scholarly journals A Complete Calculus of Monotone and Antitone Higher-Order Functions

10.29007/3n54 ◽  
2018 ◽  
Author(s):  
Thomas Icard ◽  
Lawrence Moss
Keyword(s):  
Type System ◽  
Lambda Calculus ◽  
Higher Order ◽  
Typed Lambda Calculus ◽  
Higher Order Functions

This paper adds monotonicity and antitonicity information to the typed lambda calculus, thereby providing a foundation for the Monotonicity Calculus first developed by van Benthem and others. We establish properties of the type system, propose a syntax, semantics, and proof calculus, and prove completeness for the calculus with respect to hierarchies of monotone and antitone functions over base preorders.

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.

Operations on records

1991 ◽  
Vol 1 (1) ◽  
pp. 3-48 ◽  
Author(s):  
Luca Cardelli ◽  
John C. Mitchell
Keyword(s):  
Type System ◽  
Lambda Calculus ◽  
Object Oriented ◽  
Order Type ◽  
Second Order ◽  
Typed Lambda Calculus ◽  
Semantic Models ◽  
Object Oriented Languages

We define a simple collection of operations for creating and manipulating record structures, where records are intended as finite associations of values to labels. A second-order type system over these operations supports both subtyping and polymorphism. We provide typechecking algorithms and limited semantic models.Our approach unifies and extends previous notions of records, bounded quantification, record extension, and parametrization by row-variables. The general aim is to provide foundations for concepts found in object-oriented languages, within a framework based on typed lambda-calculus.

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 Lambda calculus with algebraic simplification for reduction parallelisation: Extended study

2021 ◽  
Vol 31 ◽  
Author(s):  
AKIMASA MORIHATA
Keyword(s):  
Parallel Programming ◽  
Type System ◽  
Lambda Calculus ◽  
Extended Study ◽  
Equational Reasoning ◽  
Typed Lambda Calculus ◽  
Parallel Reduction ◽  
Algebraic Properties ◽  
Algebraic Simplification

Abstract Parallel reduction is a major component of parallel programming and widely used for summarisation and aggregation. It is not well understood, however, what sorts of non-trivial summarisations can be implemented as parallel reductions. This paper develops a calculus named λAS, a simply typed lambda calculus with algebraic simplification. This calculus provides a foundation for studying a parallelisation of complex reductions by equational reasoning. Its key feature is δ abstraction. A δ abstraction is observationally equivalent to the standard λ abstraction, but its body is simplified before the arrival of its arguments using algebraic properties such as associativity and commutativity. In addition, the type system of λAS guarantees that simplifications due to δ abstractions do not lead to serious overheads. The usefulness of λAS is demonstrated on examples of developing complex parallel reductions, including those containing more than one reduction operator, loops with conditional jumps, prefix sum patterns and even tree manipulations.

Type safe incremental rebinding

2015 ◽  
Vol 27 (2) ◽  
pp. 94-122 ◽  
Author(s):  
DAVIDE ANCONA ◽  
PAOLA GIANNINI ◽  
ELENA ZUCCA
Keyword(s):  
Type System ◽  
Lambda Calculus ◽  
Typed Lambda Calculus ◽  
Free Variables ◽  
Open Code

We extend the simply-typed lambda-calculus with a mechanism for dynamic and incremental rebinding of code. Fragments of open code which can be dynamically rebound are values. Differently from standard static binding, which is done on a positional basis, rebinding is done on a nominal basis, that is, free variables in open code are associated with names which do not obey α-equivalence. Moreover, rebinding is incremental, that is, just a subset of names can be rebound, making possible code specialization, and rebinding can even introduce new names. Finally, rebindings, which are associations between names and terms, are first-class values, and can be manipulated by operators such as overriding and renaming. We define a type system in which the type for a rebinding, in addition to specify an association between names and types (similarly to record types), is also annotated. The annotation says whether or not the domain of the rebinding having this type may contain more names than the ones that are specified in the type. We show soundness of the type system.

scholarly journals Helmholtz: A Verifier for Tezos Smart Contracts Based on Refinement Types

2021 ◽  
pp. 262-280
Author(s):  
Yuki Nishida ◽  
Hiromasa Saito ◽  
Ran Chen ◽  
Akira Kawata ◽  
Jun Furuse ◽  
...  
Keyword(s):  
Type System ◽  
Higher Order ◽  
Smart Contracts ◽  
Static Verification ◽  
Smt Solver ◽  
The Core ◽  
Verification Conditions ◽  
Refinement Types ◽  
Characteristic Features ◽  
Higher Order Functions

AbstractA smart contract is a program executed on a blockchain, based on which many cryptocurrencies are implemented, and is being used for automating transactions. Due to the large amount of money that smart contracts deal with, there is a surging demand for a method that can statically and formally verify them.This tool paper describes our type-based static verification tool Helmholtz for Michelson, which is a statically typed stack-based language for writing smart contracts that are executed on the blockchain platform Tezos. Helmholtz is designed on top of our extension of Michelson’s type system with refinement types. Helmholtz takes a Michelson program annotated with a user-defined specification written in the form of a refinement type as input; it then typechecks the program against the specification based on the refinement type system, discharging the generated verification conditions with the SMT solver Z3. We briefly introduce our refinement type system for the core calculus Mini-Michelson of Michelson, which incorporates the characteristic features such as compound datatypes (e.g., lists and pairs), higher-order functions, and invocation of another contract. Helmholtz successfully verifies several practical Michelson programs, including one that transfers money to an account and that checks a digital signature.

scholarly journals Reachability types: tracking aliasing and separation in higher-order functional programs

2021 ◽  
Vol 5 (OOPSLA) ◽  
pp. 1-32
Author(s):  
Yuyan Bao ◽  
Guannan Wei ◽  
Oliver Bračevac ◽  
Yuxuan Jiang ◽  
Qiyang He ◽  
...  
Keyword(s):  
Functional Programming ◽  
Type System ◽  
Type Systems ◽  
Higher Order ◽  
Ownership Type ◽  
Ownership Transfer ◽  
Reachability Sets ◽  
Programming Patterns ◽  
New Type ◽  
Higher Order Functions

Ownership type systems, based on the idea of enforcing unique access paths, have been primarily focused on objects and top-level classes. However, existing models do not as readily reflect the finer aspects of nested lexical scopes, capturing, or escaping closures in higher-order functional programming patterns, which are increasingly adopted even in mainstream object-oriented languages. We present a new type system, λ * , which enables expressive ownership-style reasoning across higher-order functions. It tracks sharing and separation through reachability sets, and layers additional mechanisms for selectively enforcing uniqueness on top of it. Based on reachability sets, we extend the type system with an expressive flow-sensitive effect system, which enables flavors of move semantics and ownership transfer. In addition, we present several case studies and extensions, including applications to capabilities for algebraic effects, one-shot continuations, and safe parallelization.

A general method for proving the normalization theorem for first and second order typed λ-calculi

1999 ◽  
Vol 9 (6) ◽  
pp. 719-739 ◽  
Author(s):  
VENANZIO CAPRETTA ◽  
SILVIO VALENTINI
Keyword(s):  
Type System ◽  
Lambda Calculus ◽  
Second Order ◽  
Typed Lambda Calculus ◽  
Normalization Theorem ◽  
System F ◽  
Dependent Type ◽  
General Method

In this paper we describe a method for proving the normalization property for a large variety of typed lambda calculi of first and second order, which is based on a proof of equivalence of two deduction systems. We first illustrate the method on the elementary example of simply typed lambda calculus, and then we show how to extend it to a more expressive dependent type system. Finally we use it to prove the normalization theorem for Girard's system F.

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