Theories for TC0 and Other Small Complexity Classes

We present a general method for introducing finitely axiomatizable "minimal" two-sorted theories for various subclasses of P (problems solvable in polynomial time). The two sorts are natural numbers and finite sets of natural numbers. The latter are essentially the finite binary strings, which provide a natural domain for defining the functions and sets in small complexity classes. We concentrate on the complexity class TC^0, whose problems are defined by uniform polynomial-size families of bounded-depth Boolean circuits with majority gates. We present an elegant theory VTC^0 in which the provably-total functions are those associated with TC^0, and then prove that VTC^0 is "isomorphic" to a different-looking single-sorted theory introduced by Johannsen and Pollet. The most technical part of the isomorphism proof is defining binary number multiplication in terms a bit-counting function, and showing how to formalize the proofs of its algebraic properties.Comment: 40 pages, Logical Methods in Computer Scienc

Why Philosophers Should Care About Computational Complexity

One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed case that one would be wrong. In particular, I argue that computational complexity theory---the field that studies the resources (such as time, space, and randomness) needed to solve computational problems---leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume's problem of induction, Goodman's grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest. I end by discussing aspects of complexity theory itself that could benefit from philosophical analysis.Comment: 58 pages, to appear in "Computability: G\"odel, Turing, Church, and beyond," MIT Press, 2012. Some minor clarifications and corrections; new references adde

Arithmetic and Modularity in Declarative Languages for Knowledge Representation

The past decade has witnessed the development of many important declarative languages for knowledge representation and reasoning such as answer set programming (ASP) languages and languages that extend first-order logic. Also, since these languages depend on background solvers, the recent advancements in the efficiency of solvers has positively affected the usability of such languages. This thesis studies extensions of knowledge representation (KR) languages with arithmetical operators and methods to combine different KR languages. With respect to arithmetic in declarative KR languages, we show that existing KR languages suffer from a huge disparity between their expressiveness and their computational power. Therefore, we develop an ideal KR language that captures the complexity class NP for arithmetical search problems and guarantees universality and efficiency for solving such problems. Moreover, we introduce a framework to language-independently combine modules from different KR languages. We study complexity and expressiveness of our framework and develop algorithms to solve modular systems. We define two semantics for modular systems based on (1) a model-theoretical view and (2) an operational view on modular systems. We prove that our two semantics coincide and also develop mechanisms to approximate answers to modular systems using the operational view. We augment our algorithm these approximation mechanisms to speed up the process of solving modular system. We further generalize our modular framework with supported model semantics that disallows self-justifying models. We show that supported model semantics generalizes our two previous model-theoretical and operational semantics. We compare and contrast the expressiveness of our framework under supported model semantics with another framework for interlinking knowledge bases, i.e., multi-context systems, and prove that supported model semantics generalizes and unifies different semantics of multi-context systems. Motivated by the wide expressiveness of supported models, we also define a new supported equilibrium semantics for multi-context systems and show that supported equilibrium semantics generalizes previous semantics for multi-context systems. Furthermore, we also define supported semantics for propositional programs and show that supported model semnatics generalizes the acclaimed stable model semantics and extends the two celebrated properties of rationality and minimality of intended models beyond the scope of logic programs

Characterization and computation of equilibria in infinite games

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 79-82).Broadly, we study continuous games (those with continuous strategy spaces and utility functions) with a view towards computation of equilibria. We cover all of the game-theoretic background needed to understand these results in detail. Then we present new work, which can be divided into three parts. First, it is known that arbitrary continuous games may have arbitrarily complicated equilibria, so we investigate some properties of games with polynomial utility functions and a class of games with polynomial-like utility functions called separable games. We prove new bounds on the complexity of equilibria of separable games in terms of the complexity of the utility functions. In order to measure this complexity we propose a new definition for the rank of a continuous game; when applied to the case of finite games this improves on the results known in that setting. Furthermore, we prove a characterization theorem showing that several conditions which are necessary for a game to possess a finite-dimensional representation each define the class of separable games precisely, providing evidence that separable games are the natural class of continuous games in which to study computation. The characterization theorem also provides a natural connection between separability and the notion of the rank of a game. Second, we apply this theory to give an algorithm for computing e-Nash equilibria of two-player separable games with continuous strategy spaces. While a direct comparison to corresponding algorithms for finite games is not possible, the asymptotic running time in the complexity of the game grows slower for our algorithm than for any known algorithm for finite games.(cont.) Nonetheless, as in finite games, computing e-Nash equilibria still appears to be difficult for infinite games. Third, we consider computing approximate correlated equilibria in polynomial games. To do so, we first prove several new characterizations of correlated equilibria in continuous games which may be of independent interest. Then we introduce three algorithms for approximating correlated equilibria of polynomial games arbitrarily accurately. These include two discretization algorithms for computing a sample correlated equilibrium: a naive linear programming approach called static discretization which operates without regard to the structure of the game, and a semidefinite programming approach called adaptive discretization which exploits the structure of the game to achieve far better performance in practice. The third algorithm consists of a nested sequence of semidefinite programs converging to a description of the entire set of correlated equilibria.by Noah D. Stein.S.M