AbstractWe combine Solomonoff’s approach to universal prediction with algorithmic statistics and suggest to use the computable measure that provides the best “explanation” for the observed data (in the sense of algorithmic statistics) for prediction. In this way we keep the expected sum of squares of prediction errors bounded (as it was for the Solomonoff’s predictor) and, moreover, guarantee that the sum of squares of prediction errors is bounded along any Martin-Löf random sequence. An extended abstract of this paper was presented at the 16th International Computer Science Symposium in Russia (CSR 2021) (Milovanov 2021).

AbstractThe power word problem for a group $$\varvec{G}$$ G asks whether an expression $$\varvec{u_1^{x_1} \cdots u_n^{x_n}}$$ u 1 x 1 ⋯ u n x n , where the $$\varvec{u_i}$$ u i are words over a finite set of generators of $$\varvec{G}$$ G and the $$\varvec{x_i}$$ x i binary encoded integers, is equal to the identity of $$\varvec{G}$$ G . It is a restriction of the compressed word problem, where the input word is represented by a straight-line program (i.e., an algebraic circuit over $$\varvec{G}$$ G ). We start by showing some easy results concerning the power word problem. In particular, the power word problem for a group $$\varvec{G}$$ G is $$\varvec{\textsf{uNC}^{1}}$$ uNC 1 -many-one reducible to the power word problem for a finite-index subgroup of $$\varvec{G}$$ G . For our main result, we consider graph products of groups that do not have elements of order two. We show that the power word problem in a fixed such graph product is $$\varvec{\textsf{AC} ^0}$$ AC 0 -Turing-reducible to the word problem for the free group $$\varvec{F_2}$$ F 2 and the power word problems of the base groups. Furthermore, we look into the uniform power word problem in a graph product, where the dependence graph and the base groups are part of the input. Given a class of finitely generated groups $$\varvec{\mathcal {C}}$$ C without order two elements, the uniform power word problem in a graph product can be solved in $$\varvec{\textsf{AC} ^0[\textsf{C}_=\textsf{L} ^{{{\,\textrm{UPowWP}\,}}(\mathcal {C})}]}$$ AC 0 [ C = L UPowWP ( C ) ] , where $$\varvec{{{\,\textrm{UPowWP}\,}}(\mathcal {C})}$$ UPowWP ( C ) denotes the uniform power word problem for groups from the class $$\varvec{\mathcal {C}}$$ C . As a consequence of our results, the uniform knapsack problem in right-angled Artin groups is $$\varvec{\textsf{NP}}$$ NP -complete. The present paper is a combination of the two conference papers (Lohrey and Weiß 2019b, Stober and Weiß 2022a). In Stober and Weiß (2022a) our results on graph products were wrongly stated without the additional assumption that the base groups do not have elements of order two. In the present work we correct this mistake. While we strongly conjecture that the result as stated in Stober and Weiß (2022a) is true, our proof relies on this additional assumption.

AbstractThe notion of a quasi-order generated by a homomorphism from the semigroup of all words onto a finite ordered semigroup was introduced by Bucher et al. (Theor. Comput. Sci. 40, 131–148 1985). It naturally occurred in their studies of derivation relations associated with a given set of context-free rules, and they asked a crucial question, whether the resulting relation is a well quasi-order. We answer this question in the case of the quasi-order generated by a semigroup homomorphism. We show that the answer does not depend on the homomorphism, but it is a property of its image. Moreover, we give an algebraic characterization of those finite semigroups for which we get well quasi-orders. This characterization completes the structural characterization given by Kunc (Theor. Comput. Sci. 348, 277–293 2005) in the case of semigroups ordered by equality. Compared with Kunc’s characterization, the new one has no structural meaning, and we explain why that is so. In addition, we prove that the new condition is testable in polynomial time.

AbstractA pumping lemma for a class of languages $$\varvec{\mathcal {C}}$$ C is often used to show particular languages are not in $$\varvec{\mathcal {C}}$$ C . In contrast, we show that a pumping lemma for a class of languages $$\varvec{\mathcal {C}}$$ C can be used to study the computational complexity of the predicate “$$\in \varvec{\mathcal {C}}$$ ∈ C ” via highly efficient many-one reductions. In this paper, we use extended regular expressions (EXREGs, introduced in Câmpeanu et al. (Int. J. Foundations Comput. Sci. 14(6), 1007–1018, 2003)) as an example to illustrate the proof technique and establish the complexity of the predicate “is an EXREG language” for several classes of languages. Due to the efficiency of the reductions, both productiveness (a stronger form of non-recursive enumerability) and complexity results can be obtained simultaneously. For example, we show that the predicate “is an EXREG language” is productive (hence, not recursively enumerable) for context-free grammars, and is Co-NEXPTIME-hard for context-free grammars generating bounded languages. The proof technique is easy to use and requires only a few conditions. This suggests that for any class of languages $$\varvec{\mathcal {C}}$$ C having a pumping lemma, the language class comparison problems (e.g., does a given context-free grammar generate a language in $$\varvec{\mathcal {C}}$$ C ?) are almost guaranteed to be hard. So, pumping lemmas sometimes could be “harmful” when studying computational complexity results.

AbstractWe consider a setting with agents that have preferences over alternatives and are partitioned into disjoint districts. The goal is to choose one alternative as the winner using a mechanism which first decides a representative alternative for each district based on a local election with the agents therein as participants, and then chooses one of the district representatives as the winner. Previous work showed bounds on the distortion of a specific class of deterministic plurality-based mechanisms depending on the available information about the preferences of the agents in the districts. In this paper, we first consider the whole class of deterministic mechanisms and show asymptotically tight bounds on their distortion. We then initiate the study of the distortion of randomized mechanisms in distributed voting and show bounds based on several informational assumptions, which in many cases turn out to be tight. Finally, we also experimentally compare the distortion of many different mechanisms of interest using synthetic and real-world data.

AbstractIn the well known planted clique problem, a clique (or alternatively, an independent set) of size k is planted at random in an Erdos-Renyi random G(n, p) graph, and the goal is to design an algorithm that finds the maximum clique (or independent set) in the resulting graph. We introduce a variation on this problem, where instead of planting the clique at random, the clique is planted by an adversary who attempts to make it difficult to find the maximum clique in the resulting graph. We show that for the standard setting of the parameters of the problem, namely, a clique of size $$k = \sqrt{n}$$ k = n planted in a random $$G(n, \frac{1}{2})$$ G ( n , 1 2 ) graph, the known polynomial time algorithms can be extended (in a non-trivial way) to work also in the adversarial setting. In contrast, we show that for other natural settings of the parameters, such as planting an independent set of size $$k=\frac{n}{2}$$ k = n 2 in a G(n, p) graph with $$p = n^{-\frac{1}{2}}$$ p = n - 1 2 , there is no polynomial time algorithm that finds an independent set of size k, unless NP has randomized polynomial time algorithms.

AbstractStudies of issues related to computability and computational complexity involve the use of a model of computation. Central in such a model are computational processes. Processes of this kind can be described using an imperative process algebra based on ACP (Algebra of Communicating Processes). In this paper, it is investigated whether the imperative process algebra concerned can play a role in the field of models of computation. It is demonstrated that the process algebra is suitable to describe in a mathematically precise way models of computation corresponding to existing models based on sequential, asynchronous parallel, and synchronous parallel random access machines as well as time and work complexity measures for those models.

AbstractThe descriptional complexity of basic operations on regular languages using 1-limited automata, a restricted version of one-tape Turing machines, is investigated. When simulating operations on deterministic finite automata with deterministic 1-limited automata, the sizes of the resulting devices are polynomial in the sizes of the simulated machines. The situation is different when the operations are applied to deterministic 1-limited automata: while for boolean operations the simulations remain polynomial, for product, star, and reversal they cost exponential in size. The costs for product and star do not reduce if the given machines are sweeping two-way deterministic finite automata. These bounds are tight.