feferman, 1975-=-; Feferman, 1979) and later applied to the study of purely functional languages (Feferman, 1985; Feferman, 1990). A similar extension incorporating non-local control effects was introduced in (Talcott... |

91 | Categorical logic
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- 1996
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Citation Context ...otably Hoare's logic (Apt, 1981), Dynamic logic (Harel, 1984), Reynolds Specification Logic (Reynolds, 1982), Moggi's metalanguage for computational monads (Moggi, 1991), and Pitt's Evaluation Logic (=-=Pitts, 1990-=-). All are program logics of the exogenous kind, i.e. programs appear in formulas. 2 Hoare's logic is quite weak in the sense that it cannot express termination of programs, nor their equivalence. Dyn... |

84 |
Control operators, the SECD-machine, and the *-calculus
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- 1986
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Citation Context ...entify the subexpression of an expression that is to be evaluated next, they correspond to the standard reduction strategy (left-first, call-by-value) of (Plotkin, 1975) and were first introduced in (=-=Felleisen and Friedman, 1986-=-). Definition (R): The set of reduction contexts, R, is the subset of C defined by R = ffflg + app(R; E ) + app(V; R) + [ n;m2N F m+n+1 (V m ; R; E n ) We let R range over R. An expression is either a... |

67 |
Constnictive theories of functions and classes
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- 1979
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Citation Context ...spired by the variable type systems of Feferman. These systems are two sorted theories of operations and classes initially developed for the formalization of constructive mathematics (Feferman, 1975; =-=Feferman, 1979-=-) and later applied to the study of purely functional languages (Feferman, 1985; Feferman, 1990). A similar extension incorporating non-local control effects was introduced in (Talcott, 1993). VTLoE (... |

58 |
Types and effects, towards an integration of functional and imperative programming
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- 1987
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Citation Context ...uction principles. Classes can be used to express, inter alia, the non-expansiveness of terms (Tofte, 1990). Other effects can also be represented within the system. These include read/write effects (=-=Lucassen, 1987-=-; Lucassen and Gifford, 1988; Jouvelot and Gifford, 1991) and various forms of interference (Reynolds, 1978; Reynolds, 1982). 3 The paper is divided into three parts. (I) The first part describes the ... |

50 | Polymorphic type inference and assignment
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- 1991
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Citation Context ... Then the desired functional is Y = f:letfz := mk(nil)gseq(set(z; x:f(get(z); x)); get(z)) In addition, references are problematic for polymorphic type systems (Damas, 1985; Tofte, 1988; Tofte, 1990; =-=Leroy and Wies, 1990-=-). Even when references are only allowed to contain numbers they are troublesome from a denotational point of view as illustrated by the absence of fully abstract models. For example, in (Meyer and Si... |

48 |
Idealized Algol and its specification logic
- Reynolds
- 1982
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Citation Context ... reasoning about properties of programs written in general programming languages have been proposed, most notably Hoare's logic (Apt, 1981), Dynamic logic (Harel, 1984), Reynolds Specification Logic (=-=Reynolds, 1982-=-), Moggi's metalanguage for computational monads (Moggi, 1991), and Pitt's Evaluation Logic (Pitts, 1990). All are program logics of the exogenous kind, i.e. programs appear in formulas. 2 Hoare's log... |

45 |
Fully abstract models of typed *-calculi
- Milner
- 1977
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Citation Context ...in, 1975) to computation over memory structures. As shown by (Abramsky, 1990; Abramsky, 1991; Bloom, 1990; Egidi et al., 1992; Howe, 1989; Mason, 1986b; Mason and Talcott, 1991a; Jim and Meyer, 1991; =-=Milner, 1977; On-=-g, 1988; Pitts and Stark, 1993; Smith, 1992; Talcott, 1985) operational equivalence and approximation can be characterized in various ways. Definition ( �� =): Two expressions are operationally eq... |

43 |
Ten years of Hoare’s logic: A survey–part I
- Apt
- 1981
(Show Context)
Citation Context ...this model remains an open problem. In recent years various systems for reasoning about properties of programs written in general programming languages have been proposed, most notably Hoare's logic (=-=Apt, 1981-=-), Dynamic logic (Harel, 1984), Reynolds Specification Logic (Reynolds, 1982), Moggi's metalanguage for computational monads (Moggi, 1991), and Pitt's Evaluation Logic (Pitts, 1990). All are program l... |

41 |
The Lazy Lambda Calculus: an Investigation into the Foundations of Functional Programming
- Ong
- 1988
(Show Context)
Citation Context ...omputation over memory structures. As shown by (Abramsky, 1990; Abramsky, 1991; Bloom, 1990; Egidi et al., 1992; Howe, 1989; Mason, 1986b; Mason and Talcott, 1991a; Jim and Meyer, 1991; Milner, 1977; =-=Ong, 1988; Pi-=-tts and Stark, 1993; Smith, 1992; Talcott, 1985) operational equivalence and approximation can be characterized in various ways. Definition ( �� =): Two expressions are operationally equivalent, w... |

39 |
A syntactic theory of sequential state
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- 1989
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Citation Context ...and Hieb, 1992). As well as being conceptually elegant, it has also provided the necessary tools for several key results and proofs. It provided the basis for an elegant revision of (Felleisen, 1987; =-=Felleisen and Friedman, 1989-=-) that was later published in (Felleisen and Hieb, 1992). Other notable successful uses of the technique is the type soundness proof, via subject reduction, of the imperative ML type system (Felleisen... |

35 | The Calculi of Lambda-v-cs Conversion: A Syntactic Theory of Control and State in Imperative Higher-Order Programming Languages - Felleisen - 1987 |

35 | Semantics of local variables
- O’Hearn, Tennent
- 1992
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Citation Context ... 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 (=-=O'Hearn and Tennent, 1992-=-; O'Hearn and Tennent, 1993b) a denotational semantics that overcomes some of these problems is presented. However variations on the seventh example remain problematic. Since numerous proof systems fo... |

30 | References, local variables and operational reasoning
- Mason, Talcott
(Show Context)
Citation Context ...e same cons cell, then after assignment there is a cycle starting from w and hence the class of w has changed. The usual (first-order) laws for S-expression operations hold (Mason and Talcott, 1991a; =-=Mason and Talcott, 1992a). We use x = [-=-x a ; x d ] to abbreviate the formula cons?(x) �� = tscar(x) �� = x ascar(x) �� = x d . Lemma (cons axioms): (i) letfx := cons(x a ; x d )g[[:(x �� = y)sx = [x a ; x d ]]] x fresh (ii)... |

27 |
A Theory of Binding Structures and Applications to Rewriting
- Talcott
- 1993
(Show Context)
Citation Context ...erning the corresponding notion of operational equivalence. We also discuss some of the subtleties that arise due to the presence of mutable data. Most of the results in this part are from (Mason and =-=Talcott, 1991-=-a). (II) In part two we describe the first order aspects of our logic of effects, and provide examples of their use. These examples include proofs within VTLoE of the validity of the examples of Meyer... |

26 | Inferring the equivalence of functional programs that mutate data
- Mason, Talcott
- 1992
(Show Context)
Citation Context ... of the use of VTLoE in specifying and verifying a program. Part II is independent of part III, while much of part III is orthogonal to part II. Earlier versions of part II of this paper appeared as (=-=Mason and Talcott, 1992-=-b). Earlier versions of part III of this paper appeared as (Honsell et al., 1993). 1.1.1. Part I In our language atoms, references and lambda abstractions are all first class values and as such are st... |

25 |
Trakhtenbrot, The semantics of local storage, or what makes the free-list free
- Halpern, Meyer, et al.
- 1984
(Show Context)
Citation Context ...e 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, (Halpern et al., 1984; =-=Halpern et al., 1983-=-; Sieber, 1985; Olderog, 1984; Manna and Waldinger, 1981; O'Hearn and Tennent, 1992; O'Hearn and Tennent, 1993b), these equivalences, if expressible, must be independent of these systems. More recentl... |

23 |
Polymorphic typed lambda-calculi in a type-free axiomatic framework
- Feferman
- 1990
(Show Context)
Citation Context ...tions and classes initially developed for the formalization of constructive mathematics (Feferman, 1975; Feferman, 1979) and later applied to the study of purely functional languages (Feferman, 1985; =-=Feferman, 1990-=-). A similar extension incorporating non-local control effects was introduced in (Talcott, 1993). VTLoE (Variable Type Logic of Effects) is introduced in two stages. The first stage is the first-order... |

23 |
The Semantics of Destructive Lisp
- Mason
- 1986
(Show Context)
Citation Context ...al equivalence relations defined by (Morris, 1968) and (Plotkin, 1975) to computation over memory structures. As shown by (Abramsky, 1990; Abramsky, 1991; Bloom, 1990; Egidi et al., 1992; Howe, 1989; =-=Mason, 1986-=-b; Mason and Talcott, 1991a; Jim and Meyer, 1991; Milner, 1977; Ong, 1988; Pitts and Stark, 1993; Smith, 1992; Talcott, 1985) operational equivalence and approximation can be characterized in various ... |

21 | Full Abstraction and the Context Lemma
- Jim, Meyer
- 1996
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Citation Context ...ris, 1968) and (Plotkin, 1975) to computation over memory structures. As shown by (Abramsky, 1990; Abramsky, 1991; Bloom, 1990; Egidi et al., 1992; Howe, 1989; Mason, 1986b; Mason and Talcott, 1991a; =-=Jim and Meyer, 1991; Mi-=-lner, 1977; Ong, 1988; Pitts and Stark, 1993; Smith, 1992; Talcott, 1985) operational equivalence and approximation can be characterized in various ways. Definition ( �� =): Two expressions are op... |

19 | Axiomatizing operational equivalence in the presence of side effects
- Mason, Talcott
- 1989
(Show Context)
Citation Context ...) if(y; e 0 ; e 1 ) �� = e 0 (v) (atom(x 0 ) �� = tsatom(x) �� = nil) , :(x 0 �� = x 1 ) 3.8. Memory Operation Principles This logic extends and improves the complete first order syste=-=m presented in (Mason and Talcott, 1989-=-; Mason and Talcott, 1992a). There certain reasoning principles were established as basic, and from these all others, suitably restricted, could be derived using simple equational reasoning. The syste... |

19 | Partial Objects in Type Theory
- Smith
- 1989
(Show Context)
Citation Context ...s much too rich to be considered a type system. One problem that arises is that fixed-point combinators can not be uniformly typed over all classes. This problem arises even in the absence of memory (=-=Smith, 1988-=-; Talcott, 1993). Theorem (FixTypeFails): Let Y v by any fixed-point combinator (such that f(Y v (f)) = Y v (f )). Then it is not the case that f 2 (C ! C) ) Y v (f) 2 C for all function classes C (C ... |

18 | From operational to denotational semantics
- Smith
- 1992
(Show Context)
Citation Context ... As shown by (Abramsky, 1990; Abramsky, 1991; Bloom, 1990; Egidi et al., 1992; Howe, 1989; Mason, 1986b; Mason and Talcott, 1991a; Jim and Meyer, 1991; Milner, 1977; Ong, 1988; Pitts and Stark, 1993; =-=Smith, 1992; Talcot-=-t, 1985) operational equivalence and approximation can be characterized in various ways. Definition ( �� =): Two expressions are operationally equivalent, written e 0 �� = e 1 , if for any clo... |

16 | Faithful ideal models for recursive polymorphic types
- Abadi, Pierce, et al.
- 1991
(Show Context)
Citation Context ... \Phi : (e; �� b )) ) \Phi : (x:e; �� a ! �� b )) (3) V i!n \Phi : (x i ; �� i ) ) ((\Phi : (e; �� a ! �� b )s\Phi : (e a ; �� a )) ) (\Phi : (app(e; e a ); �� b ) The =-=principle of type faithfulness (Abadi et al., 1991) e a �� = e b )-=- (( i!n \Phi : (x i ; �� i ) ) \Phi : (e a ; ��)) ) ( i!n \Phi : (x i ; �� i ) ) \Phi : (e b ; �� ))) is also a desirable property. Note that in our framework this implies that a subte... |

13 |
From denotational to operational and axiomatic semantics for ALGOL-like languages: An overview
- Trakhtenbrot, Halpern, et al.
- 1984
(Show Context)
Citation Context ...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, (=-=Halpern et al., 1984-=-; Halpern et al., 1983; Sieber, 1985; Olderog, 1984; Manna and Waldinger, 1981; O'Hearn and Tennent, 1992; O'Hearn and Tennent, 1993b), these equivalences, if expressible, must be independent of these... |

12 |
A theory of variable types. Revista Colombiana de Matem'aticas
- Feferman
- 1975
(Show Context)
Citation Context ...heories of operations and classes initially developed for the formalization of constructive mathematics (Feferman, 1975; Feferman, 1979) and later applied to the study of purely functional languages (=-=Feferman, 1985-=-; Feferman, 1990). A similar extension incorporating non-local control effects was introduced in (Talcott, 1993). VTLoE (Variable Type Logic of Effects) is introduced in two stages. The first stage is... |

12 | Reasoning about explicit and implicit representations of state
- Sabry, Field
- 1993
(Show Context)
Citation Context ...ight, 1991). The analysis of parameter passing in Algol (Crank and Felleisen, 1991; Weeks and Felleisen, 1993). The analysis of reduction calculi for Scheme-like languages (Sabry and Felleisen, 1993; =-=Fields and Sabry, 1993-=-). In 1987 Ian Mason realized that it provided the ideal notion of a normal form and symbolic evaluation needed in the completeness result presented in (Mason and Talcott, 1992a). The approach Felleis... |

12 |
Verification of programs that destructively manipulate data
- Mason
- 1988
(Show Context)
Citation Context ..., append, that concatenates its first argument with its second, copying the top level list structure of its first argument in the process. This example was first dealt with in depth in (Mason, 1986b; =-=Mason, 1988-=-) Definition (append): append(x; y) / if(eq(x; nil); y; cons(car(x); f(cdr(x); y))) The problem with this definition of append is that to perform the cons in the non-trivial case we must first compute... |

9 |
Can LCF be topped
- Bloom
- 1990
(Show Context)
Citation Context ...ce them. Our definition extends the extensional equivalence relations defined by (Morris, 1968) and (Plotkin, 1975) to computation over memory structures. As shown by (Abramsky, 1990; Abramsky, 1991; =-=Bloom, 1990-=-; Egidi et al., 1992; Howe, 1989; Mason, 1986b; Mason and Talcott, 1991a; Jim and Meyer, 1991; Milner, 1977; Ong, 1988; Pitts and Stark, 1993; Smith, 1992; Talcott, 1985) operational equivalence and a... |