Normalization by Evaluation in the Delay Monad

2021

In this paper we introduce a typed, concurrent $\lambda$-calculus with references featuring explicit substitutions for variables and references. Alongside usual safety properties, we recover strong normalization. The proof is based on a reducibility technique and an original interactive property reminiscent of the Game Semantics approach.

2021

In this paper we introduce a typed, concurrent $\lambda$-calculus with references featuring explicit substitutions for variables and references. Alongside usual safety properties, we recover strong normalization. The proof is based on a reducibility technique and an original interactive property reminiscent of the Game Semantics approach.

Lecture Notes in Computer Science, 2014

We consider a simplified version of Nakano's guarded fixed-point types in a representation by infinite type expressions, defined coinductively. Smallstep reduction is parametrized by a natural number "depth" that expresses under how many guards we may step during evaluation. We prove that reduction is strongly normalizing for any depth. The proof involves a typed inductive notion of strong normalization and a Kripke model of types in two dimensions: depth and typing context. Our results have been formalized in Agda and serve as a case study of reasoning about a language with coinductive type expressions.

2013

Container types can be modeled as instances of the Haskell MonadPlus type class which support a fold operation. In this paper we present subclasses that extend the MonadPlus type class to support a membership operator. The laws for the EMonadPlus type class specify how membership behaves with respect to the monad and monad plus operators. Using EMonads we are able write and prove properties of generic specifications of containers. In the second part of the paper we present an induction rule for monads we call bind induction. The computational content of the new induction rule is the Monad bind operator and the new proof rule is proved to be sound. Using this new proof rule we are able to extract monadic programs from proofs. We present an example that uses the rule to extract a simple monadic program from a proof of a specification. We have used the Coq theorem prover with the Coq Type Class mechanism to formalize the definitions presented here and to prove many properties of the fo...

extended abstract -We try to combine the 'syntactic composition of tree transducers' [FV98, KV01] on the one hand side and 'short cut fusion' [GLP93] on the other hand side.

Fixed Points in Computer Science, 2003

Recently, higher-rank datatypes have drawn interest in the functional pro grainming community [0ka99,0ka96,HinOl]. Rank-2 non-regular types, so-called nested datatypes, have been investigated in the context of Haskell. To define total functions which traverse nested datastructures, Bird et al. [B P99] have developed generalized folds which implement an iteration scheme and are strong enough to encode most of the known algorithms for nested datatypes. In this note, we in vestigate a scheme to overcome some limitations of iteration which we expound in the following. Since the work of Böhm et al. [BB85] it is well-known that iteration for rank-1 datatypes can be simulated in typed lambda-calculi. The easiest examples are iterative definitions of addition and multiplication for Church numerals. The iterative definition of the predecessor, however, is inefficient: It traverses the whole numeral in order to remove one constructor. Surely, taking the predecessor should run in constant time. Primitive recursion is the combination of iteration and efficient predecessor. A typical example for a prim. rec. algorithm is the natural definition of the factorial function. It is common belief that prim. rec. cannot be reduced to it eration in a computationally faithful manner. This is because no encoding of natural numbers in the polymorphic lambda-calculus (System F) seems possible which supports a constant-time predecessor operation (see Splawski and Urzy czyn [SU99]). Mendler extended System F by a scheme of prim. rec. for rank-i datatypes and proved strong normalization [Men87]. Mendler's formulation does not follow the usual category-theoretic approach with initial recursive algebras (see Geuvers [Geu92]). For rank-2 datatypes there are also examples of functions which can most naturally be implemented with prim. rec. One is redecoration for triangular ma- ti-ices which is presented below. These examples are not instances of generalized folds a Ia Bird et al., which remain within the realm of iteration but hardwire Kan extensions into the recursion scheme. Rank-2 prim. rec., which we propose in this work, seeks to extend rank-2 iteration in the same way that prim. rec. extends rank-i iteration. We achieve this by lifting Mendler's scheme of prim. rec. to rank 2. The decision for Mendler-style and against the traditional way roots in the following observation: Experiments with formulations according to the traditional style showed unnecessary but unavoidable traversals of the whole data structures in our examples. Mendler's style, however, yielded precisely the The first author is supported by the GraduiertenkoUeg "Logik in der Informatik" of the Deutsche Forschungsgemeinschaft. Both authors acknowledge financial support by ETAPS 2003. desired efficient reduction behavior. This was crucial since the only reason to incorporate prim. rec. is operational efficiency as opposed to denotational ex pressiveness. We work within the framework System F" of higher-order parametric poly morphism formulated in Curry-style, i.e., as a type assignment system for the pure lambda-calculus. For type transformers X, Y : * -* * we abbreviate the type of natural transformations VA. XA -* YA from X to Y by X C Y. Let id = .Xx.x denote the identity function. We extend the framework by a new constructor constant p and two term constants in and MRec and a new reduction rule as follows. Formation. p :((*-+*)--**-+*)-+*-+* Introduction. in VF'-'. F (pF) ç pF Elimination. MRec : VF(**) (VX. X C pF - X ç C -, F X ç C) -pF ç C Reduction. MRecs(int) -+ a id(MRecs)t The type transfomer pF : * -+ * is the least fixed-point of the constructor F : (* -÷ *) -, * -* and denotes a simultaneously defined family of types of well-founded trees, their shape depending on F. For instance, using F = ),X),A. 1 + A x X A the well-known type of polymorphic lists is recovered. The term in is the general constructor, which, in case of lists, codes together nil and cons. The term MRec establishes a scheme of primitive recursion in the style of Mendler. Typical for this style is the universally quantified constructor variable X in the type of the step term a which ensures termination without any positivity restrictions on F. During reduction, X is instantiated by pF, and the first parameter, i: X C pF, by id. The presence of a transformation i from the blank type X back into the fixed-point pF is what distinguishes Mendler-style prim. rec. from Mendler-style iteration.

2007

The decidability of equality is proved for Martin-Löf type theory with a universeá la Russell and typed betaeta-equality judgements. A corollary of this result is that the constructor for dependent function types is injective, a property which is crucial for establishing the correctness of the type-checking algorithm. The decision procedure uses normalization by evaluation, an algorithm which first interprets terms in a domain with untyped semantic elements and then extracts normal forms. The correctness of this algorithm is established using a PER-model and a logical relation between syntax and semantics. * Research partially supported by the EU coordination action TYPES (510996). † Research partially supported by VR-project Typed Lambda Calculus and Applications. sional type theories such as Coq, Agda, and Epigram, all rely on the decision algorithm for a : A using normalization: the user attempts to prove the theorem A by building a construction a, and the system checks that a is indeed a proof of A.

Electronic Notes in Theoretical Computer Science, 2009

We present a context-based approach to proving termination of evaluation in reduction semantics (i.e., a form of operational semantics with explicit representation of reduction contexts), using Tait-style reducibility predicates defined on both terms and contexts. We consider the simply typed lambda calculus as well as its extension with abortive control operators for first-class continuations under the call-by-value and the call-by-name evaluation strategies. For each of the proofs we present its computational content that takes the form of an evaluator in continuation-passing style and is an instance of normalization by evaluation.

This thesis presents a new algorithm for Normalisation by Evaluation for different type theories. This algorithm is later used to define a type-checking algorithm and to prove other meta-theoretical results about predicative type systems.

2018

Weak reduction relations in the λ-calculus are characterized by the rejection of the so-called ξ-rule, which allows arbitrary reductions under abstractions. A notable instance of weak reduction can be found in the literature under the name restricted reduction or weak λ-reduction. In this work, we attack the problem of algorithmically computing normal forms for λ, the λ-calculus with weak λ-reduction. We do so by first contrasting it with other weak systems, arguing that their notion of reduction is not strong enough to compute λ-normal forms. We observe that some aspects of weak λ-reduction prevent us from normalizing λ directly, thus devise a new, better-behaved weak calculus λ, and reduce the normalization problem for λ to that of λ. We finally define type systems for both calculi and apply Normalization by Evaluation to λ, obtaining a normalization proof for λ as a corollary. We formalize all our results in Agda, a proof-assistant based on intensional Martin-Löf Type Theory. 201...