The Complexity of Kings

A king in a directed graph is a node from which each node in the graph can be reached via paths of length at most two. There is a broad literature on tournaments (completely oriented digraphs), and it has been known for more than half a century that all tournaments have at least one king [Lan53]. Recently, kings have proven useful in theoretical computer science, in particular in the study of the complexity of the semifeasible sets [HNP98,HT05] and in the study of the complexity of reachability problems [Tan01,NT02]. In this paper, we study the complexity of recognizing kings. For each succinctly specified family of tournaments, the king problem is known to belong to $\Pi_2^p$ [HOZZ]. We prove that this bound is optimal: We construct a succinctly specified tournament family whose king problem is $\Pi_2^p$-complete. It follows easily from our proof approach that the problem of testing kingship in succinctly specified graphs (which need not be tournaments) is $\Pi_2^p$-complete. We also obtain $\Pi_2^p$-completeness results for k-kings in succinctly specified j-partite tournaments, $k,j \geq 2$, and we generalize our main construction to show that $\Pi_2^p$-completeness holds for testing k-kingship in succinctly specified families of tournaments for all $k \geq 2$

Improved Reinforcement Learning with Curriculum

Humans tend to learn complex abstract concepts faster if examples are presented in a structured manner. For instance, when learning how to play a board game, usually one of the first concepts learned is how the game ends, i.e. the actions that lead to a terminal state (win, lose or draw). The advantage of learning end-games first is that once the actions which lead to a terminal state are understood, it becomes possible to incrementally learn the consequences of actions that are further away from a terminal state - we call this an end-game-first curriculum. Currently the state-of-the-art machine learning player for general board games, AlphaZero by Google DeepMind, does not employ a structured training curriculum; instead learning from the entire game at all times. By employing an end-game-first training curriculum to train an AlphaZero inspired player, we empirically show that the rate of learning of an artificial player can be improved during the early stages of training when compared to a player not using a training curriculum.Comment: Draft prior to submission to IEEE Trans on Games. Changed paper slightl

States and Social Complexity: The Indus Valley (Harappan) Civilisation

This article explores 'statehood' and argues Indus Valley (Harappan) Civilisation was not a state like contemporary Sumer and Egypt were, despite being equally complex - hence calling for revision of the unilineal anthropological model culminating in the state

Cultivating Systems Leadership in Cross-Sector Partnerships: Lessons from the Linked Learning Regional Hubs of Excellence

Investments in the social sector have become increasingly complex, with many foundations shifting from supports for single organizations toward more systemic strategies focused on improving outcomes for entire communities. As a result, the field has become awash in regional, or place-based, investments that rely on cross-sector partnerships and networks to drive change. These efforts require coordination among stakeholders across all levels of the practice and policy continuum – from direct service providers, to nonprofit intermediaries, funders, advocacy organizations, and policymakers.It is in this context, and in the spirit of continuous learning, that The James Irvine Foundation's Linked Learning Regional Hubs of Excellence investment serves as a systems change experiment, offering insights and critical lessons that can inform others undertaking similar work. The aim of this Issue Brief – authored by Equal Measure and Harder+Company – is to contribute to field dialogue and learning about the role of leadership in complex systems change strategies, particularly those focused on producing equitable impacts in college and career readiness.Equal Measure and Harder+Company serve as the evaluators of The James Irvine Foundation's Linked Learning Regional Hubs of Excellence. They work in partnership with Jobs for the Future, the intermediary and technical assistance provider for this initiative

A Hebrew inscription on the Polish coronation sword

The present article deals with the problem of the presumably Hebrew inscription on the coronation sword of Polish kings, known as the “Szczerbiec”. Our objective is not to recall or reconstruct the complexity of its history. It is rather to expose some elements of the sword’s bizarre haps and mishaps, as far as they may cast some light on the emergence of the inscription and its meaning. The main purpose of our attempt is to present one possible consistent reading of the inscription, taking into account some historical and linguistic data. Our proposal draws from some previous reconstructions, both past and current. Finally, however, we hope to offer an original interpretation of the inscriptio

Two-variable Logic with Counting and a Linear Order

We study the finite satisfiability problem for the two-variable fragment of first-order logic extended with counting quantifiers (C2) and interpreted over linearly ordered structures. We show that the problem is undecidable in the case of two linear orders (in the presence of two other binary symbols). In the case of one linear order it is NEXPTIME-complete, even in the presence of the successor relation. Surprisingly, the complexity of the problem explodes when we add one binary symbol more: C2 with one linear order and in the presence of other binary predicate symbols is equivalent, under elementary reductions, to the emptiness problem for multicounter automata