Unary profile of lambda terms with restricted De Bruijn indices

In this paper we present an average-case analysis of closed lambda terms with restricted values of De Bruijn indices in the model where each occurrence of a variable contributes one to the size. Given a fixed integer k, a lambda term in which all De Bruijn indices are bounded by k has the following shape: It starts with k De Bruijn levels, forming the so-called hat of the term, to which some number of k-colored Motzkin trees are attached. By means of analytic combinatorics, we show that the size of this hat is constant on average and that the average number of De Bruijn levels of k-colored Motzkin trees of size n is asymptotically Θ(√ n). Combining these two facts, we conclude that the maximal non-empty De Bruijn level in a lambda term with restrictions on De Bruijn indices and of size n is, on average, also of order √ n. On this basis, we provide the average unary profile of such lambda terms.

Distribution of Variables in Lambda-Terms with Restrictions on De Bruijn Indices and De Bruijn Levels

We consider two special subclasses of lambda-terms that are restricted by a bound on the number of abstractions between a variable and its binding lambda, the so-called De-Bruijn index, or by a bound on the nesting levels of abstractions, i.e., the number of De Bruijn levels, respectively. We show that the total number of variables is asymptotically normally distributed for both subclasses of lambda-terms with mean and variance asymptotically equal to $Cn$ and $\tilde{C}n$, respectively, where the constants $C$ and $\tilde{C}$ depend on the bound that has been imposed. For the class of lambda-terms with bounded De Bruijn index we derive closed formulas for the constant. For the other class of lambda-terms that we consider, namely lambda-terms with a bounded number of De Bruijn levels, we show quantitative and distributional results on the number of variables, as well as abstractions and applications, in the different De Bruijn levels and thereby exhibit a so-called "unary profile" that attains a very interesting shape. Our results give a combinatorial explanation of an earlier discovered strange phenomenon exhibited by the counting sequence of this particular class of lambda-terms.

The full-reducing Krivine abstract machine KN simulates pure normal-order reduction in lockstep: A proof via corresponding calculus

We exploit the idea of proving properties of an abstract machine by using a corresponding semantic artefact better suited to their proof. The abstract machine is an improved version of Pierre Crégut’s full-reducing Krivine machine KN. The original version works with closed terms of the pure lambda calculus with de Bruijn indices. The improved version reduces in similar fashion but works on closures where terms may be open. The corresponding semantic artefact is a structural operational semantics of a calculus of closures whose reduction relation is purposely a reduction strategy. As shown in previous work, improved KN and the structural operational semantics ‘correspond’, i.e. both artefacts realise the same reduction strategy. In this paper, we prove in the calculus of closures that the reduction strategy simulates in lockstep (at every reduction step) the complete and standard normal-order strategy (i.e. leftmost reduction to normal form) of the pure lambda calculus. The simulation is witnessed by a substitution function from closures of the closure calculus to pure terms of the pure lambda calculus. Thus, KN also simulates normal-order in lockstep by the correspondence. This result is stronger than the known proof that KN is complete, for in the pure lambda calculus there are complete but non-standard strategies. The lockstep simulation proof consists of straightforward structural inductions, thanks to three properties of the closure calculus we call ‘index alignment’, ‘parameters-as-levels’ and ‘balanced derivations’. The first two come from KN. Thanks to these properties, a proof in a calculus of closures involving de Bruijn indices and de Bruijn levels is unproblematic. There is no lexical adjustment at binding lookup, on-the-fly alpha-conversion or recursive traversals of the term to deal with bound and free variables as in other calculi. This paper contributes to the framework for environment machines of Biernacka and Danvy a full-reducing open-terms closure calculus, its corresponding abstract machine, and a lockstep simulation proof via a substitution function.

Exploring Neural Methods for Parsing Discourse Representation Structures

Neural methods have had several recent successes in semantic parsing, though they have yet to face the challenge of producing meaning representations based on formal semantics. We present a sequence-to-sequence neural semantic parser that is able to produce Discourse Representation Structures (DRSs) for English sentences with high accuracy, outperforming traditional DRS parsers. To facilitate the learning of the output, we represent DRSs as a sequence of flat clauses and introduce a method to verify that produced DRSs are well-formed and interpretable. We compare models using characters and words as input and see (somewhat surprisingly) that the former performs better than the latter. We show that eliminating variable names from the output using De Bruijn indices increases parser performance. Adding silver training data boosts performance even further.

Proof-relevant π-calculus: a constructive account of concurrency and causality

We present a formalisation in Agda of the theory of concurrent transitions, residuation and causal equivalence of traces for the π-calculus. Our formalisation employs de Bruijn indices and dependently typed syntax, and aligns the ‘proved transitions’ proposed by Boudol and Castellani in the context of CCS with the proof terms naturally present in Agda's representation of the labelled transition relation. Our main contributions are proofs of the ‘diamond lemma’ for the residuals of concurrent transitions and a formal definition of equivalence of traces up to permutation of transitions.In the π-calculus, transitions represent propagating binders whenever their actions involve bound names. To accommodate these cases, we require a more general diamond lemma where the target states of equivalent traces are no longer identical, but are related by abraidingthat rewires the bound and free names to reflect the particular interleaving of events involving binders. Our approach may be useful for modelling concurrency in other languages where transitions carry meta-data sensitive to particular interleavings, such as dynamically allocated memory addresses.

Pure type systems with explicit substitutions

We introduce a new formulation of pure type systems (PTSs) with explicit substitution and de Bruijn indices and formally prove some of its meta-theory. Using techniques based on Normalisation by Evaluation, we prove that untyped conversion can be typed for predicative PTSs. Although this equivalence was settled by Siles and Herbelin for the conventional presentation of PTSs, we strongly conjecture that our proof method can also be applied to PTSs with η.

Counting and generating lambda terms

Lambda calculus is the basis of functional programming and higher order proof assistants. However, little is known about combinatorial properties of lambda terms, in particular, about their asymptotic distribution and random generation. This paper tries to answer questions like: How many terms of a given size are there? What is a ‘typical’ structure of a simply typable term? Despite their ostensible simplicity, these questions still remain unanswered, whereas solutions to such problems are essential for testing compilers and optimizing programs whose expected efficiency depends on the size of terms. Our approach toward the aforementioned problems may be later extended to any language with bound variables, i.e., with scopes and declarations. This paper presents two complementary approaches: one, theoretical, uses complex analysis and generating functions, the other, experimental, is based on a generator of lambda terms. Thanks to de Bruijn indices (de Bruijn, N. (1972) Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem. Indagat. Math.34(5), 381–392), we provide three families of formulas for the number of closed lambda terms of a given size and we give four relations between these numbers which have interesting combinatorial interpretations. As a by-product of the counting formulas, we design an algorithm for generating λ-terms. Performed tests provide us with experimental data, like the average depth of bound variables and the average number of head lambdas. We also create random generators for various sorts of terms. Thereafter, we conduct experiments that answer questions like: What is the ratio of simply typable terms among all terms? (Very small!) How are simply typable lambda terms distributed among all lambda terms? (A typable term almost always starts with an abstraction.) In this paper, abstractions and applications have size 1 and variables have size 0.