Complete Non-Orders and Fixed Points

In this paper, we develop an Isabelle/HOL library of order-theoretic concepts, such as various completeness conditions and fixed-point theorems. We keep our formalization as general as possible: we reprove several well-known results about complete orders, often without any property of ordering, thus complete non-orders. In particular, we generalize the Knaster - Tarski theorem so that we ensure the existence of a quasi-fixed point of monotone maps over complete non-orders, and show that the set of quasi-fixed points is complete under a mild condition - attractivity - which is implied by either antisymmetry or transitivity. This result generalizes and strengthens a result by Stauti and Maaden. Finally, we recover Kleene\u27s fixed-point theorem for omega-complete non-orders, again using attractivity to prove that Kleene\u27s fixed points are least quasi-fixed points

SO(4) multicriticality of two-dimensional Dirac fermions

We study quantum multicritical behavior in a (2+1)-dimensional Gross-Neveu-Yukawa field theory with eight-component Dirac fermions coupled to two triplets of order parameters that act as Dirac masses, and transform as $(1,0) + (0,1)$ representation under the SO(4)$\simeq$SO(3)$\times$SO(3) symmetry group. This field theory is relevant to spin-1/2 fermions on honeycomb or $\pi$-flux lattices, for example, near the transition points between an $s$-wave superconductor and a charge-density wave, on one side, and N\'eel order, on the other. Two triplets of such order parameters always allow for a common pair of two other order parameters that would complete them to the maximal set of compatible (anticommuting) orders of five. We first derive a unitary transformation in the Nambu (particle-hole) space which maps any two such triplets, possibly containing some superconducting orders, onto purely insulating order parameters. This allows one to consider a universal SO(4) Gross-Neveu-Yukawa description of the multicriticality without any Nambu doubling. We then proceed to derive the renormalization-group flow of the coupling constants at one-loop order in $4-\epsilon$ space-time dimensions, allowing also a more general set of order parameters transforming under SO($n_a$)$\times$SO($n_b$). While for $n_a=n_b > 2$ in the bosonic sector and with fermions decoupled there is a stable fixed point of the flow, the Yukawa coupling to fermions quickly leads to its elimination by a generic fixed-point collision in the relevant range of fermion flavor numbers $N_f$. This suggests the replacement of the critical behavior by a runaway flow in the physical case $n_a=n_b=3$. The structure of the RG flow at $n_a\neq n_b$ is also discussed, and some non-perturbative arguments in favor of the stability of the decoupled critical point when $n_a=3$ and $n_b=1$ in $D=2+1$ are provided.Comment: 13 pages, 4 figure

Inversion, Iteration, and the Art of Dual Wielding

The humble $\dagger$ ("dagger") is used to denote two different operations in category theory: Taking the adjoint of a morphism (in dagger categories) and finding the least fixed point of a functional (in categories enriched in domains). While these two operations are usually considered separately from one another, the emergence of reversible notions of computation shows the need to consider how the two ought to interact. In the present paper, we wield both of these daggers at once and consider dagger categories enriched in domains. We develop a notion of a monotone dagger structure as a dagger structure that is well behaved with respect to the enrichment, and show that such a structure leads to pleasant inversion properties of the fixed points that arise as a result. Notably, such a structure guarantees the existence of fixed point adjoints, which we show are intimately related to the conjugates arising from a canonical involutive monoidal structure in the enrichment. Finally, we relate the results to applications in the design and semantics of reversible programming languages.Comment: Accepted for RC 201

Topological aspects of poset spaces

We study two classes of spaces whose points are filters on partially ordered sets. Points in MF spaces are maximal filters, while points in UF spaces are unbounded filters. We give a thorough account of the topological properties of these spaces. We obtain a complete characterization of the class of countably based MF spaces: they are precisely the second-countable T_1 spaces with the strong Choquet property. We apply this characterization to domain theory to characterize the class of second-countable spaces with a domain representation.Comment: 29 pages. To be published in the Michigan Mathematical Journa

Discovering and quantifying nontrivial fixed points in multi-field models

We use the functional renormalization group and the $\epsilon$-expansion concertedly to explore multicritical universality classes for coupled $\bigoplus_i O(N_i)$ vector-field models in three Euclidean dimensions. Exploiting the complementary strengths of these two methods we show how to make progress in theories with large numbers of interactions, and a large number of possible symmetry-breaking patterns. For the three- and four-field models we find a new fixed point that arises from the mutual interaction between different field sectors, and we establish the absence of infrared-stable fixed point solutions for the regime of small $N_i$. Moreover, we explore these systems as toy models for theories that are both asymptotically safe and infrared complete. In particular, we show that these models exhibit complete renormalization group trajectories that begin and end at nontrivial fixed points.Comment: 10 pages, 6 figures; minor changes, as published in EPJ

Towards a Dichotomy for the Possible Winner Problem in Elections Based on Scoring Rules

To make a joint decision, agents (or voters) are often required to provide their preferences as linear orders. To determine a winner, the given linear orders can be aggregated according to a voting protocol. However, in realistic settings, the voters may often only provide partial orders. This directly leads to the Possible Winner problem that asks, given a set of partial votes, whether a distinguished candidate can still become a winner. In this work, we consider the computational complexity of Possible Winner for the broad class of voting protocols defined by scoring rules. A scoring rule provides a score value for every position which a candidate can have in a linear order. Prominent examples include plurality, k-approval, and Borda. Generalizing previous NP-hardness results for some special cases, we settle the computational complexity for all but one scoring rule. More precisely, for an unbounded number of candidates and unweighted voters, we show that Possible Winner is NP-complete for all pure scoring rules except plurality, veto, and the scoring rule defined by the scoring vector (2,1,...,1,0), while it is solvable in polynomial time for plurality and veto.Comment: minor changes and updates; accepted for publication in JCSS, online version available

Fixed-Parameter Algorithms for Computing Kemeny Scores - Theory and Practice

The central problem in this work is to compute a ranking of a set of elements which is "closest to" a given set of input rankings of the elements. We define "closest to" in an established way as having the minimum sum of Kendall-Tau distances to each input ranking. Unfortunately, the resulting problem Kemeny consensus is NP-hard for instances with n input rankings, n being an even integer greater than three. Nevertheless this problem plays a central role in many rank aggregation problems. It was shown that one can compute the corresponding Kemeny consensus list in f(k) + poly(n) time, being f(k) a computable function in one of the parameters "score of the consensus", "maximum distance between two input rankings", "number of candidates" and "average pairwise Kendall-Tau distance" and poly(n) a polynomial in the input size. This work will demonstrate the practical usefulness of the corresponding algorithms by applying them to randomly generated and several real-world data. Thus, we show that these fixed-parameter algorithms are not only of theoretical interest. In a more theoretical part of this work we will develop an improved fixed-parameter algorithm for the parameter "score of the consensus" having a better upper bound for the running time than previous algorithms.Comment: Studienarbei

Group orderings, dynamics, and rigidity

Let G be a countable group. We show there is a topological relationship between the space CO(G) of circular orders on G and the moduli space of actions of G on the circle; as well as an analogous relationship for spaces of left orders and actions on the line. In particular, we give a complete characterization of isolated left and circular orders in terms of strong rigidity of their induced actions of G on $S^1$ and R. As an application of our techniques, we give an explicit construction of infinitely many nonconjugate isolated points in the spaces CO(F_{2n}) of circular orders on free groups disproving a conjecture from Baik--Samperton, and infinitely many nonconjugate isolated points in the space of left orders on the pure braid group P_3, answering a question of Navas. We also give a detailed analysis of circular orders on free groups, characterizing isolated orders