A fully-automated algorithm is developed able to show that evaluation of a given untyped lambda-expression will terminate under CBV (call-by-value). The ``size-change principle'' from first-order programs is extended to arbitrary untyped lambda-expressions in two steps. The first step suffices to show CBV termination of a single, stand-alone lambda;-expression. The second suffices to show CBV termination of any member of a regular set of lambda-expressions, defined by a tree grammar. (A simple example is a minimum function, when applied to arbitrary Church numerals.) The algorithm is sound and proven so in this paper. The Halting Problem's undecidability implies that any sound algorithm is necessarily incomplete: some lambda-expressions may in fact terminate under CBV evaluation, but not be recognised as terminating.
The intensional power of the termination algorithm is reasonably high. It certifies as terminating many interesting and useful general recursive algorithms including programs with mutual recursion and parameter exchanges, and Colson's ``minimum'' algorithm. Further, our type-free approach allows use of the Y combinator, and so can identify as terminating a substantial subset of PCF.
We aim to reason about the correctness of behaviour-preserving transformations of Erlang programs. Behaviour preservation is characterised by semantic equivalence. Based upon our existing formal semantics for Core Erlang, we investigate potential definitions of suitable equivalence relations. In particular we adapt a number of existing approaches of expression equivalence to a simple functional programming language that carries the main features of sequential Core Erlang; we then examine the properties of the equivalence relations and formally establish connections between them. The results pr…
We introduce a method to evaluate untyped lambda terms by combining the theory of traversals, a term-tree traversing technique inspired from Game Semantics, with judicious use of the eta-conversion rule of the lambda calculus. The traversal theory of the simply-typed lambda calculus relies on the eta-long transform to ensure that when traversing an application, there is a subterm representing every possible operator's argument. In the untyped setting, we instead exhibit the missing operand via ad-hoc instantiation of the eta-expansion rule, which allows the traversal to proceed as if the opera…
We investigate the relationship between finite terms in λ-letrec, the λ-calculus with letrec, and the infinite λ-terms they express. We say that a lambda-letrec term expresses a lambda-term if the latter can be obtained as an infinite unfolding of the former. Unfolding is the process of substituting occurrences of function variables by the right-hand side of their definition. We consider the following questions: (i) How can we characterise those infinite λ-terms that are λ-letrec-expressible? (ii) Given two λ-letrec terms, how can we determine whether they have the same unfolding? (iii) Given …
We develop the operational semantics of an untyped probabilistic lambda-calculus with continuous distributions, as a foundation for universal probabilistic programming languages such as Church, Anglican, and Venture. Our first contribution is to adapt the classic operational semantics of lambda-calculus to a continuous setting via creating a measure space on terms and defining step-indexed approximations. We prove equivalence of big-step and small-step formulations of this distribution-based semantics. To move closer to inference techniques, we also define the sampling-based semantics of a ter…
We introduce a method for proving almost sure termination in the context of lambda calculus with continuous random sampling and explicit recursion, based on ranking supermartingales. This result is extended in three ways. Antitone ranking functions have weaker restrictions on how fast they must decrease, and are applicable to a wider range of programs. Sparse ranking functions take values only at a subset of the program's reachable states, so they are simpler to define and more flexible. Ranking functions with respect to alternative reduction strategies give yet more flexibility, and significa…
The Calculus of Audited Units (CAU) is a typed lambda calculus resulting from a computational interpretation of Artemov's Justification Logic under the Curry-Howard isomorphism; it extends the simply typed lambda calculus by providing audited types, inhabited by expressions carrying a trail of their past computation history. Unlike most other auditing techniques, CAU allows the inspection of trails at runtime as a first-class operation, with applications in security, debugging, and transparency of scientific computation. An efficient implementation of CAU is challenging: not only do the sizes …
We investigate the possibility of a semantic account of the execution time (i.e. the number of beta-steps leading to the normal form, if any) for the shuffling calculus, an extension of Plotkin's call-by-value lambda-calculus. For this purpose, we use a linear logic based denotational model that can be seen as a non-idempotent intersection type system: relational semantics. Our investigation is inspired by similar ones for linear logic proof-nets and untyped call-by-name lambda-calculus. We first prove a qualitative result: a (possibly open) term is normalizable for weak reduction (which does …
A cornerstone of the theory of lambda-calculus is that intersection types characterise termination properties. They are a flexible tool that can be adapted to various notions of termination, and that also induces adequate denotational models. Since the seminal work of de Carvalho in 2007, it is known that multi types (i.e. non-idempotent intersection types) refine intersection types with quantitative information and a strong connection to linear logic. Typically, type derivations provide bounds for evaluation lengths, and minimal type derivations provide exact bounds. De Carvalho studied call-…
In the Calculus of Dependent Lambda Eliminations (CDLE), a pure Curry-style type theory, it is possible to generically λ-encode inductive datatypes which support course-of-values (CoV) induction. We present a datatype subsystem for Cedille (an implementation of CDLE) that provides this feature to programmers through convenient notation for declaring datatypes and for defining functions over them by case analysis and fixpoint-style recursion guarded by a type-based termination checker. We demonstrate that this does not require extending CDLE by showing how datatypes and functions over them elab…
The LLMbda Calculus: AI Agents, Conversations, and Information Flow
University of Nottingham, UK（诺丁汉大学）；University of Edinburgh, UK（爱丁堡大学）；Chalmers University of Technology（查尔姆斯理工大学）；The University of Gothenburg, Sweden（哥德堡大学）
