Efficient algorithms for infinite-state recursive stochastic models and Newton’s method

Some well-studied infinite-state stochastic models give rise to systems of nonlinear equations. These systems of equations have solutions that are probabilities, generally probabilities of termination in the model. We are interested in finding efficient, preferably polynomial time, algorithms for calculating probabilities associated with these models. The chief tool we use to solve systems of polynomial equations will be Newton’s method as suggested by [EY09]. The main contribution of this thesis is to the analysis of this and related algorithms. We give polynomial-time algorithms for calculating probabilities for broad classes of models for which none were known before. Stochastic models that give rise to such systems of equations include such classic and heavily-studied models as Multi-type Branching Processes, Stochastic Context- Free Grammars(SCFGs) and Quasi Birth-Death Processes. We also consider models that give rise to infinite-state Markov Decision Processes (MDPs) by giving algorithms for approximating optimal probabilities and finding policies that give probabilities close to the optimal probability, in several classes of infinite-state MDPs. Our algorithms for analysing infinite-state MDPs rely on a non-trivial generalization of Newton’s method that works for the max/min polynomial systems that arise as Bellman optimality equations in these models. For SCFGs, which are used in statistical natural language processing, in addition to approximating termination probabilities, we analyse algorithms for approximating the probability that a grammar produces a given string, or produces a string in a given regular language. In most cases, we show that we can calculate an approximation to the relevant probability in time polynomial in the size of the model and the number of bits of desired precision. We also consider more general systems of monotone polynomial equations. For such systems we cannot give a polynomial-time algorithm, which pre-existing hardness results render unlikely, but we can still give an algorithm with a complexity upper bound which is exponential only in some parameters that are likely to be bounded for the monotone polynomial equations that arise for many interesting stochastic models

Linear-Time Model Checking Branching Processes

(Multi-type) branching processes are a natural and well-studied model for generating random infinite trees. Branching processes feature both nondeterministic and probabilistic branching, generalizing both transition systems and Markov chains (but not generally Markov decision processes). We study the complexity of model checking branching processes against linear-time omega-regular specifications: is it the case almost surely that every branch of a tree randomly generated by the branching process satisfies the omega-regular specification? The main result is that for LTL specifications this problem is in PSPACE, subsuming classical results for transition systems and Markov chains, respectively. The underlying general model-checking algorithm is based on the automata-theoretic approach, using unambiguous Büchi automata

Stochastic Context-Free Grammars, Regular Languages, and Newton's Method

We study the problem of computing the probability that a given stochastic context-free grammar (SCFG), G, generates a string in a given regular language L(D) (given by a DFA, D). This basic problem has a number of applications in statistical natural language processing, and it is also a key necessary step towards quantitative \omega-regular model checking of stochastic context-free processes (equivalently, 1-exit recursive Markov chains, or stateless probabilistic pushdown processes). We show that the probability that G generates a string in L(D) can be computed to within arbitrary desired precision in polynomial time (in the standard Turing model of computation), under a rather mild assumption about the SCFG, G, and with no extra assumption about D. We show that this assumption is satisfied for SCFG's whose rule probabilities are learned via the well-known inside-outside (EM) algorithm for maximum-likelihood estimation (a standard method for constructing SCFGs in statistical NLP and biological sequence analysis). Thus, for these SCFGs the algorithm always runs in P-time

Turku Centre for Computer Science – Annual Report 2013

Due to a major reform of organization and responsibilities of TUCS, its role, activities, and even structures have been under reconsideration in 2013. The traditional pillar of collaboration at TUCS, doctoral training, was reorganized due to changes at both universities according to the renewed national system for doctoral education. Computer Science and Engineering and Information Systems Science are now accompanied by Mathematics and Statistics in newly established doctoral programs at both University of Turku and &Aring;bo Akademi University. Moreover, both universities granted sufficient resources to their respective programmes for doctoral training in these fields, so that joint activities at TUCS can continue. The outcome of this reorganization has the potential of proving out to be a success in terms of scientific profile as well as the quality and quantity of scientific and educational results.&nbsp; International activities that have been characteristic to TUCS since its inception continue strong. TUCS&rsquo; participation in European collaboration through EIT ICT Labs Master&rsquo;s and Doctoral School is now more active than ever. The new double degree programs at MSc and PhD level between University of Turku and Fudan University in Shaghai, P.R.China were succesfully set up and are&nbsp; now running for their first year. The joint students will add to the already international athmosphere of the ICT House.&nbsp; The four new thematic reseach programmes set up acccording to the decision by the TUCS Board have now established themselves, and a number of events and other activities saw the light in 2013. The TUCS Distinguished Lecture Series managed to gather a large audience with its several prominent speakers. The development of these and other research centre activities continue, and&nbsp; new practices and structures will be initiated to support the tradition of close academic collaboration.&nbsp; The TUCS&rsquo; slogan Where Academic Tradition Meets the Exciting Future has proven true throughout these changes. Despite of the dark clouds on the national and European economic sky, science and higher education in the field have managed to retain all the key ingredients for success. Indeed, the future of ICT and Mathematics in Turku seems exciting.</p

Generalized asset integrity games

Generalized assets represent a class of multi-scale adaptive state-transition systems with domain-oblivious performance criteria. The governance of such assets must proceed without exact specifications, objectives, or constraints. Decision making must rapidly scale in the presence of uncertainty, complexity, and intelligent adversaries. This thesis formulates an architecture for generalized asset planning. Assets are modelled as dynamical graph structures which admit topological performance indicators, such as dependability, resilience, and efficiency. These metrics are used to construct robust model configurations. A normalized compression distance (NCD) is computed between a given active/live asset model and a reference configuration to produce an integrity score. The utility derived from the asset is monotonically proportional to this integrity score, which represents the proximity to ideal conditions. The present work considers the situation between an asset manager and an intelligent adversary, who act within a stochastic environment to control the integrity state of the asset. A generalized asset integrity game engine (GAIGE) is developed, which implements anytime algorithms to solve a stochastically perturbed two-player zero-sum game. The resulting planning strategies seek to stabilize deviations from minimax trajectories of the integrity score. Results demonstrate the performance and scalability of the GAIGE. This approach represents a first-step towards domain-oblivious architectures for complex asset governance and anytime planning

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum