We solve the decision problem for simply typed lambda calculus with strong binary sums, equivalently the word problem for free cartesian closed categories with binary coproducts. Our method is based on the semantical technique known as “normalization by evaluation ” and involves inverting the interpretation of the syntax into a suitable sheaf model and from this extracting appropriate unique normal forms. There is no rewriting theory involved, and the proof is completely constructive, allowing program extraction from the proof. 1

We present the spine calculus S #-#&# as an efficient representation for the linear #-calculus # #-#&# which includes unrestricted functions (#), linear functions (-#), additive pairing (&), and additive unit (#). S #-#&# enhances the representation of Church's simply typed #-calculus by enforcing extensionality and by incorporating linear constructs. This approach permits procedures such as unification to retain the efficient head access that characterizes first-order term languages without the overhead of performing #-conversions at run time. Applications lie in proof search, logic programming, and logical frameworks based on linear type theories. It is also related to foundational work on term assignment calculi for presentations of the sequent calculus. We define the spine calculus, give translations of # #-#&# into S #-#&# and vice-versa, prove their soundness and completeness with respect to typing and reductions, and show that the typable fragment of the spine calculus is strongly normalizing and admits unique canonical, i.e. ##-normal, forms.

) Delia Kesner CNRS and LRI, B at 490, Universit e Paris-Sud - 91405 Orsay Cedex, France. e-mail:Delia.Kesner@lri.fr Abstract. This paper studies confluence properties of extensional and non-extensional #-calculi with explicit substitutions, where extensionality is interpreted by #-expansion. For that, we propose a general scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. Our general scheme makes it possible to treat at the same time many well-known calculi such as ## , ## # and ## , or some other new calculi that we propose in this paper. We also show for those calculi not fitting in the general scheme that can be translated to another one fitting the scheme, such as #s , how to reason about confluence properties of their extensional and non-extensional versions. 1 Introduction The #-calculus is a convenient framework to study functional programming, where the evaluation process is modeled by #-reduction. The...

The combinatorics of classical propositional logic lies at the heart of both local and global methods of proof-search enabling the achievement of least-commitment search. Extension of such methods to the predicate calculus, or to non-classical systems, presents us with the problem of recovering this least-commitment principle in the context of non-invertible rules. One successful approach is to view the nonclassical logic as a perturbation on search in classical logic and characterize when a least-commitment (classical) search yields sufficient evidence for provability in the (non-classical) logic. This technique has been successfully applied to both local and global methods at the cost of subsidiary searches and is the analogue of the standard treatment of quantifiers via skolemization and unification. In this paper, we take a type-theoretic view of this approach for the case in which the non-classical logic is intuitionistic. We develop a system of realizers (proof-obje...

It is well known that confluence and strong normalization are preserved when combining left-linear algebraic rewriting systems with the simply typed lambda calculus. It is equally well known that confluence fails when adding either the usual extensional rule for j, or recursion together with the usual contraction rule for surjective pairing. We show that confluence and normalization are modular properties for the combination of left-linear algebraic rewriting systems with typed lambda calculi enriched with expansive extensional rules for j and surjective pairing. For that, we use a translation technique allowing to simulate expansions without expansion rules. We also show that confluence is maintained in a modular way when adding fixpoints. This result is also obtained by a simple translation technique allowing to simulate bounded recursion with fi reduction. 1 Introduction Confluence and strong normalization for the combination of lambda calculus and algebraic rewriting systems have...

We add extensional equalities for the functional and product types to the typed -calculus with not only products and terminal object, but also sums and bounded recursion (a version of recursion that does not allow recursive calls of infinite length). We provide a confluent and strongly normalizing (thus decidable) rewriting system for the calculus, that stays confluent when allowing unbounded recursion. For that, we turn the extensional equalities into expansion rules, and not into contractions as is done traditionally. We first prove the calculus to be weakly confluent, which is a more complex and interesting task than for the usual -calculus. Then we provide an effective mechanism to simulate expansions without expansion rules, so that the strong normalization of the calculus can be derived from that of the underlying, traditional, non extensional system. These results give us the confluence of the full calculus, but we also show how to deduce confluence directly form our simulation...

This paper presents an extension of the simple-typed lambda-calculus allowing iteration and case...

. Although the use of expansionary j-rewrite has become increasingly common in recent years, one area where j-contractions have until now remained the only possibility is in the more powerful type theories of the -cube. This paper rectifies this situation by applying j-expansions to the Calculus of Constructions --- we discuss some of the difficulties posed by the presence of dependent types, prove that every term rewrites to a unique long fij-normal form and deduce the decidability of fij-equality, typeability and type inhabitation as corollaries. 1 Introduction Extensional equality for the simply typed -calculus requires j-conversion, whose interpretation as a rewrite rule has traditionally been as a contraction x : T:fx ) f where x 6 2 FV(t). When combined with the usual fi-reduction, the resulting rewrite relation is strongly normalising and confluent, and thus reduction to normal form provides a decision procedure for the associated equational theory. However j-contractions beh...

This paper studies confluence of extensional and non-extensional -calculi with explicit substitutions, where extensionality is interpreted by j-expansion. For that, we propose a scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. Our method makes it possible to treat at the same time many well-known calculi such as oe , oe * , OE , s , AE , f , d and dn . Keywords: functional programming, -calculi, explicit substitutions, confluence, extensionality. 1 Introduction The -calculus is a convenient framework to study functional programming, where the evaluation process is modeled by fi-reduction. The main mechanism used to perform fi-reduction is substitution, which consists of the replacement of formal parameters by actual arguments. The correctness of substitution is guaranteed by a systematic renaming of bound variables, inconvenient which can be simply avoided in the -calculus `a la de Bruijn by using natur...

. We prove the confluence and strong normalization properties for second order lambda calculus equipped with an expansive version of j-reduction. Our proof technique, based on a simple abstract lemma and a labelled -calculus, can also be successfully used to simplify the proofs of confluence and normalization for first order calculi, and can be applied to various extensions of the calculus presented here. 1 Introduction The typed lambda calculus provides a convenient framework for studying functional programming and offers a natural formalism to deal with proofs in intuitionistic logic. It comes traditionally equipped with the fi equality (x:M)N = M [N=x] as fundamental computational mechanism, and with the j (extensional) equality x:Mx = M as a tool for reasoning about programs. This basic calculus can then be extended by adding further types, like products, unit and second order types, each coming with its own computational mechanism and/or its extensional equalities. To reason abou...