Credimus

We believe that economic design and computational complexity---while already important to each other---should become even more important to each other with each passing year. But for that to happen, experts in on the one hand such areas as social choice, economics, and political science and on the other hand computational complexity will have to better understand each other's worldviews. This article, written by two complexity theorists who also work in computational social choice theory, focuses on one direction of that process by presenting a brief overview of how most computational complexity theorists view the world. Although our immediate motivation is to make the lens through which complexity theorists see the world be better understood by those in the social sciences, we also feel that even within computer science it is very important for nontheoreticians to understand how theoreticians think, just as it is equally important within computer science for theoreticians to understand how nontheoreticians think

Self-Specifying Machines

We study the computational power of machines that specify their own acceptance types, and show that they accept exactly the languages that \manyonesharp-reduce to NP sets. A natural variant accepts exactly the languages that \manyonesharp-reduce to P sets. We show that these two classes coincide if and only if \psone = \psnnoplusbigohone, where the latter class denotes the sets acceptable via at most one question to \sharpp followed by at most a constant number of questions to \np.Comment: 15 pages, to appear in IJFC

Branching-time Model Checking of One-counter Processes

One-counter processes (OCPs) are pushdown processes which operate only on a unary stack alphabet. We study the computational complexity of model checking computation tree logic ($CTL$) over OCPs. A $PSPACE$ upper bound is inherited from the modal $mu$-calculus for this problem. First, we analyze the periodic behaviour of $CTL$ over OCPs and derive a model checking algorithm whose running time is exponential only in the number of control locations and a syntactic notion of the formula that we call leftward until depth. Thus, model checking fixed OCPs against $CTL$ formulas with a fixed leftward until depth is in $P$. This generalizes a result of the first author, Mayr, and To for the expression complexity of $CTL$\u27s fragment $EF$. Second, we prove that already over some fixed OCP, $CTL$ model checking is $PSPACE$-hard. Third, we show that there already exists a fixed $CTL$ formula for which model checking of OCPs is $PSPACE$-hard. For the latter, we employ two results from complexity theory: (i) Converting a natural number in Chinese remainder presentation into binary presentation is in logspace-uniform $NC^1$ and (ii) $PSPACE$ is $AC^0$-serializable. We demonstrate that our approach can be used to answer further open questions

Branching-time model checking of one-counter processes

One-counter processes (OCPs) are pushdown processes which operate only on a unary stack alphabet. We study the computational complexity of model checking computation tree logic (CTL) over OCPs. A PSPACE upper bound is inherited from the modal mu-calculus for this problem. First, we analyze the periodic behaviour of CTL over OCPs and derive a model checking algorithm whose running time is exponential only in the number of control locations and a syntactic notion of the formula that we call leftward until depth. Thus, model checking fixed OCPs against CTL formulas with a fixed leftward until depth is in P. This generalizes a result of the first author, Mayr, and To for the expression complexity of CTL's fragment EF. Second, we prove that already over some fixed OCP, CTL model checking is PSPACE-hard. Third, we show that there already exists a fixed CTL formula for which model checking of OCPs is PSPACE-hard. To obtain the latter result, we employ two results from complexity theory: (i) Converting a natural number in Chinese remainder presentation into binary presentation is in logspace-uniform NC^1 and (ii) PSPACE is AC^0-serializable. We demonstrate that our approach can be used to obtain further results. We show that model-checking CTL's fragment EF over OCPs is hard for P^NP, thus establishing a matching lower bound and answering an open question of the first author, Mayr, and To. We moreover show that the following problem is hard for PSPACE: Given a one-counter Markov decision process, a set of target states with counter value zero each, and an initial state, to decide whether the probability that the initial state will eventually reach one of the target states is arbitrarily close to 1. This improves a previously known lower bound for every level of the Boolean hierarchy by Brazdil et al

Generation problems

AbstractGiven a fixed computable binary operation f, we study the complexity of the following generation problem: the input consists of strings a1,…,an,b. The question is whether b is in the closure of {a1,…,an} under operation f.For several subclasses of operations we prove tight upper and lower bounds for the generation problems. For example, we prove exponential-time upper and lower bounds for generation problems of length-monotonic polynomial-time computable operations. Other bounds involve classes like NP and PSPACE.Here, the class of bivariate polynomials with positive coefficients turns out to be the most interesting class of operations. We show that many of the corresponding generation problems belong to NP. However, we do not know this for all of them, e.g., for x2+2y this is an open question. We prove NP-completeness for polynomials xaybc where a,b,c⩾1. Also, we show NP-hardness for polynomials like x2+2y. As a by-product we obtain NP-completeness of the extended sum-of-subset problem SOSc={(w1,…,wn,z):∃I⊆{1,…,n}(∑i∈Iwic=z)} for any c⩾1