Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

We define Recursive Markov Chains (RMCs), a class of finitely presented denumerable Markov chains, and we study algorithms for their analysis. Informally, an RMC consists of a collection of finite-state Markov chains with the ability to invoke each other in a potentially recursive manner. RMCs offer a natural abstract model for probabilistic programs with procedures. They generalize, in a precise sense, a number of well-studied stochastic models, including Stochastic Context-Free Grammars (SCFG) and Multi-Type Branching Processes (MT-BP).We focus on algorithms for reachability and termination analysis for RMCs: what is the probability that an RMC started from a given state reaches another target state, or that it terminatesq These probabilities are in general irrational, and they arise as (least) fixed point solutions to certain (monotone) systems of nonlinear equations associated with RMCs. We address both the qualitative problem of determining whether the probabilities are 0, 1 or in-between, and the quantitative problems of comparing the probabilities with a given bound, or approximating them to desired precision.We show that all these problems can be solved in PSPACE using a decision procedure for the Existential Theory of Reals. We provide a more practical algorithm, based on a decomposed version of multi-variate Newton's method, and prove that it always converges monotonically to the desired probabilities. We show this method applies more generally to any monotone polynomial system. We obtain polynomial-time algorithms for various special subclasses of RMCs. Among these: for SCFGs and MT-BPs (equivalently, for 1-exit RMCs) the qualitative problem can be solved in P-time; for linearly recursive RMCs the probabilities are rational and can be computed exactly in P-time.We show that our PSPACE upper bounds cannot be substantially improved without a breakthrough on long standing open problems: the square-root sum problem and an arithmetic circuit decision problem that captures P-time on the unit-cost rational arithmetic RAM model. We show that these problems reduce to the qualitative problem and to the approximation problem (to within any nontrivial error) for termination probabilities of general RMCs, and to the quantitative decision problem for termination (extinction) of SCFGs (MT-BPs).

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Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations

We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, $\Gamma$, with 3 or more players, and given $\epsilon>0$, compute an approximation within $\epsilon$ of some (actual) Nash equilibrium. We show that approximation of an actual Nash equilibrium, even to within any nontrivial constant additive factor $\epsilon<1/2$ in just one desired coordinate, is at least as hard as the long-standing square-root sum problem, as well as a more general arithmetic circuit decision problem that characterizes P-time in a unit-cost model of computation with arbitrary precision rational arithmetic; thus, placing the approximation problem in P, or even NP, would resolve major open problems in the complexity of numerical computation.

We show similar results for market equilibria: it is hard to estimate with any nontrivial accuracy the equilibrium prices in an exchange economy with a unique equilibrium, where the economy is given by explicit algebraic formulas for the excess demand functions.

We define a class, FIXP, which captures search problems that can be cast as fixed point computation problems for functions represented by algebraic circuits (straight line programs) over basis $\{+,*,-,/,\max,\min\}$ with rational constants. We show that the (exact or approximate) computation of Nash equilibria for 3 or more players is complete for FIXP. The price equilibrium problem for exchange economies with algebraic demand functions is another FIXP-complete problem. We show that the piecewise linear fragment of FIXP equals PPAD.

Many other problems in game theory, economics, and probability theory can be cast as fixed point problems for such algebraic functions. We discuss several important such problems: computing the value of Shapley's stochastic games and the simpler games of Condon, extinction probabilities of branching processes, probabilities of stochastic context-free grammars, and termination probabilities of recursive Markov chains. We show that for some of them, the approximation, or even exact computation, problem can be placed in PPAD, while for others, they are at least as hard as the square-root sum and arithmetic circuit decision problems.

/pdf/on-the-complexity-of-nash-equilibria-and-other-fixed-points-3oj9xxsrjd.pdf

On the Complexity of Nash Equilibria and Other Fixed Points

Results on algorithmic problems on strings that are given in a compressed form via straightline programs are surveyed. A straight-line program is a context-free grammar that generates exactly one string. In this way, exponential compression rates can be achieved. Among others, we study pattern matching for compressed strings, membership problems for compressed strings in various kinds of formal languages, and the problem of querying compressed strings. Applications in combinatorial group theory and computational topology and to the solution of word equations are discussed as well. Finally, extensions to compressed trees and pictures are considered.

https://www.informatik.uni-leipzig.de/~lohrey/12-SURVEY.pdf

Algorithmics on SLP-compressed strings: A survey

We reexamine, what it means to compute Nash equilibria and, more, generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Gamma, with 3 or more players, and given epsiv > 0, compute a vector x' (a mixed strategy profile) that is within distance e (say in t^) of some (exact) Nash equilibrium. We show that approximation of an (actual) Nash equilibrium for games with 3 players, even to within any non-trivial constant additive factor epsiv < 1/2 in just one desired coordinate, is at least as hard as the long standing square-root sum problem, as well as more general arithmetic circuit decision problems, and thus that even placing the approximation problem in NP would-resolve a major open problem in the complexity of numerical computation. Furthermore, we show that the (exact or approximate) computation of Nash equilibria for 3 or more players is complete for the class of search problems, which we call FIXP, that can be cast as fixed point computation problems for functions represented by algebraic circuits (straight line programs) over basis {+, *, -, /, max, min}, with rational constants. We show that the linear fragment of FIXP equals PPAD. Many problems in game theory, economics, and probability theory, can be cast as fixed point problems for such algebraic functions. We discuss several important such problems: computing the value of Shapley's stochastic games, and the simpler games of Condon, extinction probabilities of branching processes, termination probabilities of stochastic context-free grammars, and of Recursive Markov Chains. We show that for some of them, the approximation, or even exact computation, problem can be placed-in PPAD, while for others, they are at least as hard as the square-root sum and arithmetic circuit decision problems.

/pdf/on-the-complexity-of-nash-equilibria-and-other-fixed-points-2s7c8twjxr.pdf

On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract)

We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: (a) the Blum-Shub-Smale model of computation over the reals; and (b) a problem we call the “generic task of numerical computation,” which captures an aspect of doing numerical computation in floating point, similar to the “long exponent model” that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a division-free straight-line program producing an integer $N$, decide whether $N>0$. In the Blum-Shub-Smale model, polynomial-time computation over the reals (on discrete inputs) is polynomial-time equivalent to PosSLP when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture. The generic task of numerical computation is also polynomial-time equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean traveling salesman problem lies in the counting hierarchy—the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE. In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of the arithmetic circuit identity testing (ACIT) problem. In particular, we show that if $n!$ is not ultimately easy, then ACIT has subexponential complexity.

/pdf/on-the-complexity-of-numerical-analysis-pucdxq4vml.pdf

On the Complexity of Numerical Analysis

We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis. We show that both hinge on the question of understanding the complexity of the following problem, which we call PosSlp: Given a division-free straight-line program producing an integer N, decide whether N>0. We show that OrdSlp lies in the counting hierarchy, and combining our results with work of Tiwari, we show that the Euclidean Traveling Salesman Problem lies in the counting hierarchy -- the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE.

On the Complexity of Numerical Analysis.

We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis. We show that both hinge on the question of understanding the complexity of the following problem, which we call PosSLP; given a division-free straight-line program producing an integer N, decide whether N > 0. We show that PosSLP lies in the counting hierarchy, and combining our results with work of Tiwari, we show that the Euclidean traveling salesman problem lies in the counting hierarchy - the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE.

/pdf/on-the-complexity-of-numerical-analysis-17j4hh5lon.pdf

Johan Kjeldgaard-Pedersen

Papers

On the Complexity of Numerical Analysis

On the Complexity of Numerical Analysis.

On the complexity of numerical analysis

On the Complexity of Numerical Analysis