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The Theory of Functional Connections (TFC) is a functional interpolation framework founded upon the so-called constrained expression: a functional that expresses the family of all possible functions that satisfy some user-specified, linear…
Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications. They have been used in many areas, including combinatorial optimization, machine learning, and economics. In…
The idea of using unfolding as a way of computing a program semantics has been applied successfully to logic programs and has shown itself a powerful tool that provides concrete, implementable results, as its outcome is actually source…
Traditional Turing machines are semantically poor, they only concern the syntactic manipulation of symbols, discarding the mathematical semantics behind the symbols. This semantic deficiency is considered the root cause of the three major…
In this paper we regard languages and their acceptors -- such as deterministic or weighted automata, transducers, or monoids -- as functors from input categories that specify the type of the languages and of the machines to categories that…
This paper develops a correspondence relating convex hulls of fractional functions with those of polynomial functions over the same domain. Using this result, we develop a number of new reformulations and relaxations for fractional…
Functional decomposition is the process of breaking down a function $f$ into a composition $f=g(f_1,\dots,f_k)$ of simpler functions $f_1,\dots,f_k$ belonging to some class $\mathcal{F}$. This fundamental notion can be used to model…
In this paper we classify some special reducts of the countable atomless Boolean algebra which we call functional reducts. We prove that there are exactly $13$ such structures up to first order interdefinability.
Several new algorithms for deciding emptiness of Boolean combinations of regular languages and of languages of alternating automata (AFA) have been proposed recently, especially in the context of analysing regular expressions and in string…
Well-designed diagnostic tasks have played a key role in studying the failure of neural nets (NNs) to generalize systematically. Famous examples include SCAN and Compositional Table Lookup (CTL). Here we introduce CTL++, a new diagnostic…
The defunctionalization translation that eliminates higher-order functions from programs forms a key part of many compilers. However, defunctionalization for dependently-typed languages has not been formally studied. We present the first…
Natural language processing for programming aims to use NLP techniques to assist programming. It is increasingly prevalent for its effectiveness in improving productivity. Distinct from natural language, a programming language is highly…
We propose a general approach to construct cryptographic significant Boolean functions of $(r+1)m$ variables based on the additive decomposition $\mathbb{F}_{2^{rm}}\times\mathbb{F}_{2^m}$ of the finite field $\mathbb{F}_{2^{(r+1)m}}$,…
We give a new characterization of the class of rational string functions from formal language theory using order-preserving interpretations with respect to a very weak monadic programming language. This refines the known characterization of…
Functional data analysis involves data described by regular functions rather than by a finite number of real valued variables. While some robust data analysis methods can be applied directly to the very high dimensional vectors obtained…
Probabilistic context-free grammars (PCFGs) with neural parameterization have been shown to be effective in unsupervised phrase-structure grammar induction. However, due to the cubic computational complexity of PCFG representation and…
We discuss the possibility of constructing a function that validates the definition or not definition of the partial recursive functions of one variable. This is a topic in computability theory, which was first approached by Alan M. Turing…
We propose a method to control the attributes of Language Models (LMs) for the text generation task using Causal Average Treatment Effect (ATE) scores and counterfactual augmentation. We explore this method, in the context of LM…
Multivariate functions emerge naturally in a wide variety of data-driven models. Popular choices are expressions in the form of basis expansions or neural networks. While highly effective, the resulting functions tend to be hard to…
Relational machine learning programs like those developed in Inductive Logic Programming (ILP) offer several advantages: (1) The ability to model complex relationships amongst data instances; (2) The use of domain-specific relations during…