Related papers: Counting and generating lambda terms
$\lambda\upsilon$ is an extension of the $\lambda$-calculus which internalises the calculus of substitutions. In the current paper, we investigate the combinatorial properties of $\lambda\upsilon$ focusing on the quantitative aspects of…
We introduce two extensions of the $\lambda$-calculus with a probabilistic choice operator, $\Lambda_\oplus^{cbv}$ and $\Lambda_\oplus^{cbn}$, modeling respectively call-by-value and call-by-name probabilistic computation. We prove that…
Many different systems with explicit substitutions have been proposed to implement a large class of higher-order languages. Motivations and challenges that guided the development of such calculi in functional frameworks are surveyed in the…
In the quest to give a formal compositional semantics to natural languages, semanticists have started turning their attention to phenomena that have been also considered as parts of pragmatics (e.g., discourse anaphora and presupposition…
Probabilistic operational semantics for a nondeterministic extension of pure lambda calculus is studied. In this semantics, a term evaluates to a (finite or infinite) distribution of values. Small-step and big-step semantics are both…
Mathematical reasoning serves as a crucial testbed for the intelligence of large language models (LLMs), and math word problems (MWPs) are a popular type of math problems. Most MWP datasets consist of problems containing only the necessary…
In this paper, we define a realizability semantics for the simply typed $\lambda\mu$-calculus. We show that if a term is typable, then it inhabits the interpretation of its type. This result serves to give characterizations of the…
This paper formalizes a latent variable inference problem we call {\em supervised pattern discovery}, the goal of which is to find sets of observations that belong to a single ``pattern.'' We discuss two versions of the problem and prove…
Evaluating the open-ended text generation of large language models (LLMs) is challenging because of the lack of a clear ground truth and the high cost of human or LLM-based assessments. We propose a novel benchmark that evaluates LLMs using…
Existing text representations such as embeddings and bag-of-words are not suitable for rule learning due to their high dimensionality and absent or questionable feature-level interpretability. This article explores whether large language…
The objective of this paper is to develop a functional programming language for quantum computers. We develop a lambda calculus for the classical control model, following the first author's work on quantum flow-charts. We define a…
Large Language Models (LLMs) have demonstrated unprecedented capabilities in code generation. However, there remains a limited understanding of code generation errors that LLMs can produce. To bridge the gap, we conducted an in-depth…
As large language models (LLMs) transition from chat interfaces to integral components of stochastic pipelines and systems approaching general intelligence, the ability to faithfully sample from specified probability distributions has…
Random testing approaches work by generating inputs at random, or by selecting inputs randomly from some pre-defined operational profile. One long-standing question that arises in this and other testing contexts is as follows: When can we…
While a mature body of work supports the study of rewriting systems, abstract tools for Probabilistic Rewriting are still limited. In this paper we study the question of uniqueness of the result (unique limit distribution), and develop a…
Since Val Tannen's pioneer work on the combination of simply-typed lambda-calculus and first-order rewriting (LICS'88), many authors have contributed to this subject by extending it to richer typed lambda-calculi and rewriting paradigms,…
Large Language Models (LLMs) have recently been widely used for code generation. Due to the complexity and opacity of LLMs, little is known about how these models generate code. We made the first attempt to bridge this knowledge gap by…
It is well known that the length of a beta-reduction sequence of a simply typed lambda-term of order k can be huge; it is as large as k-fold exponential in the size of the lambda-term in the worst case. We consider the following relevant…
The task of inferring logical formulas from examples has garnered significant attention as a means to assist engineers in creating formal specifications used in the design, synthesis, and verification of computing systems. Among various…
The Algebraic lambda-calculus and the Linear-Algebraic lambda-calculus extend the lambda-calculus with the possibility of making arbitrary linear combinations of terms. In this paper we provide a fine-grained, System F-like type system for…