Lambda Terms Research Articles

The approach development of data extraction from lambda terms

The study's object is the process of extracting the characteristics of lambda terms, which indicate the optimality of the reduction strategy and increase the productivity of compilers and interpreters. The solution to the problem of extracting specific strategy priority data from lambda terms using Machine Learning methods was considered. Such data was extracted using the large language model Microsoft CodeBERT, which was trained to solve the problem of summarizing the software code. The resulting matrices of embeddings were used to obtain vectors of average embeddings of size 768 and a latent space of size 8 thousand. Further, vectors of average embeddings were used for cluster analysis using the DBSCAN and Hierarchical Agglomerative clustering methods. The most informative variables affecting clustering were determined. Next, the clustering results were compared with the priorities of reduction strategies, which showed the impossibility of separating terms with RI priority. A feature of the obtained results is using machine learning methods to obtain knowledge. The clustering results showed many of the same informative variables, which is explained by the similar shape of the obtained clusters. The results of comparing the clustering values with the real priority are explained by the impossibility of clearly determining the priority and the use of the Microsoft CodeBERT model, which was not trained for the analysis of lambda terms. The proposed approach can find application in the development of compilers and interpreters of functional programming languages, allowing to analyze the code and extract important data to optimize the execution of programs. The obtained data can be used to develop rules aimed at improving the efficiency of compilation and interpretation

Supervised data extraction from transformer representation of Lambda-terms

The object of this research is the process of compiler optimization, as it is essential in modern software development, particularly in functional programming languages like Lambda Calculus. Optimization strategies directly impact interpreter and compiler performance, influencing resource efficiency and program execution. While functional programming compilers have garnered less attention regarding optimization efforts than their object-oriented counterparts, Lambda Calculus’s complexity poses unique challenges. Bridging this gap requires innovative approaches like leveraging machine learning techniques to enhance optimization strategies. This study focuses on leveraging machine learning to bridge the optimization gap in functional programming, particularly within the context of Lambda Calculus. This study delves into the extraction features from Lambda terms related to reduction strategies by applying machine learning. Previous research has explored various approaches, including analyzing reduction step complexities and using sequence analysis Artificial Neural Networks (ANNs) with simplified term representation. This research aims to develop a methodology for extracting comprehensive term data and providing insights into optimal reduction priorities by employing Large Language Models (LLMs). Tasks were set to generate embeddings from Lambda terms using LLMs, train ANN models to predict reduction steps, and compare results with simplified term representations. This study employs a sophisticated blend of machine learning algorithms and deep learning models as a method of analyzing and predicting optimal reduction paths in Lambda Calculus terms. The result of this study is a method that showed improvement in determining the number of reduction steps by using embeddings. Conclusions: The findings of this research offer significant implications for further advancements in compiler and interpreter optimization. This study paves the way for future research to enhance compiler efficiency by demonstrating the efficacy of employing LLMs to prioritize normalization strategies. Using machine learning in functional programming optimization opens avenues for dynamic optimization strategies and comprehensive analysis of program features.

Cost Automata, Safe Schemes, and Downward Closures

In this work we prove decidability of the model-checking problem for safe recursion schemes against properties defined by alternating B-automata. We then exploit this result to show how to compute downward closures of languages of finite trees recognized by safe recursion schemes. Higher-order recursion schemes are an expressive formalism used to define languages of finite and infinite ranked trees by means of fixed points of lambda terms. They extend regular and context-free grammars, and are equivalent in expressive power to the simply typed λY-calculus and collapsible pushdown automata. Safety in a syntactic restriction which limits their expressive power. The class of alternating B-automata is an extension of alternating parity automata over infinite trees; it enhances them with counting features that can be used to describe boundedness properties.

A correct-by-construction conversion from lambda calculus to combinatory logic

Abstract This pearl defines a translation from well-typed lambda terms to combinatory logic, where both the preservation of types and the correctness of the translation are enforced statically.

The applicative approach to the synthesis of a data structure with the given combinatory characteristic

The paper shows that the use of a composition of objects representing a data structure, in fact, means the creation of a kind of information channels on their basis, along which the computation process spreads. In the case of applying the applicative computational technology, the computation process is started by means of the operation of applying the function object to the argument object. Two ways are presented to generate function objects — to use either lambda expressions or combinators to represent them. The first method uses an abstraction meta-operator and associated variables, which leads to the use of substitution systems with the potential for side effects. In the second method, only constant combinator objects participate in the construction of the function, and their application to the argument triggers a conversion based on rewriting rules, which does not cause a side effect. A practical solution to the problem of synthesizing a compositional data structure can be mixed, when both lambda terms and combinators are involved in the computations, which reduces the length of expressions. As a result, a data structure appears, which is composed of compositions of argument objects, equipped with a generated set of supporting function objects.

Уравнение Власова-Эйнштейна и точки Лагранжа

In classical works, equations for fields are proposed without derivation of the right-hand sides. Here we give a derivation of the right-hand sides of the Maxwell and Einstein equations in the framework of the Vlasov-Maxwell-Einstein equations from the classical, but more general principle of least action. Moreover, in the case of Friedman's model of the Universe, one possibility is obtained to explain the mysterious accelerated expansion of the Universe. The accelerated expansion of the Universe, marked by the 2011 Nobel Prize in Physics, is receiving close attention. The generally accepted explanation now is the addition of Einstein's lambda term to the relativistic action. And it is well known that in the nonrelativistic theory this corresponds to the addition of a repulsive quadratic potential.

Linear Additives

We introduce LAM, a subsystem of IMALL2 with restricted additive rules able to manage duplication linearly, called linear additive rules. LAM is presented as the type assignment system for a calculus endowed with copy constructors, which deal with substitution in a linear fashion. As opposed to the standard additive rules, the linear additive rules do not affect the complexity of term reduction: typable terms of LAM enjoy linear strong normalization. Moreover, a mildly weakened version of cut-elimination for this system is proven which takes a cubic number of steps. Finally, we define a sound translation from proofs of LAM into linear lambda terms of IMLL2, and we study its complexity.

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.

Deriving Efficient Sequential and Parallel Generators for Closed Simply-Typed Lambda Terms and Normal Forms

Contrary to several other families of lambda terms, no closed formula or generating function is known and none of the sophisticated techniques devised in analytic combinatorics can currently help with counting or generating the set of simply-typed closed lambda terms of a given size. Moreover, their asymptotic scarcity among the set of closed lambda terms makes counting them via brute force generation and type inference quickly intractable, with previous published work showing counts for them only up to size 10. By taking advantage of the synergy between logic variables, unification with occurs check and efficient backtracking in today’s Prolog systems, we climb 4 orders of magnitude above previously known counts by deriving progressively faster sequential Prolog programs that generate and/or count the set of closed simply-typed lambda terms of sizes up to 14. Similar counts for closed simply-typed normal forms are also derived up to size 14. Finally, we devise several parallel execution algorithms, based on generating code to be uniformly distributed among the available cores, that push the counts for simply typed terms up to size 15 and simply typed normal forms up to size 16. As a remarkable feature, our parallel algorithms are linearly scalable with the number of available cores.

Modeling spread, interlace and interchange of information processes with variable domains

In this paper a semantic metalanguage is developed and designed to study the occurrence, spread and safe interaction of semantic processes in information modeling systems, including cognitive interference. An approach to construe a semantic network is proposed and based on a computational model in which both nodes and arcs are information processes. Concepts are represented by intensional objects within the framework of theories without types, and they, in turn, are considered as special counterparts of typed theories. Similar mixing was used in model studies for lambda calculus. To a contrast with them, in this paper, information processes correspond to parameterized metadata objects, which are variable domain constructs. Transformations of variable domains correspond to the spread of the process. Directional transformation provides the generation of metadata targets in the form of parameterized concepts. This simulates the development of the process, which corresponds to the spread of cognitive interference and allows the interpretation of a hidden time factor. The emerging model is purely process based and provides such a conceptual framework. The possibility of coding this framework with a system of interdependent lambda terms is reflected.

Deriving Theorems in Implicational Linear Logic, Declaratively

The problem we want to solve is how to generate all theorems of a given size in the implicational fragment of propositional intuitionistic linear logic. We start by filtering for linearity the proof terms associated by our Prolog-based theorem prover for Implicational Intuitionistic Logic. This works, but using for each formula a PSPACE-complete algorithm limits it to very small formulas. We take a few walks back and forth over the bridge between proof terms and theorems, provided by the Curry-Howard isomorphism, and derive step-by-step an efficient algorithm requiring a low polynomial effort per generated theorem. The resulting Prolog program runs in O(N) space for terms of size N and generates in a few hours 7,566,084,686 theorems in the implicational fragment of Linear Intuitionistic Logic together with their proof terms in normal form. As applications, we generate datasets for correctness and scalability testing of linear logic theorem provers and training data for neural networks working on theorem proving challenges. The results in the paper, organized as a literate Prolog program, are fully replicable. Keywords: combinatorial generation of provable formulas of a given size, intuitionistic and linear logic theorem provers, theorems of the implicational fragment of propositional linear intuitionistic logic, Curry-Howard isomorphism, efficient generation of linear lambda terms in normal form, Prolog programs for lambda term generation and theorem proving.

Higher-order pattern generalization modulo equational theories

AbstractWe consider anti-unification for simply typed lambda terms in theories defined by associativity, commutativity, identity (unit element) axioms and their combinations and develop a sound and complete algorithm which takes two lambda terms and computes their equational generalizations in the form of higher-order patterns. The problem is finitary: the minimal complete set of such generalizations contains finitely many elements. We define the notion of optimal solution and investigate special restrictions of the problem for which the optimal solution can be computed in linear or polynomial time.

On Some Enumerative Problems in Lambda Calculus

Combinatorial problems related to the enumeration of lambda terms in the untyped lambda calculus and in the Church-style simply typed lambda calculus with a single atom are considered. For the case of untyped lambda calculus, a system of equations for generating functions that enumerate lambda terms is derived. For the typed lambda calculus, the inhabited types are enumerated, as well as their primitive inhabitants. Bibliography: 11 titles.

Connected Chord Diagrams and Bridgeless Maps

We present a surprisingly new connection between two well-studied combinatorial classes: rooted connected chord diagrams on one hand, and rooted bridgeless combinatorial maps on the other hand. We describe a bijection between these two classes, which naturally extends to indecomposable diagrams and general rooted maps. As an application, this bijection provides a simplifying framework for some technical quantum field theory work realized by some of the authors. Most notably, an important but technical parameter naturally translates to vertices at the level of maps. We also give a combinatorial proof to a formula which previously resulted from a technical recurrence, and with similar ideas we prove a conjecture of Hihn. Independently, we revisit an equation due to Arquès and Béraud for the generating function counting rooted maps with respect to edges and vertices, giving a new bijective interpretation of this equation directly on indecomposable chord diagrams, which moreover can be specialized to connected diagrams and refined to incorporate the number of crossings. Finally, we explain how these results have a simple application to the combinatorics of lambda calculus, verifying the conjecture that a certain natural family of lambda terms is equinumerous with bridgeless maps.

Statistical Properties of Lambda Terms

We present a quantitative, statistical analysis of random lambda terms in the De Bruijn notation. Following an analytic approach using multivariate generating functions, we investigate the distribution of various combinatorial parameters of random open and closed lambda terms, including the number of redexes, head abstractions, free variables or the De Bruijn index value profile. Moreover, we conduct an average-case complexity analysis of finding the leftmost-outermost redex in random lambda terms showing that it is on average constant. The main technical ingredient of our analysis is a novel method of dealing with combinatorial parameters inside certain infinite, algebraic systems of multivariate generating functions. Finally, we briefly discuss the random generation of lambda terms following a given skewed parameter distribution and provide empirical results regarding a series of more involved combinatorial parameters such as the number of open subterms and binding abstractions in closed lambda terms.

Evaluating lambda terms with traversals

This paper presents a method to evaluate untyped lambda terms based on a term tree-traversing technique and judicious use of eta-expansion. The traversal method is inspired by Game Semantics. As traversals explore nodes of the term tree, they dynamically eta-expand some of the subterms to locate their non-immediate arguments. A quantity called dynamic arity determines the necessary amount of eta-expansion to perform at a given point. Traversals are finitely enumerable and characterize the paths in the tree representation of the beta-normal form when it exists. Correctness of the evaluation method follows from the fact that traversals implement leftmost linear reduction, a non-standard reduction strategy based on head linear reduction.

Compaction of Church Numerals

In this study, we address the problem of compaction of Church numerals. Church numerals are unary representations of natural numbers on the scheme of lambda terms. We propose a novel decomposition scheme from a given natural number into an arithmetic expression using tetration, which enables us to obtain a compact representation of lambda terms that leads to the Church numeral of the natural number. For natural number n, we prove that the size of the lambda term obtained by the proposed method is O ( ( slog 2 n ) ( log n / log log n ) ) . Moreover, we experimentally confirmed that the proposed method outperforms binary representation of Church numerals on average, when n is less than approximately 10,000.

A Type-Driven Vector Semantics for Ellipsis with Anaphora Using Lambek Calculus with Limited Contraction

We develop a vector space semantics for verb phrase ellipsis with anaphora using type-driven compositional distributional semantics based on the Lambek calculus with limited contraction (LCC) of Jäger (Anaphora and type logical grammar, Springer, Berlin, 2006). Distributional semantics has a lot to say about the statistical collocation based meanings of content words, but provides little guidance on how to treat function words. Formal semantics on the other hand, has powerful mechanisms for dealing with relative pronouns, coordinators, and the like. Type-driven compositional distributional semantics brings these two models together. We review previous compositional distributional models of relative pronouns, coordination and a restricted account of ellipsis in the DisCoCat framework of Coecke et al. (Mathematical foundations for a compositional distributional model of meaning, 2010. arXiv:1003.4394, Ann Pure Appl Log 164(11):1079–1100, 2013). We show how DisCoCat cannot deal with general forms of ellipsis, which rely on copying of information, and develop a novel way of connecting typelogical grammar to distributional semantics by assigning vector interpretable lambda terms to derivations of LCC in the style of Muskens and Sadrzadeh (in: Amblard, de Groote, Pogodalla, Retoré (eds) Logical aspects of computational linguistics, Springer, Berlin, 2016). What follows is an account of (verb phrase) ellipsis in which word meanings can be copied: the meaning of a sentence is now a program with non-linear access to individual word embeddings. We present the theoretical setting, work out examples, and demonstrate our results with a state of the art distributional model on an extended verb disambiguation dataset.

Classical Copying versus Quantum Entanglement in Natural Language: The Case of VP-ellipsis

This paper compares classical copying and quantum entanglement in natural language by considering the case of verb phrase (VP) ellipsis. VP ellipsis is a non-linear linguistic phenomenon that requires the reuse of resources, making it the ideal test case for a comparative study of different copying behaviours in compositional models of natural language. Following the line of research in compositional distributional semantics set out by (Coecke et al., 2010) we develop an extension of the Lambek calculus which admits a controlled form of contraction to deal with the copying of linguistic resources. We then develop two different compositional models of distributional meaning for this calculus. In the first model, we follow the categorical approach of (Coecke et al., 2013) in which a functorial passage sends the proofs of the grammar to linear maps on vector spaces and we use Frobenius algebras to allow for copying. In the second case, we follow the more traditional approach that one finds in categorial grammars, whereby an intermediate step interprets proofs as non-linear lambda terms, using multiple variable occurrences that model classical copying. As a case study, we apply the models to derive different readings of ambiguous elliptical phrases and compare the analyses that each model provides.

A case study in programming coinductive proofs: Howe’s method

Bisimulation proofs play a central role in programming languages in establishing rich properties such as contextual equivalence. They are also challenging to mechanize, since they require a combination of inductive and coinductive reasoning on open terms. In this paper, we describe mechanizing the property that similarity in the call-by-name lambda calculus is a pre-congruence using Howe’s method in theBelugaformal reasoning system. The development relies on three key ingredients: (1) we give a higher order abstract syntax (HOAS) encoding of lambda terms together with their operational semantics as intrinsically typed terms, thereby avoiding not only the need to deal with binders, renaming and substitutions, but keeping all typing invariants implicit; (2) we take advantage ofBeluga’s support for representing open terms using built-in contexts and simultaneous substitutions: this allows us to directly state central definitions such as open simulation without resorting to the usual inductive closure operation and to encode very elegantly notoriously painful proofs such as the substitutivity of the Howe relation; (3) we exploit the possibility of reasoning by coinduction inBeluga’s reasoning logic. The end result is succinct and elegant, thanks to the high-level abstractions and primitivesBelugaprovides. We believe that this mechanization is a significant example that illustratesBeluga’s strength at mechanizing challenging (co)inductive proofs using HOAS encodings.