scholarly journals Computational complexity of the word problem in modal algebras

2021 ◽  
pp. 5-17
Author(s):  
Михаил Николаевич Рыбаков
Keyword(s):  
Computational Complexity ◽  
Word Problem ◽  
Complexity Class ◽  
Complete Problem ◽  
Modal Algebras

Приводится доказательство $\mathrm{PSPACE}$-полноты проблемы равенства слов в классе всех нуль-порождённых модальных алгебр, или, эквивалентно, проблемы равенства константных слов в классе всех модальных алгебр. Также рассматривается вопрос о сложности равенства слов в произвольном многообразии модальных алгебр. Доказывается, что уже проблема равенства константных слов в многообразии модальных алгебр может быть сколь угодно трудной (включая как классы сложности, так и степени неразрешимости). Показано, как построить соответствующие многообразия. The paper deals with the word problem for modal algebras. It is proved that, for the variety of all modal algebras, the word problem is $\mathrm{PSPACE}$-complete if only constant modal terms or only $0$-generated modal algebras are considered. We also consider the word problem for different varieties of modal algebras. It is proved that the word problem for a variety of modal algebras can be $C$-hard, for any complexity class or unsolvability degree $C$ containing a $C$-complete problem. It is shown how to construct such varieties.

scholarly journals The Computational Complexity of Probabilistic Planning

1998 ◽  
Vol 9 ◽  
pp. 1-36 ◽  
Author(s):  
M. L. Littman ◽  
J. Goldsmith ◽  
M. Mundhenk
Keyword(s):  
Computational Complexity ◽  
Efficient Algorithms ◽  
Boolean Satisfiability ◽  
Complexity Classes ◽  
Probabilistic Planning ◽  
Plan Evaluation ◽  
Partially Ordered ◽  
Boolean Satisfiability Problem ◽  
Complete Problem ◽  
Planning Problems

We examine the computational complexity of testing and finding small plans in probabilistic planning domains with both flat and propositional representations. The complexity of plan evaluation and existence varies with the plan type sought; we examine totally ordered plans, acyclic plans, and looping plans, and partially ordered plans under three natural definitions of plan value. We show that problems of interest are complete for a variety of complexity classes: PL, P, NP, co-NP, PP, NP^PP, co-NP^PP, and PSPACE. In the process of proving that certain planning problems are complete for NP^PP, we introduce a new basic NP^PP-complete problem, E-MAJSAT, which generalizes the standard Boolean satisfiability problem to computations involving probabilistic quantities; our results suggest that the development of good heuristics for E-MAJSAT could be important for the creation of efficient algorithms for a wide variety of problems.

Complete problems for fixed-point logics

1995 ◽  
Vol 60 (2) ◽  
pp. 517-527 ◽  
Author(s):  
Martin Grohe
Keyword(s):  
Fixed Point ◽  
Computational Complexity ◽  
Normal Form ◽  
Query Language ◽  
Main Step ◽  
Complexity Classes ◽  
Complete Problem ◽  
Complete Problems ◽  
Database Query Language ◽  
Natural Description

The notion of logical reducibilities is derived from the idea of interpretations between theories. It was used by Lovász and Gács [LG77] and Immerman [Imm87] to give complete problems for certain complexity classes and hence establish new connections between logical definability and computational complexity.However, the notion is also interesting in a purely logical context. For example, it is helpful to establish nonexpressibility results.We say that a class of τ-structures is a >complete problem for a logic under L-reductions if it is definable in [τ] and if every class definable in can be ”translated” into by L-formulae (cf. §4).We prove the following theorem:1.1. Theorem. There are complete problemsfor partial fixed-point logic andfor inductive fixed-point logic under quantifier-free reductions.The main step of the proof is to establish a new normal form for fixed-point formulae (which might be of some interest itself). To obtain this normal form we use theorems of Abiteboul and Vianu [AV91a] that show the equivalence between the fixed-point logics we consider and certain extensions of the database query language Datalog.In [Dah87] Dahlhaus gave a complete problem for least fixed-point logic. Since least fixed-point logic equals inductive fixed-point logic by a well-known result of Gurevich and Shelah [GS86], this already proves one part of our theorem.However, our class gives a natural description of the fixed-point process of an inductive fixed-point formula and hence sheds some light on completely different aspects of the logic than Dahlhaus's construction, which is strongly based on the features of least fixed-point formulae.

scholarly journals Design and implementation of aggregate functions in the DLV system

2008 ◽  
Vol 8 (5-6) ◽  
pp. 545-580 ◽  
Author(s):  
WOLFGANG FABER ◽  
GERALD PFEIFER ◽  
NICOLA LEONE ◽  
TINA DELL'ARMI ◽  
GIUSEPPE IELPA
Keyword(s):  
Computational Complexity ◽  
Logic Programming ◽  
Real World ◽  
State Of The Art ◽  
Complexity Class ◽  
Natural Manner ◽  
Design And Implementation ◽  
Finite Structures ◽  
Real World Applications ◽  
Aggregate Functions

AbstractDisjunctive logic programming (DLP) is a very expressive formalism. It allows for expressing every property of finite structures that is decidable in the complexity class ΣP2(=NPNP). Despite this high expressiveness, there are some simple properties, often arising in real-world applications, which cannot be encoded in a simple and natural manner. Especially properties that require the use of arithmetic operators (like sum, times, or count) on a set or multiset of elements, which satisfy some conditions, cannot be naturally expressed in classic DLP. To overcome this deficiency, we extend DLP by aggregate functions in a conservative way. In particular, we avoid the introduction of constructs with disputed semantics, by requiring aggregates to be stratified. We formally define the semantics of the extended language (called ), and illustrate how it can be profitably used for representing knowledge. Furthermore, we analyze the computational complexity of , showing that the addition of aggregates does not bring a higher cost in that respect. Finally, we provide an implementation of in DLV—a state-of-the-art DLP system—and report on experiments which confirm the usefulness of the proposed extension also for the efficiency of computation.

On the computational complexity of the Dirichlet Problem for Poisson's Equation

2016 ◽  
Vol 27 (8) ◽  
pp. 1437-1465 ◽  
Author(s):  
AKITOSHI KAWAMURA ◽  
FLORIAN STEINBERG ◽  
MARTIN ZIEGLER
Keyword(s):  
Dirichlet Problem ◽  
Computational Complexity ◽  
Complexity Theory ◽  
Complexity Class ◽  
Poisson’S Equation ◽  
Poisson's Equation ◽  
Precise Sense ◽  
Real Functions ◽  
Riemann Integration

The last years have seen an increasing interest in classifying (existence claims in) classical mathematical theorems according to their strength. We pursue this goal from the refined perspective of computational complexity. Specifically, we establish that rigorously solving the Dirichlet Problem for Poisson's Equation is in a precise sense ‘complete’ for the complexity class ${\#\mathcal{P}}$ and thus as hard or easy as parametric Riemann integration (Friedman 1984; Ko 1991. Complexity Theory of Real Functions).

scholarly journals FUNCTIONS ON GROUPS AND COMPUTATIONAL COMPLEXITY

2004 ◽  
Vol 14 (04) ◽  
pp. 409-429 ◽  
Author(s):  
JEAN-CAMILLE BIRGET
Keyword(s):  
Computational Complexity ◽  
Symmetric Space ◽  
Word Problem ◽  
Exponential Inequality ◽  
Isoperimetric Function ◽  
Length Functions ◽  
Finitely Presented Groups ◽  
Double Exponential ◽  
Context Free ◽  
Finitely Presented

We give some connections between various functions defined on finitely presented groups (isoperimetric, isodiametric, Todd–Coxeter radius, filling length functions, etc.), and we study the relation between those functions and the computational complexity of the word problem (deterministic time, nondeterministic time, symmetric space). We show that the isoperimetric function can always be linearly decreased (unless it is the identity map). We present a new proof of the Double Exponential Inequality, based on context-free languages.

scholarly journals On the Computational Complexity of Gossip Protocols

2017 ◽  
Author(s):  
Krzysztof R. Apt ◽  
Eryk Kopczyński ◽  
Dominik Wojtczak
Keyword(s):  
Computational Complexity ◽  
Private Information ◽  
Epistemic Logic ◽  
Distributed Programs ◽  
Distributed Protocols ◽  
Partial Correctness ◽  
Complete Problem ◽  
Gossip Protocol ◽  
Np Complete ◽  
Gossip Protocols

Gossip protocols deal with a group of communicating agents, each holding a private information, and aim at arriving at a situation in which all the agents know each other secrets. Distributed epistemic gossip protocols are particularly simple distributed programs that use formulas from an epistemic logic. Recently, the implementability of these distributed protocols was established (which means that the evaluation of these formulas is decidable), and the problems of their partial correctness and termination were shown to be decidable, but their exact computational complexity was left open. We show that for any monotonic type of calls the implementability of a distributed epistemic gossip protocol is a P^{NP}_{||}-complete problem, while the problems of its partial correctness and termination are in coNP^{NP}.

scholarly journals A Multivariate Complexity Analysis of Lobbying in Multiple Referenda

2014 ◽  
Vol 50 ◽  
pp. 409-446 ◽  
Author(s):  
R. Bredereck ◽  
J. Chen ◽  
S. Hartung ◽  
S. Kratsch ◽  
R. Niedermeier ◽  
...  
Keyword(s):  
Computational Complexity ◽  
Complexity Analysis ◽  
Central Question ◽  
Logarithmic Factor ◽  
Fixed Parameter Tractable ◽  
Complete Problem ◽  
Fixed Parameter ◽  
Matrix Modification ◽  
Simple Matrix ◽  
Limited Nondeterminism

Assume that each of n voters may or may not approve each of m issues. If an agent (the lobby) may influence up to k voters, then the central question of the NP-hard Lobbying problem is whether the lobby can choose the voters to be influenced so that as a result each issue gets a majority of approvals. This problem can be modeled as a simple matrix modification problem: Can one replace k rows of a binary n x m-matrix by k all-1 rows such that each column in the resulting matrix has a majority of 1s? Significantly extending on previous work that showed parameterized intractability (W[2]-completeness) with respect to the number k of modified rows, we study how natural parameters such as n, m, k, or the "maximum number of 1s missing for any column to have a majority of 1s" (referred to as "gap value g") govern the computational complexity of Lobbying. Among other results, we prove that Lobbying is fixed-parameter tractable for parameter m and provide a greedy logarithmic-factor approximation algorithm which solves Lobbying even optimally if m < 5. We also show empirically that this greedy algorithm performs well on general instances. As a further key result, we prove that Lobbying is LOGSNP-complete for constant values g>0, thus providing a first natural complete problem from voting for this complexity class of limited nondeterminism.

scholarly journals Dense Complete Set For NP

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Formal Language ◽  
Polynomial Function ◽  
Complexity Class ◽  
Complete Problem ◽  
Complete Set ◽  
Np Complete ◽  
Dense Language ◽  
Binary Strings

A sparse language is a formal language such that the number of strings of length $n$ is bounded by a polynomial function of $n$. We create a class with the opposite definition, that is a class of languages that are dense instead of sparse. We define a dense language on $m$ as a formal language (a set of binary strings) where there exists a positive integer $n_{0}$ such that the counting of the number of strings of length $n \geq n_{0}$ in the language is greater than or equal to $2^{n - m}$ where $m$ is a real number and $0 < m \leq 1$. We call the complexity class of all dense languages on $m$ as $DENSE(m)$. We prove that there exists an $\textit{NP--complete}$ problem that belongs to $DENSE(m)$ for every possible value of $0 < m \leq 1$.

Networks of Evolutionary Processors

2011 ◽  
pp. 78-114 ◽  
Author(s):  
Carlos Martin-Vide ◽  
Victor Mitrana
Keyword(s):  
Computational Complexity ◽  
Linear Time ◽  
Complexity Class ◽  
Hybrid Networks ◽  
Computational Power ◽  
Complete Problems ◽  
Computational Aspects ◽  
Np Complete ◽  
Traditional Class

The goal of this chapter is to survey, in a systematic and uniform way, the main results regarding different computational aspects of hybrid networks of evolutionary processors viewed both as generating and accepting devices, as well as solving problems with these mechanisms. We first show that generating hybrid networks of evolutionary processors are computationally complete. The same computational power is reached by accepting hybrid networks of evolutionary processors. Then, we define a computational complexity class of accepting these networks and prove that this class equals the traditional class NP. In another section, we present a few NP-complete problems and recall how they can be solved in linear time by accepting networks of evolutionary processors with linearly bounded resources (nodes, rules, symbols). Finally, we discuss some possible directions for further research.

REFINING KNOWN RESULTS ON THE GENERALIZED WORD PROBLEM FOR FREE GROUPS

1992 ◽  
Vol 02 (02) ◽  
pp. 221-236 ◽  
Author(s):  
IAIN A. STEWART
Keyword(s):  
Word Problem ◽  
Free Groups ◽  
Complexity Classes ◽  
Finitely Generated ◽  
Complete Problem ◽  
Complete Problems

We refine the known result that the generalized word problem for finitely-generated subgroups of free groups is complete for P via logspace reductions and show that by restricting the lengths of the words in any instance and by stipulating that all words must be conjugates then we obtain complete problems for the complexity classes NSYMLOG, NL, and P. The proofs of our results range greatly: some are complexity-theoretic in nature (for example, proving completeness by reducing from another known complete problem), some are combinatorial, and one involves the characterization of complexity classes as problems describable in some logic.

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