Information, Architecture, Complexity

The study of the relationship between information, architecture and complexity can be accomplished through the study of patterns of relationships, opening up the field for the understanding of architecture as organization.     

DRM, Complexity, and Correctness

Steve Bellovin looks at the complex code behind Microsoft Vista and its DRM mechanisms. Increased amounts of code add to insecurity, but the real danger with DRM is with increased interaction among different pieces of code. A lot of new mechanisms have been introduced; more seriously, a lot of new communications paths and dependencies have been introduced. Worst of all, these paths and mechanisms are solving a new problem, one with which the profession has very little experience. Did Microsoft get it right?     

Complexity,convexity, and unimodality

A class of polygons termedunimodal is introduced. LetP = P₁,p ₂,...,p _n be a simplen-vertex polygon. Given a fixed vertex or edge, several definitions of the distance between the fixed vertex or edge and any other vertex or edge are considered. For a fixed vertex (edge), a distance measure defines a distance function as the remaining vertices (edges) are traversed in order. If for every vertex (edge) ofP a specified distance function is unimodal thenP is a unimodal polygon in the corresponding sense. Relationships between unimodal polygons, in several senses, andconvex polygons are established. Several properties are derived for unimodal polygons when the distance measure is the euclidean distance between vertices of the polygons. These properties lead to very simple 0(n) algorithms for solving a variety of problems that occur in computational geometry and pattern recognition. Furthermore, these algorithms establish that convexity is not the key factor in obtaining linear-time-complexity for solving these problems. The paper closes with several open questions in this area.     

Incompleteness,Complexity, Randomness and Beyond

Gödel's Incompleteness Theorems have the same scientific status as Einstein's principle of relativity, Heisenberg's uncertainty principle, and Watson and Crick's double helix model of DNA. Our aim is to discuss some new faces of the incompleteness phenomenon unveiled by an information-theoretic approach to randomness and recent developments in quantum computing.     

Complexity, class dynamics, and distance learning

Classroom participants learn early on that each classroom has its own dynamic comprised of personalities, motivation levels, skills, and other variables. This paper explores features of complexity theory—nonlinearity and emergent self-organization—relevant to dynamics in physical or virtual classrooms. These central notions of complexity theory and their importance in composition classrooms help explain why students in virtual classrooms are often less successful than their physical classroom counterparts in negotiating the eddies of virtual interactions. The paper closes with a brief consideration of how teachers can interrogate all the elements of teaching and classroom context (whether physical or virtual) to influence the emergent dynamic of our classrooms.     

Hardness Hypotheses, Derandomization, and Circuit Complexity

We consider hypotheses about nondeterministic computation that have been studied in different contexts and shown to have interesting consequences:

Space Complexity Vs. Query Complexity

Combinatorial property testing deals with the following relaxation of decision problems: Given a fixed property and an input x, one wants to decide whether x satisfies the property or is “far” from satisfying it. The main focus of property testing is in identifying large families of properties that can be tested with a certain number of queries to the input. In this paper we study the relation between the space complexity of a language and its query complexity. Our main result is that for any space complexity s(n) ≤ log n there is a language with space complexity O(s(n)) and query complexity 2^Ω(s(n)). Our result has implications with respect to testing languages accepted by certain restricted machines. Alon et al. [FOCS 1999] have shown that any regular language is testable with a constant number of queries. It is well known that any language in space o(log log n) is regular, thus implying that such languages can be so tested. It was previously known that there are languages in space O(log n) that are not testable with a constant number of queries and Newman [FOCS 2000] raised the question of closing the exponential gap between these two results. A special case of our main result resolves this problem as it implies that there is a language in space O(log log n) that is not testable with a constant number of queries. It was also previously known that the class of testable properties cannot be extended to all context-free languages. We further show that one cannot even extend the family of testable languages to the class of languages accepted by single counter machines.      

Nash equilibria: Complexity,symmetries, and approximation

We survey recent joint work with Christos Papadimitriou and Paul Goldberg on the computational complexity of Nash equilibria. We show that finding a Nash equilibrium in normal form games is computationally intractable, but in an unusual way. It does belong to the class NP; but Nash’s theorem, showing that a Nash equilibrium always exists, makes the possibility that it is also NP-complete rather unlikely. We show instead that the problem is as hard computationally as finding Brouwer fixed points, in a precise technical sense, giving rise to a new complexity class called PPAD. The existence of the Nash equilibrium was established via Brouwer’s fixed-point theorem; hence, we provide a computational converse to Nash’s theorem.To alleviate the negative implications of this result for the predictive power of the Nash equilibrium, it seems natural to study the complexity of approximate equilibria: an efficient approximation scheme would imply that players could in principle come arbitrarily close to a Nash equilibrium given enough time. We review recent work on computing approximate equilibria and conclude by studying how symmetries may affect the structure and approximation of Nash equilibria. Nash showed that every symmetric game has a symmetric equilibrium. We complement this theorem with a rich set of structural results for a broader, and more interesting class of games with symmetries, called anonymous games.     

Complexity,convexity and combinations of theories

We restrict our attention to decidable quantifier-free theories, such as the quantifier-free theory of integers under addition, the quantifier-free theory of arrays under storing and selecting, or the quantifier-free theory of list structure under cons, car and cdr. We consider combinations of such theories: theories whose sets of symbols are the union of the sets of the symbols of the individual theories and whose set of axioms is the union of the sets of axioms of the individual theories. We give a general technique for determining the complexity of decidable combinations of theories, and show, for example, that the satisfiability problem for the quantifier-free theory of integers, arrays, list structure and uninterpreted function symbols under +, ≤, store, select, cons, car and cdr is NP-complete. We next consider the complexity of the satisfiability problem for formulas already in disjunctive normal form: why some combinations of theories admit deterministic polynomial time decision procedures while for others the problem is NP-hard. Our analysis hinges on the question of whether the theories being combined are convex; that is, whether any conjunction of literals in the theory can entail a proper disjunction of equalities between variables. This leads to a discussion of the role that case analysis plays in deciding combinations of theories.     

Choosing,Agreeing, and Eliminating in Communication Complexity

We consider several questions inspired by the direct-sum problem in (two-party) communication complexity. In all questions, there are k fixed Boolean functions f ₁,…,f _k and each of Alice and Bob has k inputs, x ₁,…,x _k and y ₁,…,y _k , respectively. In the eliminate problem, Alice and Bob should output a vector σ₁,…,σ _k such that f _i (x _i , y _i ) ≠ σ _i for at least one i (i.e., their goal is to eliminate one of the 2 ^k output vectors); in the choose problem, Alice and Bob should return (i, f _i (x _i , y _i )), for some i (i.e., they choose one instance to solve), and in the agree problem they should return f _i (x _i , y _i ), for some i (i.e., if all the k Boolean values agree then this must be the output). The question, in each of the three cases, is whether one can do better than solving one (say, the first) instance. We study these three problems and prove various positive and negative results. In particular, we prove that the randomized communication complexity of eliminate, of k instances of the same function f, is characterized by the randomized communication complexity of solving one instance of f.     

SOCIETAL SYSTEMS Planning,Policy and Complexity

The contribution of systems theory is reexamined from an organizational view. Its influence on artificial intelligence (and on computer science) is examined through the examples of analogical problem solving and cooperative distributed problem solving.     

Complexity,Algorithms, Programs,Systems: The Shifting Focus

We investigate the changing relationship between the small research community of theoretical computer scientists and the much larger community of computer users, in particular, the technology transfer problem of how to exploit theoretical insights that can lead to better products. Our recommendation can be summarized in four points:1 The computing community is impressed by usable tools and by little else. Although a powerful theorem or ån elegant algorithm may be a useful tool for a fellow theoretician, by and large, the only tools directly usable by the general computing community are systems. No systems, no impact!2 System development means programming-in-the-large, but the algorithms research community so far has learned only how to program in-the-small.3 Algorithm researchers must enshrine their algorithms not merely in individual elegant programs, but collectively in useful application packages aimed at some identifiable user group.4 Since the development of software systems easily turns into a full-time activity that requires different skills from those of algorithms research, we must strive to develop techniques that lets a small group of algorithm researchers develop simply structured, open-ended systems whose kernel can be implemented with an effort of the order of 1 man-year. Low-complexity systems is the goal!     

Complexity,Fractal Dimension for Quantum States

The complexities in information dynamics are reviewed and their examples are given. The fractal dimensions of a quantum state are discussed from a general point of view of complexity. It is shown trough a model that the fractal dimensions of a state provide measures for order structure of chaotic systems.     

Arithmetic Complexity, Kleene Closure, and Formal Power Series

The aim of this paper is to use formal power series techniques to study the structure of small arithmetic complexity classes such as GapNC ¹ and GapL. More precisely, we apply the formal power series operations of inversion and root extraction to these complexity classes. We define a counting version of Kleene closure and show that it is intimately related to inversion and root extraction within GapNC ¹ and GapL. We prove that Kleene closure, inversion, and root extraction are all hard operations in the following sense: there is a language in AC ⁰ for which inversion and root extraction are GapL-complete and Kleene closure is NLOG-complete, and there is a finite set for which inversion and root extraction are GapNC ¹ -complete and Kleene closure is NC ¹ -complete, with respect to appropriate reducibilities. The latter result raises the question of classifying finite languages so that their inverses fall within interesting subclasses of GapNC ¹ , such as GapAC ⁰ . We initiate work in this direction by classifying the complexity of the Kleene closure of finite languages. We formulate the problem in terms of finite monoids and relate its complexity to the internal structure of the monoid. Some results in this paper show properties of complexity classes that are interesting independent of formal power series considerations, including some useful closure properties and complete problems for GapL.