Related papers: A Lambda Term Representation Inspired by Linear Or…
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…
We study how large language models (LLMs) reason about memorized knowledge through simple binary relations such as equality ($=$), inequality ($<$), and inclusion ($\subset$). Unlike in-context reasoning, the axioms (e.g., $a < b, b < c$)…
We show that any multiple-valued function can be represented by a linear lambda term typed in a second-order polymorphic type system, using two distinct styles. The first is a circuit style, which mimics combinational circuits in switching…
The logic programming paradigm provides the basis for a new intensional view of higher-order notions. This view is realized primarily by employing the terms of a typed lambda calculus as representational devices and by using a richer form…
We introduce a new symbolic representation based on an original generalization of counter abstraction. Unlike classical counter abstraction (used in the analysis of parameterized systems with unordered or unstructured topologies) the new…
This paper proposes an alternative to standard first-order logic that seeks greater naturalness, generality, and semantic self-containment. The system removes the first-order restriction, avoids type hierarchies, and dispenses with external…
Traditional neural embeddings represent concepts as points, excelling at similarity but struggling with higher-level reasoning and asymmetric relationships. We introduce a novel paradigm: embedding concepts as linear subspaces. This…
Formal deductive systems are very common in computer science. They are used to represent logics, programming languages, and security systems. Moreover, writing programs that manipulate them and that reason about them is important and…
Permissive-Nominal Logic (PNL) is an extension of first-order predicate logic in which term-formers can bind names in their arguments. This allows for direct axiomatisations with binders, such as of the lambda-binder of the lambda-calculus…
The representation space of pretrained Language Models (LMs) encodes rich information about words and their relationships (e.g., similarity, hypernymy, polysemy) as well as abstract semantic notions (e.g., intensity). In this paper, we…
The key to any nameless representation of syntax is how it indicates the variables we choose to use and thus, implicitly, those we discard. Standard de Bruijn representations delay discarding maximally till the leaves of terms where one is…
In neural network models of language, words are commonly represented using context-invariant representations (word embeddings) which are then put in context in the hidden layers. Since words are often ambiguous, representing the…
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.…
Vision-language models (VLMs) transform environment percepts into vision-language semantics interpretable by LLMs. However, completing complex tasks often requires reasoning about information beyond what is currently perceived. We propose…
We propose a principle for exploring context in machine learning models. Starting with a simple assumption that each observation may or may not depend on its context, a conditional probability distribution is decomposed into two parts:…
This paper introduces context algebras and demonstrates their application to combining logical and vector-based representations of meaning. Other approaches to this problem attempt to reproduce aspects of logical semantics within new…
We extend the {\lambda}-calculus with constructs suitable for relational and functional-logic programming: non-deterministic choice, fresh variable introduction, and unification of expressions. In order to be able to unify…
The Edinburgh Logical Framework (LF) is a dependently type lambda calculus that can be used to encode formal systems. The versatility of LF allows specifications to be constructed also about the encoded systems. The Twelf system exploits…
We extend the framework of variational autoencoders to represent transformations explicitly in the latent space. In the family of hierarchical graphical models that emerges, the latent space is populated by higher order objects that are…
Logical relations are one of the most powerful techniques in the theory of programming languages, and have been used extensively for proving properties of a variety of higher-order calculi. However, there are properties that cannot be…