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This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NP-complete sets look like? To what extent are the properties of particular NP-complete sets, e.g., SAT, shared by all NP-complete sets? If there are are structural differences between NP-complete sets, what are they and what explains the differences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need first to formalize NP-completeness. There are a number of competing definitions of NP-completeness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of m-reduction, also known as polynomial-time manyone reduction and Karp reduction. A set A is m-reducible to B if and only if there is a (total) polynomial-time computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1

This work presents a unied pattern-based epistemological framework, called a Knowledge Manifold, for the description and extraction of knowledge from information. Within this framework it also presents the metaphor of the Garden Of Knowledge as a constructive example. Any type of KM is defined in terms of its objective calibration protocols - procedures that are implemented on top of the participating subjective knowledge-patches. They are the procedures of agreement and obedience that characterize the coherence of any type of interaction, and which are used here in order to formalize the concept of participator consciousness in terms of the inverse-direct limit duality of Category Theory.

Abstract. Proofs of termination are essential for establishing the correct behavior of computing systems. There are various ways of establishing termination, but the most general involves the use of ordinals. An example of a theorem proving system in which ordinals are used to prove termination is ACL2. In ACL2, every function defined must be shown to terminate using the ordinals up to ɛ0. We use a compact notation for the ordinals up to ɛ0 (exponentially more succinct than the one used by ACL2) and define efficient algorithms for ordinal addition, subtraction, multiplication, and exponentiation. In this paper we describe our notation and algorithms, prove their correctness, and analyze their complexity. 1

Abstract. Ordinals form the basis for termination proofs in ACL2. Currently, ACL2 uses a rather inefficient representation for the ordinals up to ɛ0 and provides limited support for reasoning about them. We present algorithms for ordinal arithmetic on an exponentially more compact representation than the one used by ACL2. The algorithms have been implemented and numerous properties of the arithmetic operators have been mechanically verified, thereby greatly extending ACL2’s ability to reason about the ordinals. We describe how to use the libraries containing these results, which are currently distributed with ACL2 version 2.7. 1

Abstract. We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom, Π1 1 comprehension, is needed to prove such basic facts as the existence of the weak-∗ closure of any norm-closed subspace of ℓ1 = c ∗ 0. This is in contrast to earlier work [6, 4, 7, 23, 22] in which theorems of separable Banach space theory were proved in very weak subsystems of second order arithmetic, subsystems which are conservative over Primitive Recursive Arithmetic for Π0 2 sentences. En route to our main results, we prove the Krein- ˇ Smulian theorem in ACA0, and we give a new, elementary proof of a result of McGehee on weak- ∗ sequential closure ordinals. 1.

Abstract We describe a method to permit the user of a mathematical logic to write elegant logical definitions while allowing sound and efficient execution. We focus on the ACL2 logic and automated reasoning environment. ACL2 is used by industrial researchers to describe microprocessor designs and other complicated digital systems. Properties of the designs can be formally established with the theorem prover. But because ACL2 is also a functional programming language, the formal models can be executed as simulation engines. We implement features that afford these dual applications, namely formal proof and execution on industrial test suites. In particular, the features allow the user to install, in a logically sound way, alternative executable counterparts for logically-defined functions. These alternatives are often much more efficient than the logically equivalent terms they replace. We discuss several applications of these features. 1 Introduction This paper is about a way to permit the functional programmer to prove efficientprograms correct. The idea is to allow the provision of two definitions of the program: an elegant definition that supports effective reasoning by a mechanizedtheorem prover, and an efficient definition for evaluation. A bridge of this sort,

how progress is repeatedly stifled by certain ways of thinking until some quantum leap ushers in a new era. In addition to allowing a firsthand look at the mathematical mindscape of the time, no other method would show so clearly the evolution of mathematical rigor and the conception of what constitutes an acceptable proof. Thus most homework assignments focus on gaps and difficult points in the original texts. Since mathematics is not created in a social vacuum, we supplement the mathematical content with cultural, biographical, and mathematical history, as well as a variety of prose readings, ranging from Plato's dialogue Socrates and the Slave Boy to modern writings such as an excerpt on "Mathematics and the End of the World" from [8]. They form the basis of regular class discussions. Two good sources for such readings are [11, 18]. To encourage student involvement, the discussions are led by one or two students, and everybody is expected to contribute. As the finale

Abstract. Termination proofs are of critical importance for establishing the correct behavior of both transformational and reactive computing systems. A general setting for establishing termination proofs involves the use of the ordinal numbers, an extension of the natural numbers into the transfinite which were introduced by Cantor in the nineteenth century and are at the core of modern set theory. We present the first comprehensive treatment of ordinal arithmetic on compact ordinal notations and give efficient algorithms for various operations, including addition, subtraction, multiplication, and exponentiation. Using the ACL2 theorem proving system, we implemented our ordinal arithmetic algorithms, mechanically verified their correctness, and developed a library of theorems that can be used to significantly automate reasoning involving the ordinals. To enable users of the ACL2 system to fully utilize our work required that we modify ACL2, e.g., we replaced the underlying representation of the ordinals and added a large library of definitions and theorems. Our modifications are available starting with ACL2 version 2.8. 1.

Our goal in this book is to build sofware tools that automatically search for proofs of program termination in mathematical logic. However, before delving directly into strategies for automation, we must first introduce some notation and establish a basic foundation in the areas of program semantics, logic and set theory. We must also discuss how programs can be proved terminating using manual techniques. The concepts and notation introduced in this chapter will be used throughout the remainder of the book. 1.1 Program termination and well-founded relations For the purpose of this book it is convenient to think of the text of a computer program as representing a relation that specifies the possible transitions that the program can make between configurations during execution. We call this the program’s transition relation. Program executions can be thought of as traversals starting from a starting configuration and then moving from configuration to configuration as allowed by the transition relation. A program is called terminating if all the executions allowed by the transition relation are finite. We call a program non-terminating if the transition relation allows for at least one infinite execution. Treating programs as relations is conveinant for our purpose, as in this setting proving program termination is equivliant to proving the program’s transition relation well-founded — thus giving us access to the numerous well established techniques from mathematical logic used to establish well-foundedness. In the next few sections we define some notation, discuss our representation for program configurations, and give some basic results related 3 4