Light Logics and the Call-by-Value Lambda Calculus
📝 Abstract
The so-called light logics have been introduced as logical systems enjoying quite remarkable normalization properties. Designing a type assignment system for pure lambda calculus from these logics, however, is problematic. In this paper we show that shifting from usual call-by-name to call-by-value lambda calculus allows regaining strong connections with the underlying logic. This will be done in the context of Elementary Affine Logic (EAL), designing a type system in natural deduction style assigning EAL formulae to lambda terms.
💡 Analysis
The so-called light logics have been introduced as logical systems enjoying quite remarkable normalization properties. Designing a type assignment system for pure lambda calculus from these logics, however, is problematic. In this paper we show that shifting from usual call-by-name to call-by-value lambda calculus allows regaining strong connections with the underlying logic. This will be done in the context of Elementary Affine Logic (EAL), designing a type system in natural deduction style assigning EAL formulae to lambda terms.
📄 Content
The so-called light logics [13,1,2] have been introduced as logical counterparts of complexity classes, namely polynomial and elementary time functions. After their introduction, they have been shown to be relevant for optimal reduction [10,11], programming language design [2,16] and set theory [15]. However, proof languages for these logics, designed through the Curry-Howard correspondence, are syntactically quite complex and can hardly be proposed as programming languages. An interesting research challenge is the design of type systems assigning light logics formulae to pure lambda-terms, forcing the class of typable terms to enjoy the same remarkable properties which can be proved for the logical systems. The mismatch between β-reduction in the lambda-calculus and cut-elimination in logical systems, however, makes it difficult to both getting the subject reduction property and inheriting the complexity properties from the logic, as discussed in [6]. Indeed, β-reduction is more permissive than the restrictive copying discipline governing calculi directly derived from light logics. Consider, for example, the following expression in Λ LA (see [16]):
let M be !x in N This rewrites to N {x/P } if M is !P , but is not a redex if M is, say, an application. It is not possible to map this mechanism into pure lambda calculus. The solution proposed by Baillot and Terui [6] in the context of Light Affine Logic (LAL, see [1,2]) consists in defining a type-system which is strictly more restrictive than the one induced by the logic. In this way, they both achieve subject reduction and a strong notion of polynomial time soundness. Now, notice that mapping the above let expression to the application (λx.N )M
is not meaningless if we shift from the usual call-by-name lambda calculus to the call-byvalue lambda calculus, where (λx.N )M is not necessarily a redex. In this paper, we make the best of this idea, introducing a type assignment system, that we call ETAS, assigning formulae of Elementary Affine Logic (EAL) to lambda-terms. ETAS enjoys the following remarkable properties:
• The language of types coincides with the language of EAL formulae.
• Every proof of EAL can be mapped into a type derivation in ETAS.
• (Call-by-value) subject reduction holds.
• Elementary bounds can be given on the length of any reduction sequence involving a typable term. A similar bound holds on the size of terms involved in the reduction. • Type inference is decidable and the principal typings can be inferred in polynomial time. The basic idea underlying ETAS consists in partitioning premises into three classes, depending on whether they are used once, or more than once, or they are in an intermediate status. We believe this approach can work for other light logics too, and some hints will be given.
The proposed system is the first one satisfying the above properties for light logics. A notion of typability for lambda calculus has been defined in [10,11,7] for EAL, and in [4] for LAL. Type inference has been proved to be decidable. In both cases, however, the notion of typability is not preserved by β-reduction.
Noticeably, the proposed approach can be extended to Light Affine Logic and Soft Affine Logic (SAL, see [5,14]).
A preliminary version of the present paper is [9]: here some results have been improved. In particular a new type inference algorithm is presented, and its complexity is analyzed: it turns out that our type inference algorithm for EAL has a complexity of the same order than the type inference for simple types. Moreover some discussions about possible extensions of this method have been added.
The paper is organized as follows: in Section 2 a comparison with existing work is made, in Section 3 some preliminary notions about EAL and lambda calculus are recalled, in Section 4 the ETAS system is introduced, and in Section 5 and 6 its main properties, namely complexity bounds and a type inference algorithm, are explained. Section 7 presents two possible extensions, allowing to reach completeness for elementary functions, and in Section 8 some hints on how to apply our idea to other light logics are given. Section 9 contains a short summary of the obtained results.
This work is not the first contribution on type systems derived from light logics. We should mention works on (principal) type inference for Elementary Affine Logic and Light Affine Logic by Baillot, Coppola, Martini and Ronchi Della Rocca [10,4,11]. There, the goal was basically proving decidability of type inference. The proposed type systems were the ones directly induced from logical systems. Typable lambda terms can be efficiently reduced using Lamping’s abstract algorithm, although basic properties like subject reduction and complexity bounds were not necessarily verified.
Baillot and Terui [6] proposed a type system inspired by light logics and enjoying subject reduction and polynomial time normalization, called Dual Light Affine Logic (DLAL). The u
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