scholarly journals Decomposing Probabilistic Lambda-Calculi

2020 ◽  
pp. 136-156
Author(s):  
Ugo Dal Lago ◽  
Giulio Guerrieri ◽  
Willem Heijltjes
Keyword(s):  
Lambda Calculus ◽  
Strong Normalization ◽  
Reduction Strategy ◽  
Fine Grained ◽  
Probabilistic Evaluation

AbstractA notion of probabilistic lambda-calculus usually comes with a prescribed reduction strategy, typically call-by-name or call-by-value, as the calculus is non-confluent and these strategies yield different results. This is a break with one of the main advantages of lambda-calculus: confluence, which means results are independent from the choice of strategy. We present a probabilistic lambda-calculus where the probabilistic operator is decomposed into two syntactic constructs: a generator, which represents a probabilistic event; and a consumer, which acts on the term depending on a given event. The resulting calculus, the Probabilistic Event Lambda-Calculus, is confluent, and interprets the call-by-name and call-by-value strategies through different interpretations of the probabilistic operator into our generator and consumer constructs. We present two notions of reduction, one via fine-grained local rewrite steps, and one by generation and consumption of probabilistic events. Simple types for the calculus are essentially standard, and they convey strong normalization. We demonstrate how we can encode call-by-name and call-by-value probabilistic evaluation.

Termination of rewriting in the Calculus of Constructions

2003 ◽  
Vol 13 (2) ◽  
pp. 339-414 ◽  
Author(s):  
DARIA WALUKIEWICZ-CHRZĄSZCZ
Keyword(s):  
Lambda Calculus ◽  
Term Rewriting ◽  
Higher Order ◽  
Strong Normalization ◽  
Term Rewriting System ◽  
Calculus Of Constructions ◽  
Path Ordering ◽  
Higher Order Functions ◽  
Rewriting Rules ◽  
Recursive Path

We show how to incorporate rewriting into the Calculus of Constructions and we prove that the resulting system is strongly normalizing with respect to beta and rewrite reductions. An important novelty of this paper is the possibility to define rewriting rules over dependently typed function symbols. We prove strong normalization for any term rewriting system, such that all function symbols satisfy the, so called, star dependency condition, and every rule is accepted by the Higher Order Recursive Path Ordering (which is an extension of the method created by Jouannaud and Rubio for the setting of the simply typed lambda calculus). The proof of strong normalization is done by using a typed version of reducibility candidates due to Coquand and Gallier. Our criterion is general enough to accept definitions by rewriting of many well-known higher order functions, for example dependent recursors for inductive types or proof carrying functions. This makes it a very good candidate for inclusion in a proof assistant based on the Curry-Howard isomorphism.

scholarly journals A Study of Syntactic and Semantic Artifacts and its Application to Lambda Definability, Strong Normalization, and Weak Normalization in the Presence of...

2008 ◽  
Vol 15 (3) ◽  
Author(s):  
Johan Munk
Keyword(s):  
Lambda Calculus ◽  
Expressive Power ◽  
Strong Normalization ◽  
Syntactic Theory ◽  
Abstract Machines ◽  
Value Representation

Church's lambda-calculus underlies the syntax (i.e., the form) and the semantics (i.e., the meaning) of functional programs. This thesis is dedicated to studying man-made constructs (i.e., artifacts) in the lambda calculus. For example, one puts the expressive power of the lambda calculus to the test in the area of lambda definability. In this area, we present a course-of-value representation bridging Church numerals and Scott numerals. We then turn to weak and strong normalization using Danvy et al.'s syntactic and functional correspondences. We give a new account of Felleisen and Hieb's syntactic theory of state, and of abstract machines for strong normalization due to Curien, Crégut, Lescanne, and Kluge.

The full-reducing Krivine abstract machine KN simulates pure normal-order reduction in lockstep: A proof via corresponding calculus

2019 ◽  
Vol 29 ◽  
Author(s):  
ÁLVARO GARCÍA-PÉREZ ◽  
PABLO NOGUEIRA
Keyword(s):  
Operational Semantics ◽  
Lambda Calculus ◽  
Order Reduction ◽  
Normal Order ◽  
Abstract Machine ◽  
Reduction Strategy ◽  
Structural Operational Semantics ◽  
Standard Normal ◽  
De Bruijn ◽  
De Bruijn Indices

AbstractWe exploit the idea of proving properties of an abstract machine by using a corresponding semantic artefact better suited to their proof. The abstract machine is an improved version of Pierre Crégut’s full-reducing Krivine machine KN. The original version works with closed terms of the pure lambda calculus with de Bruijn indices. The improved version reduces in similar fashion but works on closures where terms may be open. The corresponding semantic artefact is a structural operational semantics of a calculus of closures whose reduction relation is purposely a reduction strategy. As shown in previous work, improved KN and the structural operational semantics ‘correspond’, i.e. both artefacts realise the same reduction strategy. In this paper, we prove in the calculus of closures that the reduction strategy simulates in lockstep (at every reduction step) the complete and standard normal-order strategy (i.e. leftmost reduction to normal form) of the pure lambda calculus. The simulation is witnessed by a substitution function from closures of the closure calculus to pure terms of the pure lambda calculus. Thus, KN also simulates normal-order in lockstep by the correspondence. This result is stronger than the known proof that KN is complete, for in the pure lambda calculus there are complete but non-standard strategies. The lockstep simulation proof consists of straightforward structural inductions, thanks to three properties of the closure calculus we call ‘index alignment’, ‘parameters-as-levels’ and ‘balanced derivations’. The first two come from KN. Thanks to these properties, a proof in a calculus of closures involving de Bruijn indices and de Bruijn levels is unproblematic. There is no lexical adjustment at binding lookup, on-the-fly alpha-conversion or recursive traversals of the term to deal with bound and free variables as in other calculi. This paper contributes to the framework for environment machines of Biernacka and Danvy a full-reducing open-terms closure calculus, its corresponding abstract machine, and a lockstep simulation proof via a substitution function.

Perpetuality in a named lambda calculus with explicit substitutions

2001 ◽  
Vol 11 (1) ◽  
pp. 47-90 ◽  
Author(s):  
EDUARDO BONELLI
Keyword(s):  
Lambda Calculus ◽  
Strong Normalization ◽  
Reduction Strategies ◽  
Explicit Substitutions ◽  
Type Substitution

We study perpetuality in the calculus of explicit substitutions λx. A reduction is called perpetual if it preserves the possibility of infinite reduction sequences. We then take a look at applications of this study: an inductive characterization of the λx-strongly normalizing terms, two perpetual reduction strategies for λx and finally a proof of strong normalization of a polymorphic lambda calculus with explicit substitutions Fes. To complete the study of Fes, the property of subject reduction is shown to hold by extending type assignments of the typing rules to allow non-pure types (types with possible occurrences of the type substitution operator).

Remarks on the Church-Rosser Property

1990 ◽  
Vol 55 (1) ◽  
pp. 106-112
Author(s):  
E. G. K. López-Escobar
Keyword(s):  
Lambda Calculus ◽  
Direct Proof ◽  
Propositional Calculus ◽  
Second Order ◽  
Strong Normalization ◽  
Normalization Theorem ◽  
Intuitionistic Propositional Calculus ◽  
The Church

AbstractA reduction algebra is defined as a set with a collection of partial unary functions (called reduction operators). Motivated by the lambda calculus, the Church-Rosser property is defined for a reduction algebra and a characterization is given for those reduction algebras satisfying CRP and having a measure respecting the reductions. The characterization is used to give (with 20/20 hindsight) a more direct proof of the strong normalization theorem for the impredicative second order intuitionistic propositional calculus.

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