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Abstract There is no single canonical polynomial-time version of the Axiom of Choice (AC); several statements of AC that are equivalent in Zermelo-Fraenkel (ZF) set theory are already inequivalent from a constructive point of view, and are similarly inequivalent from a complexity-theoretic point of view. In this paper we show that many classical formulations of AC, when restricted to polynomial time in natural ways, are equivalent to standard complexity-theoretic hypotheses, including several that were of interest to Selman. This provides a unified view of these hypotheses, and we hope provides additional motivation for studying some of the lesser-known hypotheses that appear here. Additionally, because several classical forms of AC are formulated in terms of cardinals, we develop a theory of polynomial-time cardinality. Nerode & Remmel (Contemp. Math.106, 1990 and Springer Lec. Notes Math. 1432, 1990) developed a related theory, but restricted to unary sets. Downey (Math. Reviews MR1071525) suggested that such a theory over larger alphabets could have interesting connections to more standard complexity questions, and we illustrate some of those connections here. The connections between AC, cardinality, and complexity questions also allow us to highlight some of Selman’s work. We hope this paper is more of a beginning than an end, introducing new concepts and raising many new questions, ripe for further research. 

The Cohn-Umans (FOCS '03) group-theoretic framework for matrix multiplication produces fast matrix multiplication algorithms from three subsets of a finite group G satisfying a simple combinatorial condition (the Triple Product Property). The complexity of such an algorithm then depends on the representation theory of G. In this paper we extend the group-theoretic framework to the setting of infinite groups. In particular, this allows us to obtain constructions in Lie groups, with favorable parameters, that are provably impossible in finite groups of Lie type (Blasiak, Cohn, Grochow, Pratt, and Umans, ITCS '23). Previously the Lie group setting was investigated purely as an analogue of the finite group case; a key contribution in this paper is a fully developed framework for obtaining bona fide matrix multiplication algorithms directly from Lie group constructions. As part of this framework, we introduce "separating functions" as a necessary new design component, and show that when the underlying group is G = GL_n, these functions are polynomials with their degree being the key parameter. In particular, we show that a construction with "half-dimensional" subgroups and optimal degree would imply ω = 2. We then build up machinery that reduces the problem of constructing optimal-degree separating polynomials to the problem of constructing a single polynomial (and a corresponding set of group elements) in a ring of invariant polynomials determined by two out of the three subgroups that satisfy the Triple Product Property. This machinery combines border rank with the Lie algebras associated with the Lie subgroups in a critical way. We give several constructions illustrating the main components of the new framework, culminating in a construction in a special unitary group that achieves separating polynomials of optimal degree, meeting one of the key challenges. The subgroups in this construction have dimension approaching half the ambient dimension, but (just barely) too slowly. We argue that features of the classical Lie groups make it unlikely that constructions in these particular groups could produce nontrivial bounds on ω unless they prove ω = 2. One way to get ω = 2 via our new framework would be to lift our existing construction from the special unitary group to GL_n, and improve the dimension of the subgroups from (dim G)/2 - Θ(n) to (dim G)/2 - o(n). 

The relationship between the thermodynamic and computational properties of physical systems has been a major theoretical interest since at least the 19th century. It has also become of increasing practical importance over the last half-century as the energetic cost of digital devices has exploded. Importantly, real-world computers obey multiple physical constraints on how they work, which affects their thermodynamic properties. Moreover, many of these constraints apply to both naturally occurring computers, like brains or Eukaryotic cells, and digital systems. Most obviously, all such systems must finish their computation quickly, using as few degrees of freedom as possible. This means that they operate far from thermal equilibrium. Furthermore, many computers, both digital and biological, are modular, hierarchical systems with strong constraints on the connectivity among their subsystems. Yet another example is that to simplify their design, digital computers are required to be periodic processes governed by a global clock. None of these constraints were considered in 20th-century analyses of the thermodynamics of computation. The new field of stochastic thermodynamics provides formal tools for analyzing systems subject to all of these constraints. We argue here that these tools may help us understand at a far deeper level just how the fundamental thermodynamic properties of physical systems are related to the computation they perform. 

We study the complexity of isomorphism problems for d-way arrays, or tensors, under natural actions by classical groups such as orthogonal, unitary, and symplectic groups. These problems arise naturally in statistical data analysis and quantum information. We study two types of complexity-theoretic questions. First, for a fixed action type (isomorphism, conjugacy, etc.), we relate the complexity of the isomorphism problem over a classical group to that over the general linear group. Second, for a fixed group type (orthogonal, unitary, or symplectic), we compare the complexity of the isomorphism problems for different actions. Our main results are as follows. First, for orthogonal and symplectic groups acting on 3-way arrays, the isomorphism problems reduce to the corresponding problems over the general linear group. Second, for orthogonal and unitary groups, the isomorphism problems of five natural actions on 3-way arrays are polynomial-time equivalent, and the d-tensor isomorphism problem reduces to the 3-tensor isomorphism problem for any fixed d >= 3. For unitary groups, the preceding result implies that LOCC classification of tripartite quantum states is at least as difficult as LOCC classification of d-partite quantum states for any d. Lastly, we also show that the graph isomorphism problem reduces to the tensor isomorphism problem over orthogonal and unitary groups. 

In this paper, we show that computing canonical labelings of graphs of bounded rank-width is in TC². Our approach builds on the framework of Köbler & Verbitsky (CSR 2008), who established the analogous result for graphs of bounded treewidth. Here, we use the framework of Grohe & Neuen (ACM Trans. Comput. Log., 2023) to enumerate separators via split-pairs and flip functions. In order to control the depth of our circuit, we leverage the fact that any graph of rank-width k admits a rank decomposition of width ≤ 2k and height O(log n) (Courcelle & Kanté, WG 2007). This allows us to utilize an idea from Wagner (CSR 2011) of tracking the depth of the recursion in our computation. Furthermore, after splitting the graph into connected components, it is necessary to decide isomorphism of said components in TC¹. To this end, we extend the work of Grohe & Neuen (ibid.) to show that the (6k+3)-dimensional Weisfeiler-Leman (WL) algorithm can identify graphs of rank-width k using only O(log n) rounds. As a consequence, we obtain that graphs of bounded rank-width are identified by FO + C formulas with 6k+4 variables and quantifier depth O(log n). Prior to this paper, isomorphism testing for graphs of bounded rank-width was not known to be in NC. 

Genome rearrangement is a common model for molecular evolution. In this paper, we consider the Pairwise Rearrangement problem, which takes as input two genomes and asks for the number of minimum-length sequences of permissible operations transforming the first genome into the second. In the Single Cut-and-Join model (Bergeron, Medvedev, & Stoye, J. Comput. Biol. 2010), Pairwise Rearrangement is #P-complete (Bailey, et. al., COCOON 2023), which implies that exact sampling is intractable. In order to cope with this intractability, we investigate the parameterized complexity of this problem. We exhibit a fixed-parameter tractable algorithm with respect to the number of components in the adjacency graph that are not cycles of length 2 or paths of length 1. As a consequence, we obtain that Pairwise Rearrangement in the Single Cut-and-Join model is fixed-parameter tractable by distance. Our results suggest that the number of nontrivial components in the adjacency graph serves as the key obstacle for efficient sampling. 

In this paper, we explore the descriptive complexity theory of finite groups by examining the power of the second Ehrenfeucht-Fraisse bijective pebble game in Hella's (Ann. Pure Appl. Log., 1989) hierarchy. This is a Spoiler-Duplicator game in which Spoiler can place up to two pebbles each round. While it trivially solves graph isomorphism, it may be nontrivial for finite groups, and other ternary relational structures. We first provide a novel generalization of Weisfeiler-Leman (WL) coloring, which we call 2-ary WL. We then show that the 2-ary WL is equivalent to the second Ehrenfeucht-Fraisse bijective pebble game in Hella's hierarchy. Our main result is that, in the pebble game characterization, only O(1) pebbles and O(1) rounds are sufficient to identify all groups without Abelian normal subgroups (a class of groups for which isomorphism testing is known to be in P; Babai, Codenotti, & Qiao, ICALP 2012). In particular, we show that within the first few rounds, Spoiler can force Duplicator to select an isomorphism between two such groups at each subsequent round. By Hella's results (ibid.), this is equivalent to saying that these groups are identified by formulas in first-order logic with generalized 2-ary quantifiers, using only O(1) variables and O(1) quantifier depth.