Hardness of Equations over Finite Solvable Groups Under the Exponential Time Hypothesis

Goldmann and Russell (2002) initiated the study of the complexity of the equation satisfiability problem in finite groups by showing that it is in ? for nilpotent groups while it is ??-complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups of Fitting length two. In this work, we present the first lower bounds for the equation satisfiability problem in finite solvable groups: under the assumption of the exponential time hypothesis, we show that it cannot be in ? for any group of Fitting length at least four and for certain groups of Fitting length three. Moreover, the same hardness result applies to the equation identity problem

Intermediate problems in modular circuits satisfiability

In arXiv:1710.08163 a generalization of Boolean circuits to arbitrary finite algebras had been introduced and applied to sketch P versus NP-complete borderline for circuits satisfiability over algebras from congruence modular varieties. However the problem for nilpotent (which had not been shown to be NP-hard) but not supernilpotent algebras (which had been shown to be polynomial time) remained open. In this paper we provide a broad class of examples, lying in this grey area, and show that, under the Exponential Time Hypothesis and Strong Exponential Size Hypothesis (saying that Boolean circuits need exponentially many modular counting gates to produce boolean conjunctions of any arity), satisfiability over these algebras have intermediate complexity between $\Omega(2^{c\log^{h-1} n})$ and $O(2^{c\log^h n})$, where $h$ measures how much a nilpotent algebra fails to be supernilpotent. We also sketch how these examples could be used as paradigms to fill the nilpotent versus supernilpotent gap in general. Our examples are striking in view of the natural strong connections between circuits satisfiability and Constraint Satisfaction Problem for which the dichotomy had been shown by Bulatov and Zhuk

Efficiently Detecting Torsion Points and Subtori

Suppose X is the complex zero set of a finite collection of polynomials in Z[x_1,...,x_n]. We show that deciding whether X contains a point all of whose coordinates are d_th roots of unity can be done within NP^NP (relative to the sparse encoding), under a plausible assumption on primes in arithmetic progression. In particular, our hypothesis can still hold even under certain failures of the Generalized Riemann Hypothesis, such as the presence of Siegel-Landau zeroes. Furthermore, we give a similar (but UNconditional) complexity upper bound for n=1. Finally, letting T be any algebraic subgroup of (C^*)^n we show that deciding whether X contains T is coNP-complete (relative to an even more efficient encoding),unconditionally. We thus obtain new non-trivial families of multivariate polynomial systems where deciding the existence of complex roots can be done unconditionally in the polynomial hierarchy -- a family of complexity classes lying between PSPACE and P, intimately connected with the P=?NP Problem. We also discuss a connection to Laurent's solution of Chabauty's Conjecture from arithmetic geometry.Comment: 21 pages, no figures. Final version, with additional commentary and references. Also fixes a gap in Theorems 2 (now Theorem 1.3) regarding translated subtor

Satisfiability of Circuits and Equations over Finite Malcev Algebras

We show that the satisfiability of circuits over finite Malcev algebra A is NP-complete or A is nilpotent. This strengthens the result from our earlier paper [Idziak and Krzaczkowski, 2018] where nilpotency has been enforced, however with the use of a stronger assumption that no homomorphic image of A has NP-complete circuits satisfiability. Our methods are moreover strong enough to extend our result of [Idziak et al., 2021] from groups to Malcev algebras. Namely we show that tractability of checking if an equation over such an algebra A has a solution enforces its nice structure: A must have a nilpotent congruence ? such that also the quotient algebra A/? is nilpotent. Otherwise, if A has no such congruence ? then the Exponential Time Hypothesis yields a quasipolynomial lower bound. Both our results contain important steps towards a full characterization of finite algebras with tractable circuit satisfiability as well as equation satisfiability

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

Hidden Translation and Translating Coset in Quantum Computing

We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in $\Z_{p}^{n}$, whenever $p$ is a fixed prime. For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Translating Coset in a finite solvable group $G$ is reducible to instances of Translating Coset in $G/N$ and $N$, for appropriate normal subgroups $N$ of $G$. Our self-reducibility framework combined with Kuperberg's subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.Comment: Journal version: change of title and several minor update

Computational complexity of the landscape I

We study the computational complexity of the physical problem of finding vacua of string theory which agree with data, such as the cosmological constant, and show that such problems are typically NP hard. In particular, we prove that in the Bousso-Polchinski model, the problem is NP complete. We discuss the issues this raises and the possibility that, even if we were to find compelling evidence that some vacuum of string theory describes our universe, we might never be able to find that vacuum explicitly. In a companion paper, we apply this point of view to the question of how early cosmology might select a vacuum.Comment: JHEP3 Latex, 53 pp, 2 .eps figure