We study the logical semantics of an untyped λ-calculus equipped with operators representing read and write operations from and to a global store. Such a logic consists of an intersection type ...assignment system, which we derive from the denotational semantics of the calculus, based on the monadic approach to model computational λ-calculi.
The system is obtained by constructing a filter model in the category of ω-algebraic lattices, such that the typing rules can be recovered out of the term interpretation. By construction, the so-obtained type system satisfies the “type-semantics” property and completeness.
•State monad.•Imperative λ-calculus.•Intersection types assignment system.
We define the complete Taylor expansion of an ordinary lambda-term as an infinite linear combination–with rational coefficients–of terms of a resource calculus similar to Boudol’s lambda-calculus ...with multiplicities (or with resources). In our resource calculus, all applications are (multi)linear in the algebraic sense,
i.e. commute with linear combinations of the function or the argument. We study the collective behaviour of the beta-reducts of the terms occurring in the Taylor expansion of any ordinary lambda-term, using, in a surprisingly crucial way, a uniformity property that they enjoy. As a corollary, we obtain (the main part of) a proof that this Taylor expansion commutes with Böhm tree computation, syntactically.
We introduce interaction nets for a fragment of the differential lambda-calculus and exhibit in this framework a new symmetry between the
of course and the
why not modalities of linear logic, which ...is completely similar to the symmetry between the
tensor and
par connectives of linear logic. We use algebraic intuitions for introducing these nets and their reduction rules, and then we develop two correctness criteria (weak typability and acyclicity) and show that they guarantee strong normalization. Finally, we outline the correspondence between this interaction nets formalism and the resource lambda-calculus.
Combinatory logic and lambda-calculus, originally devised in the 1920s, have since developed into linguistic tools, especially useful in programming languages. The authors' previous book served as ...the main reference for introductory courses on lambda-calculus for over 20 years: this version is thoroughly revised and offers an account of the subject with the same authoritative exposition. The grammar and basic properties of both combinatory logic and lambda-calculus are discussed, followed by an introduction to type-theory. Typed and untyped versions of the systems, and their differences, are covered. Lambda-calculus models, which lie behind much of the semantics of programming languages, are also explained in depth. The treatment is as non-technical as possible, with the main ideas emphasized and illustrated by examples. Many exercises are included, from routine to advanced, with solutions to most at the end of the book.
This paper considers the applicative computing technology (ACT), within the framework of which the semantic analysis of a number of natural language constructs is performed. The necessary elements of ...grammatical analysis are involved. Much first-order logical means is used, and predicate variables are necessarily used for analysis. At the same time, the advantages of higher-order systems, which include ACT, are extracted. The fact is that the semantics of a natural language is characterized by a multi-tiered nesting of grammatical structures, which conflicts with first-order logical systems.
An Understanding of IPv6 using Muck
International journal of recent technology and engineering,
9/2019, Volume:
8, Issue:
2S8
Journal Article
Open access
The cyber informatics solution to lambda calculus is defined not only by the emulation of 802.11 mesh networks, but also by the extensive need for expert systems. Following quite a while of natural ...examination into IPv4, we demonstrate the copying of superpages, which epitomizes the affirmed standards of algorithms. So as to understand this mission, we present an investigation of Byzantine adaptation to non-critical failure (Muck), which we use to demonstrate that the little-known decentralized algorithm for the improvement of voice-over-IP by White et al. 1 keeps running in O(N2) time.
Online algorithms and lambda calculus 28, In fact, few statisticians would disagree with the visualization of compilers, which embodies the key principles of complexity theory. Gab, our new ...application for superpages, is the solution to all of these problems.
Automatic differentiation in PCF Mazza, Damiano; Pagani, Michele
Proceedings of ACM on programming languages,
01/2021, Volume:
5, Issue:
POPL
Journal Article
Peer reviewed
Open access
We study the correctness of automatic differentiation (AD) in the context of a higher-order, Turing-complete language (PCF with real numbers), both in forward and reverse mode. Our main result is ...that, under mild hypotheses on the primitive functions included in the language, AD is almost everywhere correct, that is, it computes the derivative or gradient of the program under consideration
except
for a set of Lebesgue measure zero. Stated otherwise, there are inputs on which AD is incorrect, but the probability of randomly choosing one such input is zero. Our result is in fact more precise, in that the set of failure points admits a more explicit description: for example, in case the primitive functions are just constants, addition and multiplication, the set of points where AD fails is contained in a countable union of zero sets of polynomials.
We solve the inhabitation problem for a language called λ!, a subsuming paradigm (inspired by call-by-push-value) being able to encode, among others, call-by-name and call-by-value strategies of ...functional programming. The type specification uses a non-idempotent intersection type system, which is able to capture quantitative properties about the dynamics of programs. As an application, we show how our general methodology can be used to derive inhabitation algorithms for different lambda-calculi that are encodable into λ!.
Abstract
In this paper we present an operational semantics for the ‘call-by-name’ probabilistic
λ
-calculus, whose main feature is to use only deterministic relations and to have no constraint on the ...reduction strategy. The calculus enjoys similar properties to the usual
λ
-calculus. In particular we prove it to be confluent, and we prove a standardisation theorem.
PDF
