Decisive Markov Chains

We consider qualitative and quantitative verification problems for infinite-state Markov chains. We call a Markov chain decisive w.r.t. a given set of target states F if it almost certainly eventually reaches either F or a state from which F can no longer be reached. While all finite Markov chains are trivially decisive (for every set F), this also holds for many classes of infinite Markov chains. Infinite Markov chains which contain a finite attractor are decisive w.r.t. every set F. In particular, this holds for probabilistic lossy channel systems (PLCS). Furthermore, all globally coarse Markov chains are decisive. This class includes probabilistic vector addition systems (PVASS) and probabilistic noisy Turing machines (PNTM). We consider both safety and liveness problems for decisive Markov chains, i.e., the probabilities that a given set of states F is eventually reached or reached infinitely often, respectively. 1. We express the qualitative problems in abstract terms for decisive Markov chains, and show an almost complete picture of its decidability for PLCS, PVASS and PNTM. 2. We also show that the path enumeration algorithm of Iyer and Narasimha terminates for decisive Markov chains and can thus be used to solve the approximate quantitative safety problem. A modified variant of this algorithm solves the approximate quantitative liveness problem. 3. Finally, we show that the exact probability of (repeatedly) reaching F cannot be effectively expressed (in a uniform way) in Tarski-algebra for either PLCS, PVASS or (P)NTM.Comment: 32 pages, 0 figure

Simple characterizations of P(#P) and complete problems

In this paper, P(#P) and PF(#P) are characterized in terms of a largely different computation structure, where P(#P) (resp., PF(#P)) is the class of sets (resp., functions) that are polynomial-time Turing reducible to #P functions. Let MidP be the class of functions that give the medians in the outputs of metric Turing machines, where a metric Turing machine is a polynomial-time bounded nondeterministic Turing machine such that each branch writes a binary number on an output tape. Then it is shown that every function in PF(#P) is polynomial-time one-Turing reducible to a function in MidP and MidP ⊆ PF (#P); that is, PF(#P) = PF(MidP[1]). Furthermore, it is shown that for all sets L, L is in P(#P) if and only if there is a function F ∈ MidP, such that for every string x, x ∈ L, iff F(x) is odd. Thus the problem of computing medians in the outputs of metric Turing machines captures the computational complexity of P(#P) and PF(#P). As applications of the results, several natural polynomial-time many-one complete problems for P(#P) are shown, for example, given an undirected graph with integer edge weights, checking that the parity of the middle cost among all the simple circuits is complete for P(#P)

Parameterised Counting in Logspace

Logarithmic space-bounded complexity classes such as L and NL play a central role in space-bounded computation. The study of counting versions of these complexity classes have lead to several interesting insights into the structure of computational problems such as computing the determinant and counting paths in directed acyclic graphs. Though parameterised complexity theory was initiated roughly three decades ago by Downey and Fellows, a satisfactory study of parameterised logarithmic space-bounded computation was developed only in the last decade by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space-bounded models developed by Elberfeld, Stockhusen and Tantau. They defined the operators paraW and paraβ for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators paraW and paraβ by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, paraW[1] and paraβtail. Then, we consider counting versions of all four operators and apply them to the class L. We obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0, 1)-matrices is # paraβtailL-hard and can be written as the difference of two functions in # paraβtailL. These problems exhibit the richness of the introduced counting classes. Our results further indicate interesting structural characteristics of these classes. For example, we show that the closure of # paraβtailL under parameterised logspace parsimonious reductions coincides with # paraβL. In other words, in the setting of read-once access to nondeterministic bits, tail-nondeterminism coincides with unbounded nondeterminism modulo parameterised reductions. Initiating the study of closure properties of these parameterised logspace counting classes, we show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions. Finally, we want to emphasise the significance of this topic by providing a promising outlook highlighting several open problems and directions for further research

On relation classes and solution relations

Die Dissertation On Relation Classes and Solution Relations ist in dem Gebiet der strukturellen Komplexitätstheorie einzuordnen. In einem ersten Teil wird die vollständige Inklusionsstruktur zwischen verschiedenen Relationenklassen aufgeklärt. Ein Großteil der Ergebnisse wird dabei mit Hilfe der Operatorenmethode erzielt. Im zweiten Teil werden Lösungsrelationen und easy-Sprachen betrachtet. Es wird der Fragestellung nachgegangen, welche Probleme durch eine vorgegebene Klasse von Relationen gelöst werden können

Parameterized Complexities of Dominating and Independent Set Reconfiguration

We settle the parameterized complexities of several variants of independent set reconfiguration and dominating set reconfiguration, parameterized by the number of tokens. We show that both problems are XL-complete when there is no limit on the number of moves and XNL-complete when a maximum length ? for the sequence is given in binary in the input. The problems are known to be XNLP-complete when ? is given in unary instead, and W[1]- and W[2]-hard respectively when ? is also a parameter. We complete the picture by showing membership in those classes. Moreover, we show that for all the variants that we consider, token sliding and token jumping are equivalent under pl-reductions. We introduce partitioned variants of token jumping and token sliding, and give pl-reductions between the four variants that have precise control over the number of tokens and the length of the reconfiguration sequence