Combining decision procedures for the reals

We address the general problem of determining the validity of boolean combinations of equalities and inequalities between real-valued expressions. In particular, we consider methods of establishing such assertions using only restricted forms of distributivity. At the same time, we explore ways in which "local" decision or heuristic procedures for fragments of the theory of the reals can be amalgamated into global ones. Let Tadd[Q] be the first-order theory of the real numbers in the language of ordered groups, with negation, a constant 1, and function symbols for multiplication by rational constants. Let Tmult[Q] be the analogous theory for the multiplicative structure, and let T[Q] be the union of the two. We show that although T[Q] is undecidable, the universal fragment of T[Q] is decidable. We also show that terms of T[Q]can fruitfully be put in a normal form. We prove analogous results for theories in which Q is replaced, more generally, by suitable subfields F of the reals. Finally, we consider practical methods of establishing quantifier-free validities that approximate our (impractical) decidability results.Comment: Will appear in Logical Methods in Computer Scienc

Learning the Right Lessons from Iraq

Foreign policy experts and policy analysts are misreading the lessons of Iraq. The emerging conventional wisdom holds that success could have been achieved in Iraq with more troops, more cooperation among U.S. government agencies, and better counterinsurgency doctrine. To analysts who share these views, Iraq is not an example of what not to do but of how not to do it. Their policy proposals aim to reform the national security bureaucracy so that we will get it right the next time. The near-consensus view is wrong and dangerous. What Iraq demonstrates is a need for a new national security strategy, not better tactics and tools to serve the current one. By insisting that Iraq was ours to remake were it not for the Bush administration's mismanagement, we ignore the limits on our power that the war exposes and in the process risk repeating our mistake. The popular contention that the Bush administration's failures and errors in judgment can be attributed to poor planning is also false. There was ample planning for the war, but it conflicted with the Bush administration's expectations. To the extent that planning failed, therefore, the lesson to draw is not that the United States national security establishment needs better planning, but that it needs better leaders. That problem is solved by elections, not bureaucratic tinkering. The military gives us the power to conquer foreign countries, but not the power to run them. Because there are few good reasons to take on missions meant to resuscitate failed governments, terrorism notwithstanding, the most important lesson from the war in Iraq should be a newfound appreciation for the limits of our power

Constitutional Law-Civil Rights-Solitary Confinement of Prisoner\u27s Based on Religious Belief

Plaintiff prisoner brought an action in a federal district court under the Civil Rights Act to enjoin the defendant, a New York state prison warden, from further subjecting him to solitary confinement because of his religious beliefs. The district court refused to take jurisdiction on the ground that solitary confinement involved state prison discipline which was reviewable only in state courts. On appeal, held, reversed, one judge dissenting. A complaint by a prisoner against a state prison official which charges violation of a preferred freedom by religious persecution states a claim under the Civil Rights Act which the district court must entertain. Pierce v. LaVallee, 293 F.2d 233 (2d Cir. 1961)

Private Insurance as a Solution to the Driver-Guest Dilemm

The duty of the driver of an automobile to his nonpaying passenger, and liability arising from the breach of that duty, has long presented a troublesome area of litigation for the courts and the parties involved. Application of standards unsuited for the peculiar risks of automotive transportation has produced inadequate compensation in some cases and excessive recoveries in others. Meanwhile, trial calendars are overcrowded with personal injury litigation, and insurance companies must bear the awards of sympathetic juries and those resulting from collusion between passenger and driver. The over-all expense of this method of determination of liability, far too little of which actually goes to compensate the injured plaintiff, is passed on to the public in the form of higher insurance premiums. It is the purpose of this comment to review the approaches heretofore used in defining the driver\u27s duty, and to offer a solution which, by eliminating the inherent difficulties in these approaches, might afford a fairer and more workable means of compensating the injured guest

Electing the Supreme Court

Symposium: Congressional Power in the Shadow of the Rehnquist Court: Strategies for the Future held at Indiana University Law School, February 1-2, 2002

Computational complexity of real functions

AbstractRecursive analysis, the theory of computation of functions on real numbers, has been studied from various aspects. We investigate the computational complexity of real functions using the methods of recursive function theory. Partial recursive real functions are defined and their domains are characterized as the recursively open sets. We define the time complexity of recursive real continuous functions and show that the time complexity and the modulus of uniform continuity of a function are closely related. We study the complexity of the roots and the differentiability of polynomial time computable real functions. In particular, a polynomial time computable real function may have a root of arbitrarily high complexity and may be nowhere differentiable. The concepts of the space complexity and nondeterministic computation are used to study the complexity of the integrals and the maximum values of real functions. These problems are shown to be related to the “P=?NP” and the “P=?PSPACE” questions