The cost of usage in the λ-calculus

Abstract-A new "inductive" approach to standardization for the λ-calculus has been recently introduced by Xi, allowing him to establish a double-exponential upper bound |M | 2 |σ| for the length of the standard reduction relative to an arbitrary reduction σ originated in M . In this paper we refine Xi's analysis, obtaining much better bounds, especially for computations producing small normal forms. For instance, for terms reducing to a boolean, we are able to prove that the length of the standard reduction is at most a mere factorial of the length of the shortest reduction sequence. The methodological innovation of our approach is that instead to try to count the cost for producing something, as customary, we count the cost of consuming things. The key observation is that the part of a λ-term that is needed to produce the normal form (or an arbitrary rigid prefix) may rapidly augment along a computation, but can only decrease very slowly (actually, linearly)

A machine-checked proof of the Standardization Theorem in Lambda Calculus using multiple substitution

Incluye bibliografía y anexos.Incluye archivos complementarios.En el siguiente trabajo se presenta la formalización en Agda del Teorema de Estandarización para el Cálculo Lambda. Se presenta un corolario del Teorema de Estandarización junto con la demostración del teorema de reducción más izquierdista (leftmost reduction). Si bien existen formalizaciones de estos resultados, lo innovador que se desarrolla en la presente tesis, consiste en el apego de la formalización del Cálculo Lambda a su presentación usual (variables con nombres). El Cálculo Lambda constituye el fundamento teórico de lenguajes de programación funcionales. En la primera parte de este trabajo se reseña la formalización del Cálculo Lambda que se utiliza como base para el desarrollo de los capítulos siguientes e introduce ciertas definiciones propias. En el tercer capítulo se presenta lo más sustancial de la tesis: las definiciones de diferentes nociones de reducción que permiten probar el Teorema de Estandarización que será en términos de una secuencia de reducción estándar. Primero se presentan las nociones de contracción del redex más izquierdista y luego la noción de reducción del redex en la cabecera (de una aplicación). Se demuestra una definición inductiva equivalente a secuencias de reducción estándar. En este capítulo se van desplegando los lemas necesarios para obtener el teorema de estandarización. En el cuarto capítulo de la tesis se aprovecha el Teorema de Estandarización para probar que si un término tiene forma normal, esta es alcanzable por la estrategia de reducción leftmost-outermost. Concluye con la comparación de este desarrollo con otras formalizaciones del mismo resultado y propone algunas líneas para continuarlo

Extending Implicit Computational Complexity and Abstract Machines to Languages with Control

The Curry-Howard isomorphism is the idea that proofs in natural deduction can be put in correspondence with lambda terms in such a way that this correspondence is preserved by normalization. The concept can be extended from Intuitionistic Logic to other systems, such as Linear Logic. One of the nice conseguences of this isomorphism is that we can reason about functional programs with formal tools which are typical of proof systems: such analysis can also include quantitative qualities of programs, such as the number of steps it takes to terminate. Another is the possiblity to describe the execution of these programs in terms of abstract machines. In 1990 Griffin proved that the correspondence can be extended to Classical Logic and control operators. That is, Classical Logic adds the possiblity to manipulate continuations. In this thesis we see how the things we described above work in this larger context

Automated Reasoning

This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book

Standardization and Confluence in Pure Lambda-Calculus Formalized for the Matita Theorem Prover

We present a formalization of pure ?-calculus for the Matita interactive theorem prover, including the proofs of two relevant results in reduction theory: the confluence theorem and the standardization theorem. The proof of the latter is based on a new approach recently introduced by Xi and refined by Kashima that, avoiding the notion of development and having a neat inductive structure, is particularly suited for formalization in theorem provers

Beta-Conversion, Efficiently

Type-checking in dependent type theories relies on conversion, i.e. testing given lambda-terms for equality up to beta-evaluation and alpha-renaming. Computer tools based on the lambda-calculus currently implement conversion by means of algorithms whose complexity has not been identified, and in some cases even subject to an exponential time overhead with respect to the natural cost models (number of evaluation steps and size of input lambda-terms). This dissertation shows that in the pure lambda-calculus it is possible to obtain conversion algorithms with bilinear time complexity when evaluation is carried following evaluation strategies that generalize Call-by-Value to the stronger case required by conversion