Bayesian Logic Programs

Bayesian networks provide an elegant formalism for representing and reasoning about uncertainty using probability theory. Theyare a probabilistic extension of propositional logic and, hence, inherit some of the limitations of propositional logic, such as the difficulties to represent objects and relations. We introduce a generalization of Bayesian networks, called Bayesian logic programs, to overcome these limitations. In order to represent objects and relations it combines Bayesian networks with definite clause logic by establishing a one-to-one mapping between ground atoms and random variables. We show that Bayesian logic programs combine the advantages of both definite clause logic and Bayesian networks. This includes the separation of quantitative and qualitative aspects of the model. Furthermore, Bayesian logic programs generalize both Bayesian networks as well as logic programs. So, many ideas developedComment: 52 page

Qualitative Analysis of POMDPs with Temporal Logic Specifications for Robotics Applications

We consider partially observable Markov decision processes (POMDPs), that are a standard framework for robotics applications to model uncertainties present in the real world, with temporal logic specifications. All temporal logic specifications in linear-time temporal logic (LTL) can be expressed as parity objectives. We study the qualitative analysis problem for POMDPs with parity objectives that asks whether there is a controller (policy) to ensure that the objective holds with probability 1 (almost-surely). While the qualitative analysis of POMDPs with parity objectives is undecidable, recent results show that when restricted to finite-memory policies the problem is EXPTIME-complete. While the problem is intractable in theory, we present a practical approach to solve the qualitative analysis problem. We designed several heuristics to deal with the exponential complexity, and have used our implementation on a number of well-known POMDP examples for robotics applications. Our results provide the first practical approach to solve the qualitative analysis of robot motion planning with LTL properties in the presence of uncertainty

Belief as Willingness to Bet

We investigate modal logics of high probability having two unary modal operators: an operator $K$ expressing probabilistic certainty and an operator $B$ expressing probability exceeding a fixed rational threshold $c\geq\frac 12$. Identifying knowledge with the former and belief with the latter, we may think of $c$ as the agent's betting threshold, which leads to the motto "belief is willingness to bet." The logic $\mathsf{KB.5}$ for $c=\frac 12$ has an $\mathsf{S5}$ $K$ modality along with a sub-normal $B$ modality that extends the minimal modal logic $\mathsf{EMND45}$ by way of four schemes relating $K$ and $B$, one of which is a complex scheme arising out of a theorem due to Scott. Lenzen was the first to use Scott's theorem to show that a version of this logic is sound and complete for the probability interpretation. We reformulate Lenzen's results and present them here in a modern and accessible form. In addition, we introduce a new epistemic neighborhood semantics that will be more familiar to modern modal logicians. Using Scott's theorem, we provide the Lenzen-derivative properties that must be imposed on finite epistemic neighborhood models so as to guarantee the existence of a probability measure respecting the neighborhood function in the appropriate way for threshold $c=\frac 12$. This yields a link between probabilistic and modal neighborhood semantics that we hope will be of use in future work on modal logics of qualitative probability. We leave open the question of which properties must be imposed on finite epistemic neighborhood models so as to guarantee existence of an appropriate probability measure for thresholds $c\neq\frac 12$.Comment: Removed date from v1 to avoid confusion on citation/reference, otherwise identical to v

Probabilistic modal {\mu}-calculus with independent product

The probabilistic modal {\mu}-calculus is a fixed-point logic designed for expressing properties of probabilistic labeled transition systems (PLTS's). Two equivalent semantics have been studied for this logic, both assigning to each state a value in the interval [0,1] representing the probability that the property expressed by the formula holds at the state. One semantics is denotational and the other is a game semantics, specified in terms of two-player stochastic parity games. A shortcoming of the probabilistic modal {\mu}-calculus is the lack of expressiveness required to encode other important temporal logics for PLTS's such as Probabilistic Computation Tree Logic (PCTL). To address this limitation we extend the logic with a new pair of operators: independent product and coproduct. The resulting logic, called probabilistic modal {\mu}-calculus with independent product, can encode many properties of interest and subsumes the qualitative fragment of PCTL. The main contribution of this paper is the definition of an appropriate game semantics for this extended probabilistic {\mu}-calculus. This relies on the definition of a new class of games which generalize standard two-player stochastic (parity) games by allowing a play to be split into concurrent subplays, each continuing their evolution independently. Our main technical result is the equivalence of the two semantics. The proof is carried out in ZFC set theory extended with Martin's Axiom at an uncountable cardinal

Qualitative Logics and Equivalences for Probabilistic Systems

We investigate logics and equivalence relations that capture the qualitative behavior of Markov Decision Processes (MDPs). We present Qualitative Randomized CTL (QRCTL): formulas of this logic can express the fact that certain temporal properties hold over all paths, or with probability 0 or 1, but they do not distinguish among intermediate probability values. We present a symbolic, polynomial time model-checking algorithm for QRCTL on MDPs. The logic QRCTL induces an equivalence relation over states of an MDP that we call qualitative equivalence: informally, two states are qualitatively equivalent if the sets of formulas that hold with probability 0 or 1 at the two states are the same. We show that for finite alternating MDPs, where nondeterministic and probabilistic choices occur in different states, qualitative equivalence coincides with alternating bisimulation, and can thus be computed via efficient partition-refinement algorithms. On the other hand, in non-alternating MDPs the equivalence relations cannot be computed via partition-refinement algorithms, but rather, they require non-local computation. Finally, we consider QRCTL*, that extends QRCTL with nested temporal operators in the same manner in which CTL* extends CTL. We show that QRCTL and QRCTL* induce the same qualitative equivalence on alternating MDPs, while on non-alternating MDPs, the equivalence arising from QRCTL* can be strictly finer. We also provide a full characterization of the relation between qualitative equivalence, bisimulation, and alternating bisimulation, according to whether the MDPs are finite, and to whether their transition relations are finitely-branching.Comment: The paper is accepted for LMC

IST Austria Technical Report

We consider partially observable Markov decision processes (POMDPs), that are a standard framework for robotics applications to model uncertainties present in the real world, with temporal logic specifications. All temporal logic specifications in linear-time temporal logic (LTL) can be expressed as parity objectives. We study the qualitative analysis problem for POMDPs with parity objectives that asks whether there is a controller (policy) to ensure that the objective holds with probability 1 (almost-surely). While the qualitative analysis of POMDPs with parity objectives is undecidable, recent results show that when restricted to finite-memory policies the problem is EXPTIME-complete. While the problem is intractable in theory, we present a practical approach to solve the qualitative analysis problem. We designed several heuristics to deal with the exponential complexity, and have used our implementation on a number of well-known POMDP examples for robotics applications. Our results provide the first practical approach to solve the qualitative analysis of robot motion planning with LTL properties in the presence of uncertainty

IST Austria Technical Report

We consider partially observable Markov decision processes (POMDPs), that are a standard framework for robotics applications to model uncertainties present in the real world, with temporal logic specifications. All temporal logic specifications in linear-time temporal logic (LTL) can be expressed as parity objectives. We study the qualitative analysis problem for POMDPs with parity objectives that asks whether there is a controller (policy) to ensure that the objective holds with probability 1 (almost-surely). While the qualitative analysis of POMDPs with parity objectives is undecidable, recent results show that when restricted to finite-memory policies the problem is EXPTIME-complete. While the problem is intractable in theory, we present a practical approach to solve the qualitative analysis problem. We designed several heuristics to deal with the exponential complexity, and have used our implementation on a number of well-known POMDP examples for robotics applications. Our results provide the first practical approach to solve the qualitative analysis of robot motion planning with LTL properties in the presence of uncertainty

Explicit Bayesian analysis for process tracing: guidelines, opportunities, and caveats

Bayesian probability holds the potential to serve as an important bridge between qualitative and quantitative methodology. Yet whereas Bayesian statistical techniques have been successfully elaborated for quantitative research, applying Bayesian probability to qualitative research remains an open frontier. This paper advances the burgeoning literature on Bayesian process tracing by drawing on expositions of Bayesian “probability as extended logic” from the physical sciences, where probabilities represent rational degrees of belief in propositions given the inevitably limited information we possess. We provide step-by-step guidelines for explicit Bayesian process tracing, calling attention to technical points that have been overlooked or inadequately addressed, and we illustrate how to apply this approach with the first systematic application to a case study that draws on multiple pieces of detailed evidence. While we caution that efforts to explicitly apply Bayesian learning in qualitative social science will inevitably run up against the difficulty that probabilities cannot be unambiguously specified, we nevertheless envision important roles for explicit Bayesian analysis in pinpointing the locus of contention when scholars disagree on inferences, and in training intuition to follow Bayesian probability more systematically

IST Austria Thesis

This dissertation concerns the automatic verification of probabilistic systems and programs with arrays by statistical and logical methods. Although statistical and logical methods are different in nature, we show that they can be successfully combined for system analysis. In the first part of the dissertation we present a new statistical algorithm for the verification of probabilistic systems with respect to unbounded properties, including linear temporal logic. Our algorithm often performs faster than the previous approaches, and at the same time requires less information about the system. In addition, our method can be generalized to unbounded quantitative properties such as mean-payoff bounds. In the second part, we introduce two techniques for comparing probabilistic systems. Probabilistic systems are typically compared using the notion of equivalence, which requires the systems to have the equal probability of all behaviors. However, this notion is often too strict, since probabilities are typically only empirically estimated, and any imprecision may break the relation between processes. On the one hand, we propose to replace the Boolean notion of equivalence by a quantitative distance of similarity. For this purpose, we introduce a statistical framework for estimating distances between Markov chains based on their simulation runs, and we investigate which distances can be approximated in our framework. On the other hand, we propose to compare systems with respect to a new qualitative logic, which expresses that behaviors occur with probability one or a positive probability. This qualitative analysis is robust with respect to modeling errors and applicable to many domains. In the last part, we present a new quantifier-free logic for integer arrays, which allows us to express counting. Counting properties are prevalent in array-manipulating programs, however they cannot be expressed in the quantified fragments of the theory of arrays. We present a decision procedure for our logic, and provide several complexity results

Verifying nondeterministic probabilistic channel systems against ω\omega-regular linear-time properties

Lossy channel systems (LCSs) are systems of finite state automata that communicate via unreliable unbounded fifo channels. In order to circumvent the undecidability of model checking for nondeterministic LCSs, probabilistic models have been introduced, where it can be decided whether a linear-time property holds almost surely. However, such fully probabilistic systems are not a faithful model of nondeterministic protocols. We study a hybrid model for LCSs where losses of messages are seen as faults occurring with some given probability, and where the internal behavior of the system remains nondeterministic. Thus the semantics is in terms of infinite-state Markov decision processes. The purpose of this article is to discuss the decidability of linear-time properties formalized by formulas of linear temporal logic (LTL). Our focus is on the qualitative setting where one asks, e.g., whether a LTL-formula holds almost surely or with zero probability (in case the formula describes the bad behaviors). Surprisingly, it turns out that -- in contrast to finite-state Markov decision processes -- the satisfaction relation for LTL formulas depends on the chosen type of schedulers that resolve the nondeterminism. While all variants of the qualitative LTL model checking problem for the full class of history-dependent schedulers are undecidable, the same questions for finite-memory scheduler can be solved algorithmically. However, the restriction to reachability properties and special kinds of recurrent reachability properties yields decidable verification problems for the full class of schedulers, which -- for this restricted class of properties -- are as powerful as finite-memory schedulers, or even a subclass of them.Comment: 39 page