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With the recent success of embeddings in natural language processing, research has been conducted into applying similar methods to code analysis. Most works attempt to process the code directly or use a syntactic tree representation,…
computable functions are defined by abstract finite deterministic algorithms on many-sorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable…
We propose a framework for reasoning about programs that manipulate coinductive data as well as inductive data. Our approach is based on using equational programs, which support a seamless combination of computation and reasoning, and using…
We present abstract acceleration techniques for computing loop invariants for numerical programs with linear assignments and conditionals. Whereas abstract interpretation techniques typically over-approximate the set of reachable states…
Explanations of cognitive behavior often appeal to computations over representations. What does it take for a system to implement a given computation over suitable representational vehicles within that system? We argue that the language of…
In-context learning (ICL) has garnered significant attention for its ability to grasp functions/tasks from demonstrations. Recent studies suggest the presence of a latent task/function vector in LLMs during ICL. Merullo et al. (2024) showed…
Unifying theories distil common features of programming languages and design methods by means of algebraic operators and their laws. Several practical concerns --- e.g., improvement of a program, conformance of code with design, correctness…
Semantic interpretability in Reinforcement Learning (RL) enables transparency and verifiability of decision-making. Achieving semantic interpretability in reinforcement learning requires (1) a feature space composed of human-understandable…
The purpose of this paper is to clarify the relationship between various conditions implying essential undecidability: our main result is that there exists a theory $T$ in which all partially recursive functions are representable, yet $T$…
Recent work by Hermanns et al. and Kattenbelt et al. has extended counterexample-guided abstraction refinement (CEGAR) to probabilistic programs. These approaches are limited to predicate abstraction. We present a novel technique, based on…
We consider sets/relations/computations defined by *Elementary Inference Systems* I, which are obtained from Smullyan's *elementary formal systems* using Gentzen's notation for inference rules, and proof trees for atoms P(t_1,...,t_n),…
To solve hard problems, AI relies on a variety of disciplines such as logic, probabilistic reasoning, machine learning and mathematical programming. Although it is widely accepted that solving real-world problems requires an integration…
Axiomatic type theory is a dependent type theory without computation rules. The term equality judgements that usually characterise these rules are replaced by computation axioms, i.e., additional term judgements that are typed by identity…
We present a new manifestation of G\"odel's second incompleteness theorem and discuss its foundational significance, in particular with respect to Hilbert's program. Specifically, we consider a proper extension of Peano arithmetic…
We advocate a declarative approach to proving properties of logic programs. Total correctness can be separated into correctness, completeness and clean termination; the latter includes non-floundering. Only clean termination depends on the…
Developing suitable formal semantics can be of great help in the understanding, design and implementation of a programming language, and act as a guide for software development tools like analyzers or partial evaluators. In this sense, full…
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…
The Abstraction and Reasoning Corpus (ARC) is a general artificial intelligence benchmark that is currently unsolvable by any Machine Learning method, including Large Language Models (LLMs). It demands strong generalization and reasoning…
Enabling reinforcement learning (RL) agents to leverage a knowledge base while learning from experience promises to advance RL in knowledge intensive domains. However, it has proven difficult to leverage knowledge that is not manually…
We investigate partial functions and computability theory from within a constructive, univalent type theory. The focus is on placing computability into a larger mathematical context, rather than on a complete development of computability…