scholarly journals SyTeCi: automating contextual equivalence for higher-order programs with references

2020 ◽  
Vol 4 (POPL) ◽  
pp. 1-28 ◽  
Author(s):  
Guilhem Jaber
Keyword(s):  
Higher Order ◽  
Contextual Equivalence

Mechanizing proofs with logical relations – Kripke-style

2018 ◽  
Vol 28 (9) ◽  
pp. 1606-1638 ◽  
Author(s):  
ANDREW CAVE ◽  
BRIGITTE PIENTKA
Keyword(s):  
Case Studies ◽  
Operational Semantics ◽  
Lambda Calculus ◽  
Equational Theory ◽  
Higher Order ◽  
Abstract Syntax ◽  
Completeness Proof ◽  
Typed Lambda Calculus ◽  
Contextual Equivalence ◽  
The Right

Proofs with logical relations play a key role to establish rich properties such as normalization or contextual equivalence. They are also challenging to mechanize. In this paper, we describe two case studies using the proof environmentBeluga: First, we explain the mechanization of the weak normalization proof for the simply typed lambda-calculus; second, we outline how to mechanize the completeness proof of algorithmic equality for simply typed lambda-terms where we reason about logically equivalent terms. The development of these proofs inBelugarelies on three key ingredients: (1) we encode lambda-terms together with their typing rules, operational semantics, algorithmic and declarative equality using higher order abstract syntax (HOAS) thereby avoiding the need to manipulate and deal with binders, renaming and substitutions, (2) we take advantage ofBeluga's support for representing derivations that depend on assumptions and first-class contexts to directly state inductive properties such as logical relations and inductive proofs, (3) we exploitBeluga's rich equational theory for simultaneous substitutions; as a consequence, users do not need to establish and subsequently use substitution properties, and proofs are not cluttered with references to them. We believe these examples demonstrate thatBelugaprovides the right level of abstractions and primitives to mechanize challenging proofs using HOAS encodings. It also may serve as a valuable benchmark for other proof environments.

scholarly journals A Complete, Co-Inductive Syntactic Theory of Sequential Control and State

2007 ◽  
Vol 14 (4) ◽  
Author(s):  
Kristian Støvring ◽  
Søren B. Lassen
Keyword(s):  
Normal Form ◽  
Lambda Calculus ◽  
Higher Order ◽  
Syntactic Theory ◽  
Sequential Control ◽  
Contextual Equivalence

We present a new co-inductive syntactic theory, eager normal form bisimilarity, for the untyped call-by-value lambda calculus extended with continuations and mutable references.<br /> <br />We demonstrate that the associated bisimulation proof principle is easy to use and that it is a powerful tool for proving equivalences between recursive imperative higher-order programs.<br /> <br />The theory is modular in the sense that eager normal form bisimilarity for each of the calculi extended with continuations and/or mutable references is a fully abstract extension of eager normal form bisimilarity for its sub-calculi. For each calculus, we prove that eager normal form bisimilarity is a congruence and is sound with respect to contextual equivalence. Furthermore, for the calculus with both continuations and mutable references, we show that eager normal form bisimilarity is complete: it coincides with contextual equivalence.

scholarly journals Short cut fusion is correct

2003 ◽  
Vol 13 (4) ◽  
pp. 797-814 ◽  
Author(s):  
PATRICIA JOHANN
Keyword(s):  
Data Structures ◽  
Formal Proof ◽  
Higher Order ◽  
Transformation Rule ◽  
Local Transformation ◽  
Fusion Technique ◽  
Intermediate Data ◽  
List Processing ◽  
Contextual Equivalence ◽  
Recent Demonstration

Fusion is the process of removing intermediate data structures from modularly constructed functional programs. Short cut fusion is a particular fusion technique which uses a single, local transformation rule to fuse compositions of list-processing functions. Short cut fusion has traditionally been treated purely syntactically, and justifications for it have appealed either to intuition or to “free theorems” – even though the latter have not been known to hold in languages supporting higher-order polymorphic functions and fixpoint recursion. In this paper we use Pitts' recent demonstration that contextual equivalence in such languages is parametric to provide the first formal proof of the correctness of short cut fusion for them. In particular, we show that programs which have undergone short cut fusion are contextually equivalent to their unfused counterparts.

scholarly journals Domain Theory for Concurrency

2003 ◽  
Vol 10 (43) ◽  
Author(s):  
Mikkel Nygaard ◽  
Glynn Winskel
Keyword(s):  
Linear Logic ◽  
Operational Semantics ◽  
Denotational Semantics ◽  
Higher Order ◽  
The Other ◽  
Domain Theory ◽  
Parallel Composition ◽  
Concurrent Computation ◽  
Contextual Equivalence

A simple domain theory for concurrency is presented. Based on a categorical model of linear logic and associated comonads, it highlights the role of linearity in concurrent computation. Two choices of comonad yield two expressive metalanguages for higher-order processes, both arising from canonical constructions in the model. Their denotational semantics are fully abstract with respect to contextual equivalence. One language derives from an exponential of linear logic; it supports a straightforward operational semantics with simple proofs of soundness and adequacy. The other choice of comonad yields a model of affine-linear logic, and a process language with a tensor operation to be understood as a parallel composition of independent processes. The domain theory can be generalised to presheaf models, providing a more refined treatment of nondeterministic branching. The article concludes with a discussion of a broader programme of research, towards a fully fledged domain theory for concurrency.

Contextual equivalence for inductive definitions with binders in higher order typed functional programming

2013 ◽  
Vol 23 (6) ◽  
pp. 658-700
Author(s):  
MATTHEW R. LAKIN ◽  
ANDREW M. PITTS
Keyword(s):  
Functional Programming ◽  
Operational Semantics ◽  
Higher Order ◽  
European Symposium ◽  
Proof Search ◽  
Inductive Definitions ◽  
University Of Cambridge ◽  
Meta Programming ◽  
Contextual Equivalence ◽  
Phd Thesis

AbstractCorrect handling of names and binders is an important issue in meta-programming. This paper presents an embedding of constraint logic programming into the αML functional programming language, which provides a provably correct means of implementing proof search computations over inductive definitions involving names and binders modulo α-equivalence. We show that the execution of proof search in the αML operational semantics is sound and complete with regard to the model-theoretic semantics of formulae, and develop a theory of contextual equivalence for the subclass of αML expressions which correspond to inductive definitions and formulae. In particular, we prove that αML expressions, which denote inductive definitions, are contextually equivalent precisely when those inductive definitions have the same model-theoretic semantics. This paper is a revised and extended version of the conference paper (Lakin, M. R. & Pitts, A. M. (2009) Resolving inductive definitions with binders in higher-order typed functional programming. InProceedings of the 18th European Symposium on Programming (ESOP 2009), Castagna, G. (ed), Lecture Notes in Computer Science, vol. 5502. Berlin, Germany: Springer-Verlag, pp. 47–61) and draws on material from the first author's PhD thesis (Lakin, M. R. (2010)An Executable Meta-Language for Inductive Definitions with Binders. University of Cambridge, UK).

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

Quantitative EPMA of nitrogen : A tricky element in the electron-probe microanalyzer

1992 ◽  
Vol 50 (2) ◽  
pp. 1622-1623
Author(s):  
G.F. Bastin ◽  
H.J.M. Heijligers
Keyword(s):  
Background Subtraction ◽  
Electron Probe ◽  
Detection System ◽  
Higher Order ◽  
Light Elements ◽  
Strong Suppression ◽  
Electron Probe Microanalyzer ◽  
Quantitative Epma ◽  
Peak Count ◽  
Lead Stearate

Among the ultra-light elements B, C, N, and O nitrogen is the most difficult element to deal with in the electron probe microanalyzer. This is mainly caused by the severe absorption that N-Kα radiation suffers in carbon which is abundantly present in the detection system (lead-stearate crystal, carbonaceous counter window). As a result the peak-to-background ratios for N-Kα measured with a conventional lead-stearate crystal can attain values well below unity in many binary nitrides . An additional complication can be caused by the presence of interfering higher-order reflections from the metal partner in the nitride specimen; notorious examples are elements such as Zr and Nb. In nitrides containing these elements is is virtually impossible to carry out an accurate background subtraction which becomes increasingly important with lower and lower peak-to-background ratios. The use of a synthetic multilayer crystal such as W/Si (2d-spacing 59.8 Å) can bring significant improvements in terms of both higher peak count rates as well as a strong suppression of higher-order reflections.

Sign in / Sign up

Export Citation Format

Share Document