## Computational Interpretations of Linear Logic (1993)

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Venue: | Theoretical Computer Science |

Citations: | 281 - 3 self |

### BibTeX

@ARTICLE{Abramsky93computationalinterpretations,

author = {Samson Abramsky},

title = {Computational Interpretations of Linear Logic},

journal = {Theoretical Computer Science},

year = {1993},

volume = {111},

pages = {3--57}

}

### Years of Citing Articles

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### Abstract

We study Girard's Linear Logic from the point of view of giving a concrete computational interpretation of the logic, based on the Curry-Howard isomorphism. In the case of Intuitionistic Linear Logic, this leads to a refinement of the lambda calculus, giving finer control over order of evaluation and storage allocation, while maintaining the logical content of programs as proofs, and computation as cut-elimination.

### Citations

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Citation Context ...first or the second projection, and this is the only part of the input we will ever see. So the additive conjunction A & B appears as a kind of choice---an "external" choice in the terminolo=-=gy of CSP [Hoa85]-=-, since it is made by the consumer of the datum. ffl The disjoint sum (additive disjunction) A \Phi B appears as an internal choice, since it is at the discretion of the producer of the datum whether ... |

3217 |
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Citation Context ...ourse(P )sQQ \Gamma! Cut(OfCourse(oe(PsQ)); Q): As the commutative conversions are not part of our calculus, we need a more elaborate argument. We shall use a technique inspired by concurrency theory =-=[Mil89]-=-. We say that a relation R ` PE 2 is a simulation if it satisfies PRQ & P \Gamma! R =) 9P 0 ; Q 0 :(R \Gamma! P 0 & Q \Gamma! + Q 0 & P 0 RQ 0 ): We need to establish some notation. We can write OfCou... |

1566 | The definition of Standard ML
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Citation Context ... canonical forms. Note that any abstraction x:t is canonical, since we only evaluate programs, and t may not be closed. This exactly mirrors what is done in actual eager evaluation languages, e.g. ML =-=[MTH90]-=-. Evaluation relation: t+c u+d ht; ui+hc; di t+hc; di fst(t)+c t+hc; di snd(t)+d x:t+x:t t+x:v u+c v[c=x]+d tu+d t+c inl(t)+inl(c) u+d inr(u)+inr(d) t+inl(c) u[c=x]+d case t of inl(x) ) u j inr(y) ) v... |

1332 | A Calculus of Communicating Systems - Milner - 1982 |

1311 | Introduction to Functional Programming - Bird, Wadler - 1988 |

1045 |
Introduction to Linear and Nonlinear Programming
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Citation Context ...I learnt from their presentations. The computational interpretation of Intuitionistic Linear Logic described in Section 3 clearly owes much to previous work by Yves Lafont [Laf88] and Soren Holmstrom =-=[Hol88]-=-. However, the key ingredients seem to me to be either new, or clarified and simplified in an essential fashion by comparison with these previous works. Similar remarks apply to the discussion of prag... |

619 | Linear logic - Girard - 1987 |

536 |
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Citation Context ...menting the copying of the !-type by sharing. With a little additional work, we can ensure that the standard function type A ) B = !A \Gammaffi B is implemented by the standard call-by-need technique =-=[PJ87]. We shall-=- briefly describe how this can be done. Firstly, we introduce a new instruction upd, and a new form of value ocv(v), representing the "consolidation" of a value into a !-closure. The compila... |

453 | The chemical abstract machine
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Citation Context ...ection 6 we introduce proof expressions as a notation for proofs in Classical Linear Logic, and present a concurrent operational semantics in the style of Berry and Boudol's Chemical Abstract Machine =-=[BB90]-=-. The basic theoretical properties of the proof-expression calculus are established in Section 7. Finally, a parallel implementation is sketched in Section 8. Acknowledgements It is a great pleasure t... |

341 |
Intuitionistic type theory
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Citation Context ...n Section 2, we review the connections between Intuitionistic Logic and typed -calculus (functional programming), and describe the operational semantics of -calculus in a style inspired by Martin-Lof =-=[ML84], and currently (as -=-"Natural" or "Relational" Semantics) widely used in programming language specification [Kah87, MTH90]. We give the correspondence for both the Natural Deduction and Sequent Calculu... |

332 |
The mechanical evaluation of expressions
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Citation Context ...sible applications to static program analysis and optimization, and give a detailed description of a (sequential) implementation of the linear term calculus, in terms of a variant of the SECD machine =-=[Lan64]-=-. Then in Section 5 we establish some of the basic theoretical properties of the calculus, in its second-order propositional version. We turn to Classical Linear Logic in sections 6--8. This should be... |

315 |
Functions as Processes
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(Show Context)
Citation Context ...ceeding in lockstep. (So coequations could be modelled by "membranes" in Berry and Boudol's terminology; but we shall not pursue this idea.) We distinguish between two kinds of rule for the =-=CHAM (cf. [Mil90]): structu-=-ral rules, which describe the "magical mixing" of the solution; and reaction rules, which describe the actual computation steps. Structural Rules There are two basic structural rules: ffl t ... |

302 | Natural semantics - Kahn - 1987 |

292 |
Abstract types have existential types
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- 1985
(Show Context)
Citation Context ...ion relation. Stripping off the types again, this establishes the Proposition. The appearance of 9 in the hypotheses of this proposition should not be a surprise. 9 provides "information hiding&q=-=uot; (cf. [MP85]-=-), and the information hidden may well include the use of lazy types. Theorem 7.20 If PE 2 ` \Theta;st : \Gamma, \Gamma does not contain any occurrences of &, ! or 9, and \Theta;st+P , then P is cut-f... |

284 |
Interpretation fonctionelle et elimination des coupures de l'arithmetique d'ordre superieur
- Girard
- 1972
(Show Context)
Citation Context ... one of the two rules corresponding to (\Phi \Gamma L) is applicable. We now turn to Convergence. Our proof is a simplified and suitably modified version of Girard's original proof of SN for System F =-=[Gir72]-=-; see [Gal90] for a good exposition. The idea is to use the evaluation relation to give a realizability semantics for types. We take a semantic type to be a set of closed linear terms, i.e. a subset o... |

232 |
Principal type schemes for functional programs
- Damas, Milner
- 1982
(Show Context)
Citation Context ...o sequent proofs or proof nets [Gir87]. Again, if we extend the calculus with general recursion, Strong Normalization will definitely be lost, while Convergence can be refined into Semantic Soundness =-=[DM82]-=- plus Computational Adequacy [MC88]. ffl As already explained, the evaluation relation reflects the intrinsic computational content of the Linear types, and so is the natural object of study. ffl The ... |

202 | A Basis for a Mathematical Theory of Computation
- McCarthy
- 1963
(Show Context)
Citation Context ...terms into two groups: constructors (corresponding to introduction rules) and destructorss(corresponding to elimination rules). (This classification of course goes right back to the pioneers McCarthy =-=[McC63]-=- and Landin [Lan64], although they lacked the prooftheoretic perspective.) Constructors produce information (pieces of structured data); destructorssconsume it. The basic unit of computation (reductio... |

170 |
Call-by-name, call-by-value and the lambda-calculus
- Plotkin
- 1975
(Show Context)
Citation Context ...es not enforce a strategy on us. This is not too surprising, since extensionalsdifferences between these strategies only show up in the -calculus in the presence of non-terminating programs (see e.g. =-=[Plo75]-=-), and under the strict correspondence of typed programs with proofs, we have Strong Normalization, so that all programs---and indeed all strategies---terminate. (However, it is one of the most notabl... |

156 |
The Lambda Calculus, its Syntax and Semantics. NorthHolland, 2nd edition
- Barendregt
- 1984
(Show Context)
Citation Context ... where \Theta corresponds to the uses of the Cut rule in \Pi. To ensure that suitable linearity constraints are satisfied, we shall adopt the following Name Convention (cf. the Variable Convention in =-=[Bar84]-=-): different names are introduced for each instance of the Axiom, With and Of Course rules. Proof Expression Assignment for CLL Axiom `; x : A ? ; x : A Exchange ` \Theta; \Gamma; t : A; u : B; \Delta... |

142 |
Miranda: A non-strict functional language with polymorphic types
- TURNER
- 1985
(Show Context)
Citation Context ...ections 6--8.) We will follow the very lucid exposition of the (standard) SECD machine given in [Hen80] fairly closely. The machine is based on a list-structured store. We shall use Turner's notation =-=[Tur85]-=- for list operations: [] for the empty list x : l for infix cons [x 1 ; : : : ; x n ] for x 1 : \Delta \Delta \Delta : x n : []. The objects manipulated by the machine are inductively defined as follo... |

140 |
Proofs and Types, volume 7 of Cambridge Tracts
- Girard, Lafont, et al.
- 1989
(Show Context)
Citation Context ...mselves draw inspiration from the mathematical structure of the language. 1 The paradigm followed by Girard in seeking to apply Linear Logic to computation is the "Curry-Howard" isomorphism =-=(see e.g. [GLT89]-=-), in which propositions (or logical formulae) are interpreted as types, proofs as programs, and the process of normalization or cut-elimination as computation. This paradigm has been a cornerstone of... |

133 | Linear types can change the world - Wadler - 1990 |

126 |
A correspondence between ALGOL 60 and Church’s lambda notation
- Landin
- 1965
(Show Context)
Citation Context ...pes. One might say that this gives a rational reconstruction, in logical terms, of the standard method for implementing lazy evaluation on top of an eager evaluation strategy, as introduced by Landin =-=[Lan65]-=-, and used in the SECD and CAM machines [Hen80, Hue90]. This idea is standardly modelled in denotational semantics by lifting [Plo85], i.e. A ) B = A ? * B where A * B is the type of partial (or alter... |

118 | Functional Programming - Field, Harrison - 1988 |

117 |
editors. Abstract Interpretation of Declarative Languages
- Abramsky, Hankin
- 1987
(Show Context)
Citation Context ...) evaluation; of course, the justification for these "improved" typings is that e.g. in K 0 , x is actually used. But this is exactly the kind of information that strictness analysis tries t=-=o extract [AH87]-=-. There are other uses for the information made explicit by the Linear types. If we know that a value is never shared (as it will not be unless it is of a !-type) then we can safely update it in place... |

117 |
The functional programming language Haskell
- Fasel, Hudak, et al.
- 1992
(Show Context)
Citation Context ...lowing the philosophy that types are compile-time constraints, and are not used in the actual computation. (This reflects current practice in languages such as ML [MTH90], Miranda [Tur85] and Haskell =-=[HW88]-=-.) The syntax of formulas is extended with propositional variables ff, fi, fl and the universal quantifier 8ff:A. The system ILL (i.e. the sequent formulation of Linear Logic as given in Section 3) is... |

100 |
Towards a geometry of interaction
- Girard
- 1989
(Show Context)
Citation Context ...claims made on its behalf. It should be said that this programme runs somewhat counter to that advocated by Girard. He has adopted the methodological principle of avoiding the "bureaucracy" =-=of syntax [Gir89b], aiming instead for-=- a "geometrical" view of computation, exemplified by the "Geometry of Interaction" [Gir89a], which interprets cut-elimination in Linear Logic by iterations of operators in C -algeb... |

91 |
The linear abstract machine
- Lafont
- 1988
(Show Context)
Citation Context ...ourse this is based on what I learnt from their presentations. The computational interpretation of Intuitionistic Linear Logic described in Section 3 clearly owes much to previous work by Yves Lafont =-=[Laf88]-=- and Soren Holmstrom [Hol88]. However, the key ingredients seem to me to be either new, or clarified and simplified in an essential fashion by comparison with these previous works. Similar remarks app... |

76 |
The theory and practice of transforming call-by-need into call-by-value
- Mycroft
- 1980
(Show Context)
Citation Context ... a term of type A \Gammaffi B must evaluate to an abstraction, which when applied to any argument will evaluate it exactly once. Evaluation of the argument exactly once is the slogan of call-by-value =-=[Myc80]-=-. Finally, any consumer of a term of type A \Phi B will evaluate it to a term of the form inl(t) or inr(t), and then use t in evaluating the appropriate arm of a case statement; so once again, eager e... |

74 |
Functional programming: Application and implementation
- Henderson
- 1980
(Show Context)
Citation Context ..., the potential for parallel execution is much greater for Classical Linear Logic, which is treated in Sections 6--8.) We will follow the very lucid exposition of the (standard) SECD machine given in =-=[Hen80]-=- fairly closely. The machine is based on a list-structured store. We shall use Turner's notation [Tur85] for list operations: [] for the empty list x : l for infix cons [x 1 ; : : : ; x n ] for x 1 : ... |

71 | Three approaches to type structure - Reynolds - 1985 |

51 |
The occam 2 Reference Manual
- Ltd
(Show Context)
Citation Context ... them. In particular, by linearity channels are used exactly once, so to determine the "strength" of the connection between terms t and u, we can simply count N (t) " N (u). (Contrast t=-=his with occam [LTD88]-=- or CSP [Hoa85], where what matters is not the number of channels two processes have in common, but the number of times the channels will be used.) A mapping algorithm could then attempt to optimize l... |

35 | Geometry of interaction 1: Interpretation of system f - Girard - 1989 |

28 |
Lectures on Predomains and Partial Functions. Notes for a course at CSLI
- Plotkin
- 1985
(Show Context)
Citation Context ...on on top of an eager evaluation strategy, as introduced by Landin [Lan65], and used in the SECD and CAM machines [Hen80, Hue90]. This idea is standardly modelled in denotational semantics by lifting =-=[Plo85]-=-, i.e. A ) B = A ? * B where A * B is the type of partial (or alternatively strict) functions. This account requires the presence of divergent programs; the Linear decomposition A ) B = !A \Gammaffi B... |

26 |
On Girard's "candidats de reductibilit'es
- Gallier
- 1990
(Show Context)
Citation Context ...wo rules corresponding to (\Phi \Gamma L) is applicable. We now turn to Convergence. Our proof is a simplified and suitably modified version of Girard's original proof of SN for System F [Gir72]; see =-=[Gal90]-=- for a good exposition. The idea is to use the evaluation relation to give a realizability semantics for types. We take a semantic type to be a set of closed linear terms, i.e. a subset of T = T ? . W... |

20 |
Types in lambda calculi and programming languages
- Barendregt, Hemerik
(Show Context)
Citation Context ...ve the form x 1 : A 1 ; : : : ; x n : A n ` t : A where the x i are distinct variables, and t is a term. We present this system (and all others in this paper) in the style of Curry rather than Church =-=[BH90]-=-; i.e. terms have no embedded types, and can have many types assigned to them. This choice is made for technical convenience rather than necessity, and certainly does not reflect any ideological commi... |

7 |
Semantical paradigms
- Meyer, Cosmadakis
- 1988
(Show Context)
Citation Context ...87]. Again, if we extend the calculus with general recursion, Strong Normalization will definitely be lost, while Convergence can be refined into Semantic Soundness [DM82] plus Computational Adequacy =-=[MC88]-=-. ffl As already explained, the evaluation relation reflects the intrinsic computational content of the Linear types, and so is the natural object of study. ffl The proofs of Determinacy and Convergen... |

4 | editors. Programs as Data Objects - Ganzinger, Jones - 1985 |

4 | A typed calculus based on a fragment of linear logic - Solitro - 1989 |

3 |
Complexity analysis for a lazy higher order language
- Sands
- 1989
(Show Context)
Citation Context ... "resource polynomials ". This leads to a term assignment in which exactly the polynomial-time computable 16 functions are typable. Again, there should be connections with "complexity a=-=nalysis" as in [San89]. The gene-=-ral framework suggested by these ideas might be called "Logic-based Program Analysis", by analogy with the already well-established subject of Semantics-Based Program Analysis [AH87, GJ86]. ... |

2 | The judgement calculus for intuitionistic linear logic: proof theory and semantics - Valentini - 1989 |

1 | Linear types can change the world! Draft paper - Wadler - 1989 |