Uniformity, Universality, and Computability Theory

We prove a number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalb\'an, Reimann and Slaman. This notion is a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual notion. With this additional uniformity hypothesis, we can answer many questions concerning how countable groups, probability measures, the subset relation, and increasing unions interact with universality. For many natural classes of countable Borel equivalence relations, we can also classify exactly which are uniformly universal. We also show the existence of refinements of Martin's ultrafilter on Turing invariant Borel sets to the invariant Borel sets of equivalence relations that are much finer than Turing equivalence. For example, we construct such an ultrafilter for the orbit equivalence relation of the shift action of the free group on countably many generators. These ultrafilters imply a number of structural properties for these equivalence relations.Comment: 61 Page

Classifying word problems of finitely generated algebras via computable reducibility

We contribute to a recent research program which aims at revisiting the study of the complexity of word problems, a major area of research in combinatorial algebra, through the lens of the theory of computably enumerable equivalence relations (ceers), which has considerably grown in recent times. To pursue our analysis, we rely on the most popular way of assessing the complexity of ceers, that is via computable reducibility on equivalence relations, and its corresponding degree structure (the c-degrees). On the negative side, building on previous work of Kasymov and Khoussainov, we individuate a collection of c-degrees of ceers which cannot be realized by the word problem of any finitely generated algebra of finite type. On the positive side, we show that word problems of finitely generated semigroups realize a collection of c-degrees which embeds rich structures and is large in several reasonable ways

The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.Comment: Association for Symbolic Logi

Computability Theory

Computability and computable enumerability are two of the fundamental notions of mathematics. Interest in effectiveness is already apparent in the famous Hilbert problems, in particular the second and tenth, and in early 20th century work of Dehn, initiating the study of word problems in group theory. The last decade has seen both completely new subareas develop as well as remarkable growth in two-way interactions between classical computability theory and areas of applications. There is also a great deal of work on algorithmic randomness, reverse mathematics, computable analysis, and in computable structure theory/computable model theory. The goal of this workshop is to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work

Benchmarking and Explaining Large Language Model-based Code Generation: A Causality-Centric Approach

While code generation has been widely used in various software development scenarios, the quality of the generated code is not guaranteed. This has been a particular concern in the era of large language models (LLMs)- based code generation, where LLMs, deemed a complex and powerful black-box model, is instructed by a high-level natural language specification, namely a prompt, to generate code. Nevertheless, effectively evaluating and explaining the code generation capability of LLMs is inherently challenging, given the complexity of LLMs and the lack of transparency. Inspired by the recent progress in causality analysis and its application in software engineering, this paper launches a causality analysis-based approach to systematically analyze the causal relations between the LLM input prompts and the generated code. To handle various technical challenges in this study, we first propose a novel causal graph-based representation of the prompt and the generated code, which is established over the fine-grained, human-understandable concepts in the input prompts. The formed causal graph is then used to identify the causal relations between the prompt and the derived code. We illustrate the insights that our framework can provide by studying over 3 popular LLMs with over 12 prompt adjustment strategies. The results of these studies illustrate the potential of our technique to provide insights into LLM effectiveness, and aid end-users in understanding predictions. Additionally, we demonstrate that our approach provides actionable insights to improve the quality of the LLM-generated code by properly calibrating the prompt