Lambda Calculus with Types

Abstract

This handbook with exercises reveals in formalisms, hitherto mainly used for hardware and software design and verification, unexpected mathematical beauty. The lambda calculus forms a prototype universal programming language, which in its untyped version is related to Lisp, and was treated in the first author's classic The Lambda Calculus (1984). The formalism has since been extended with types and used in functional programming (Haskell, Clean) and proof assistants (Coq, Isabelle, HOL), used in designing and verifying IT products and mathematical proofs. In this book, the authors focus on three classes of typing for lambda terms: simple types, recursive types and intersection types. It is in these three formalisms of terms and types that the unexpected mathematical beauty is revealed. The treatment is authoritative and comprehensive, complemented by an exhaustive bibliography, and numerous exercises are provided to deepen the readers' understanding and increase their confidence using types.

Logical Entity Level Sentiment Analysis

“…Besides variables, functional abstraction and functional application, which follows from simply typed λ-expressions cf. [2], the following structures are available:…”

“…An illustration of this is shown by the potentially unbound number of relative clauses allowed in English, e.g. (2), where breakfast is described as best, however one would need to use a window size of at least 9 to detect this relation, which is arguably larger then normally considered ( [11] only considers up to tri-grams).…”

“…(2) We present a formal logical approach for entity level sentiment analysis that utilizes machine learning techniques for efficient syntactical tagging and performs a deep structural analysis of the syntactical properties of texts in order to yield precise results.…”

A Proposal for Broad Spectrum Proof Certificates

“…When computation is determinate (i.e., when expressions have at most one value) this observation has been called the Poincaré principle [8,10]. In rich type systems, such as those found in functional Pure Type System [7], computation is dominated by β-reductions: communicating a proof in that setting does not need to communicate the trace of such β-reductions.…”

The Theory of Calculi with Explicit Substitutions Revisited

“…We remark that for every Λes-term one has C(t) → * λes t. Also, when t is typable in λes, then also C(t) is typable in λes (just change the use of subs by abs followed by app). By Lemma 11 the term L(C(t)) = C(t) is also typable in simply typed λ-calculus and thus C(t) ∈ SN β [5]. We get C(t) ∈ SN λes by Corollary 4 so that t ∈ SN λes .…”