The decision problem of modal product logics with a diagonal, and faulty counter machines

In the propositional modal (and algebraic) treatment of two-variable first-order logic equality is modelled by a `diagonal' constant, interpreted in square products of universal frames as the identity (also known as the `diagonal') relation. Here we study the decision problem of products of two arbitrary modal logics equipped with such a diagonal. As the presence or absence of equality in two-variable first-order logic does not influence the complexity of its satisfiability problem, one might expect that adding a diagonal to product logics in general is similarly harmless. We show that this is far from being the case, and there can be quite a big jump in complexity, even from decidable to the highly undecidable. Our undecidable logics can also be viewed as new fragments of first- order logic where adding equality changes a decidable fragment to undecidable. We prove our results by a novel application of counter machine problems. While our formalism apparently cannot force reliable counter machine computations directly, the presence of a unique diagonal in the models makes it possible to encode both lossy and insertion-error computations, for the same sequence of instructions. We show that, given such a pair of faulty computations, it is then possible to reconstruct a reliable run from them

Bespoke, Tailored, and Off-the-Rack Bankruptcy: A Response to Professor Coordes\u27s \u27Bespoke Bankruptcy\u27

Toward the end of every semester that I teach bankruptcy, I let my students vote on which “non-traditional” insolvency regimes they would like to study, including municipal bankruptcy, sovereign bankruptcy, and financial institutions. What I am really trying to do is convey to the students that the default procedures and substantive rules in Chapters 7 and 11 of the U.S. Bankruptcy Code do not apply to all types of enterprises

State Bans on Debtors\u27 Prisons and Criminal Justice Debt

Since the 1990s, and increasingly in the wake of the Great Recession, many municipalities, forced to operate under tight budgetary constraints, have turned to the criminal justice system as an untapped revenue stream. Raising the specter of the debtors\u27 prisons once prevalent in the United States, Imprisonment for failure to pay debts owed to the state has provoked growing concern over the year

Bankruptcy & the Benefit Corporation

As pressure grows for money-making businesses to prioritize social responsibility, the benefit corporation - a recent innovation in corporate governance - promises to require the directors of socially minded businesses to balance public benefit with shareholder interests. But will that promise survive the crucible of financial distress? While most discussions of the benefit corporation give only passing treatment to insolvency (or ignore it altogether), this Article provides the first complete analysis of how bankruptcy principles would apply to benefit corporations, informed by the practical context of out-of-court workouts and negotiations that take place in the shadow of the bankruptcy laws. After analyzing three normative models, including an innovative application of the channeling function of law, this Article answers that the benefit corporation\u27s key innovations should persist in bankruptcy. But with the reticulated provisions of creditor-debtor law and the Bankruptcy Code, the Article warns that the application of that principle is complicated and provides a detailed map of some of the major considerations - and pitfalls

The New American Debtors\u27 Prisons

State by state, Americans abolished imprisonment for debt in the first half of the nineteenth century. In forty-one states, the abolition of debtors\u27 prisons eventually took the form of constitutional bans. But debtors\u27 prisons are back, in the form of imprisonment for nonpayment of criminal fines, fees, and costs. While the new debtors\u27 prisons are not historically or doctrinally continuous with the old, some aspects of them offend the same pragmatic and moral principles that compelled the abolition of the old debtors\u27 prisons. Indeed, the same constitutional texts that abolished the old debtors\u27 prisons constitute checks on the new today. As the criminal law literature grapples with debtors\u27 prisons through more traditional doctrinal avenues, this Article engages with the metaphor head-on and asks how the old bans on debtors\u27 prisons should be interpreted for a new era of mass incarceration.

Pausing for reflection: Re-evaluating Bach's use of the fermata

The fermata sign placed above a note is nowadays used more or less exclusively to signify a pause of indeterminate length on the affected note. Within Bach's works, the fermata makes appearances in connection with chorales in both the vocal works and chorale preludes for organ. The fermata also occurs in secular pieces such as the suites and Well-Tempered Clavier. Various viewpoints as to the meaning of the fermata have developed, including calls to ignore them as pauses altogether, to signify breathing points, to hold the affected note for twice its length followed by a rest and so on. In the organ chorale preludes, there is sufficient movement in other parts for Koopman (2003) and others to assume that treating the fermata symbol as a pause would be impossible. Others have also claimed that other I?"'- and 18*

"A Barber-Surgeon’s > Instrument Case: Seeing the Iconography of Thomas Becket through a > Netherlandish Lens"

The triple anniversary in 2020 of Thomas Becket’s birth, death and translation has been an occasion to review and revisit many of the artefacts associated with the saint and his cult in England and across Europe. Many of these are items directly associated with his veneration in churches or in private devotions, but one object which served in neither capacity is an instrument case currently in the collection of the Worshipful Company of Barbers in London. This unusual object has been studied for its fine silver work, and possible royal associations, but little academic attention has so far been paid to the some of the iconography, particularly that of the scene of the murder of Thomas Becket depicted on the back of the box, the side to be worn against the body. In this article, we show how seemingly unusual elements in the iconography draw on particularly Flemish representations of Becket’s murder that, to date, have received little attention in Anglophone scholarship. From this, we discuss this scene and its significance in understanding the role the iconography may have been intended to serve, and the interplay between the decorative schema and what the surgeon thought about his own role with regard to the use of the case and its tools

MUL-Tree Pruning for Consistency and Compatibility

A multi-labelled tree (or MUL-tree) is a rooted tree leaf-labelled by a set of labels, where each label may appear more than once in the tree. We consider the MUL-tree Set Pruning for Consistency problem (MULSETPC), which takes as input a set of MUL-trees and asks whether there exists a perfect pruning of each MUL-tree that results in a consistent set of single-labelled trees. MULSETPC was proven to be NP-complete by Gascon et al. when the MUL-trees are binary, each leaf label is used at most three times, and the number of MUL-trees is unbounded. To determine the computational complexity of the problem when the number of MUL-trees is constant was left as an open problem. Here, we resolve this question by proving a much stronger result, namely that MULSETPC is NP-complete even when there are only two MUL-trees, every leaf label is used at most twice, and every MUL-tree is either binary or has constant height. Furthermore, we introduce an extension of MULSETPC that we call MULSETPComp, which replaces the notion of consistency with compatibility, and prove that MULSETPComp is NP-complete even when there are only two MUL-trees, every leaf label is used at most thrice, and every MUL-tree has constant height. Finally, we present a polynomial-time algorithm for instances of MULSETPC with a constant number of binary MUL-trees, in the special case where every leaf label occurs exactly once in at least one MUL-tree