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Bove and Capretta's popular method for justifying function definitions by general recursive equations is based on the observation that any structured general recursion equation defines an inductive subset of the intended domain (the "domain…
We explore several concepts for analyzing the intuitive notion of computational irreducibility and we propose a robust formal definition, first in the field of cellular automata and then in the general field of any computable function f…
Several formal systems, such as resolution and minimal model semantics, provide a framework for logic programming. In this paper, we will survey the use of structural proof theory as an alternative foundation. Researchers have been using…
We present an elaboration of inductive definitions down to a universe of datatypes. The universe of datatypes is an internal presentation of strictly positive families within type theory. By elaborating an inductive definition -- a…
Program slicing provides explanations that illustrate how program outputs were produced from inputs. We build on an approach introduced in prior work by Perera et al., where dynamic slicing was defined for pure higher-order functional…
Structural recursion is a common technique used by programmers in modern languages and is taught to introductory computer science students. But what about its dual, structural corecursion? Structural corecursion is an elegant technique,…
Transparency, user trust, and human comprehension are popular ethical motivations for interpretable machine learning. In support of these goals, researchers evaluate model explanation performance using humans and real world applications.…
Traditionally, formal languages are defined as sets of words. More recently, the alternative coalgebraic or coinductive representation as infinite tries, i.e., prefix trees branching over the alphabet, has been used to obtain compact and…
Iterative refinement -- start with a random guess, then iteratively improve the guess -- is a useful paradigm for representation learning because it offers a way to break symmetries among equally plausible explanations for the data. This…
Abduction is one of the most important forms of reasoning; it has been successfully applied to several practical problems such as diagnosis. In this paper we investigate whether the computational complexity of abduction can be reduced by an…
A standard informal method for analyzing the asymptotic complexity of a program is to extract a recurrence that describes its cost in terms of the size of its input, and then to compute a closed-form upper bound on that recurrence. We give…
The paper gives a soundness and completeness proof for the implicative fragment of intuitionistic calculus with respect to the semantics of computability logic, which understands intuitionistic implication as interactive algorithmic…
Recurrence equations lie at the heart of many computational paradigms including dynamic programming, graph analysis, and linear solvers. These equations are often expensive to compute and much work has gone into optimizing them for…
The fundamental challenge of drawing causal inference is that counterfactual outcomes are not fully observed for any unit. Furthermore, in observational studies, treatment assignment is likely to be confounded. Many statistical methods have…
We give new proofs of soundness (all representable functions on base types lies in certain complexity classes) for Elementary Affine Logic, LFPL (a language for polytime computation close to realistic functional programming introduced by…
Inference metaprogramming enables effective probabilistic programming by supporting the decomposition of executions of probabilistic programs into subproblems and the deployment of hybrid probabilistic inference algorithms that apply…
We propose a call-by-value lambda calculus extended with a new construct inspired by abductive inference and motivated by the programming idioms of machine learning. Although syntactically simple the abductive construct has a complex and…
Resolution lies at the foundation of both logic programming and type class context reduction in functional languages. Terminating derivations by resolution have well-defined inductive meaning, whereas some non-terminating derivations can be…
An inductive inference system for proving validity of formulas in the initial algebra $T_{\mathcal{E}}$ of an order-sorted equational theory $\mathcal{E}$ is presented. It has 20 inference rules, but only 9 of them require user interaction;…
Affordances, a foundational concept in human-computer interaction and design, have traditionally been explained by direct-perception theories, which assume that individuals perceive action possibilities directly from the environment.…