We develop a pseudo-metric analogue of bisimulation for generalized semi-Markov processes. The kernel of this pseudo-metric corresponds to bisimulation; thus we have extended bisimulation for continuous-time probabilistic processes to a much broader class of distributions than exponential distributions. This pseudo-metric gives a useful handle on approximate reasoning in the presence of numerical information -- such as probabilities and time -- in the model. We give a fixed point characterization of the pseudo-metric. This makes available coinductive reasoning principles for reasoning about distances. We demonstrate that our approach is insensitive to potentially ad hoc articulations of distance by showing that it is intrinsic to an underlying uniformity. We provide a logical characterization of this uniformity using a real-valued modal logic. We show that several quantitative properties of interest are continuous with respect to the pseudo-metric. Thus, if two processes are metrically close, then observable quantitative properties of interest are indeed close.
We study which standard operators of probabilistic process calculi allow for compositional reasoning with respect to bisimulation metric semantics. We argue that uniform continuity (generalizing the earlier proposed property of non-expansiveness) captures the essential nature of compositional reasoning and allows now also to reason compositionally about recursive processes. We characterize the distance between probabilistic processes composed by standard process algebra operators. Combining these results, we demonstrate how compositional reasoning about systems specified by continuous process …
This paper presents a wp-style calculus for obtaining expectations on the outcomes of (mutually) recursive probabilistic programs. We provide several proof rules to derive one-- and two--sided bounds for such expectations, and show the soundness of our wp-calculus with respect to a probabilistic pushdown automaton semantics. We also give a wp-style calculus for obtaining bounds on the expected runtime of recursive programs that can be used to determine the (possibly infinite) time until termination of such programs.
We consider multi-agent systems where agents actions and beliefs are determined aleatorically, or "by the throw of dice". This system consists of possible worlds that assign distributions to independent random variables, and agents who assign probabilities to these possible worlds. We present a novel syntax and semantics for such system, and show that they generalise Modal Logic. We also give a sound and complete calculus for reasoning in the base semantics, and a sound calculus for the full modal semantics, that we conjecture to be complete. Finally we discuss some application to reasoning ab…
We propose a formalization of the three-tier causal hierarchy of association, intervention, and counterfactuals as a series of probabilistic logical languages. Our languages are of strictly increasing expressivity, the first capable of expressing quantitative probabilistic reasoning -- including conditional independence and Bayesian inference -- the second encoding do-calculus reasoning for causal effects, and the third capturing a fully expressive do-calculus for arbitrary counterfactual queries. We give a corresponding series of finitary axiomatizations complete over both structural causal m…
Expectation is a central notion in probability theory. The notion of expectation also makes sense for other notions of uncertainty. We introduce a propositional logic for reasoning about expectation, where the semantics depends on the underlying representation of uncertainty. We give sound and complete axiomatizations for the logic in the case that the underlying representation is (a) probability, (b) sets of probability measures, (c) belief functions, and (d) possibility measures. We show that this logic is more expressive than the corresponding logic for reasoning about likelihood in the cas…
Probabilistic transition system specifications using the rule format ntmuft-ntmuxt provide structural operational semantics for Segala-type systems and guarantee that probabilistic bisimilarity is a congruence. Probabilistic bisimilarity is for many applications too sensitive to the exact probabilities of transitions. Approximate bisimulation provides a robust semantics that is stable with respect to implementation and measurement errors of probabilistic behavior. We provide a general method to quantify how much a process combinator expands the approximate bisimulation distance. As a direct ap…
Model checking probabilistic CTL properties of Markov decision processes with convex uncertainties has been recently investigated by Puggelli et al. Such model checking algorithms typically suffer from the state space explosion. In this paper, we address probabilistic bisimulation to reduce the size of such an MDP while preserving the probabilistic CTL properties it satisfies. In particular, we discuss the key ingredients to build up the operations of parallel composition for composing interval MDP components at run-time. More precisely, we investigate how the parallel composition operator for…
Free University of Bozen-Bolzano（博尔扎诺自由大学）；University of Osnabrück（奥斯纳布吕克大学）
A central question for knowledge representation is how to encode and handle uncertain knowledge adequately. We introduce the probabilistic description logic ALCP that is designed for representing context-dependent knowledge, where the actual context taking place is uncertain. ALCP allows the expression of logical dependencies on the domain and probabilistic dependencies on the possible contexts. In order to draw probabilistic conclusions, we employ the principle of maximum entropy. We provide reasoning algorithms for this logic, and show that it satisfies several desirable properties of probab…
University of Parma（帕尔马大学）；University of Ferrara（费拉拉大学）
While there exist several reasoners for Description Logics, very few of them can cope with uncertainty. BUNDLE is an inference framework that can exploit several OWL (non-probabilistic) reasoners to perform inference over Probabilistic Description Logics. In this chapter, we report the latest advances implemented in BUNDLE. In particular, BUNDLE can now interface with the reasoners of the TRILL system, thus providing a uniform method to execute probabilistic queries using different settings. BUNDLE can be easily extended and can be used either as a standalone desktop application or as a librar…
The probability theory is a well-studied branch of mathematics, in order to carry out formal reasoning about probability. Thus, it is important to have a logic, both for computation of probabilities and for reasoning about probabilities, with a well-defined syntax and semantics. Both current approaches, based on Nilsson's probability structures/logics, and on linear inequalities in order to reason about probabilities, have some weak points. In this paper we have presented the complete revision of both approaches. We have shown that the full embedding of Nilsson'probabilistic structure into pro…
Probabilistic coupling is a powerful tool for analyzing pairs of probabilistic processes. Roughly, coupling two processes requires finding an appropriate witness process that models both processes in the same probability space. Couplings are powerful tools proving properties about the relation between two processes, include reasoning about convergence of distributions and stochastic dominance---a probabilistic version of a monotonicity property. While the mathematical definition of coupling looks rather complex and cumbersome to manipulate, we show that the relational program logic pRHL---the …
Arizona State University（亚利桑那州立大学）；Texas Tech University（德克萨斯理工大学）
This paper develops a declarative language, P-log, that combines logical and probabilistic arguments in its reasoning. Answer Set Prolog is used as the logical foundation, while causal Bayes nets serve as a probabilistic foundation. We give several non-trivial examples and illustrate the use of P-log for knowledge representation and updating of knowledge. We argue that our approach to updates is more appealing than existing approaches. We give sufficiency conditions for the coherency of P-log programs and show that Bayes nets can be easily mapped to coherent P-log programs.
This paper describes diff-SAT, an Answer Set and SAT solver which combines regular solving with the capability to use probabilistic clauses, facts and rules, and to sample an optimal world-view (multiset of satisfying Boolean variable assignments or answer sets) subject to user-provided probabilistic constraints. The sampling process minimizes a user-defined differentiable objective function using a gradient descent based optimization method called Differentiable Satisfiability Solving ($\partial\mathrm{SAT}$) respectively Differentiable Answer Set Programming ($\partial\mathrm{ASP}$). Use cas…
This paper presents Tyche, a Python library to facilitate probabilistic reasoning in uncertain worlds through the construction, querying, and learning of belief models. Tyche uses aleatoric description logic (ADL), which provides computational advantages in its evaluation over other description logics. Tyche belief models can be succinctly created by defining classes of individuals, the probabilistic beliefs about them (concepts), and the probabilistic relationships between them (roles). We also introduce a method of observation propagation to facilitate learning from complex ADL observations.…
This paper presents a wp-style calculus for obtaining bounds on the expected run-time of probabilistic programs. Its application includes determining the (possibly infinite) expected termination time of a probabilistic program and proving positive almost-sure termination - does a program terminate with probability one in finite expected time? We provide several proof rules for bounding the run-time of loops, and prove the soundness of the approach with respect to a simple operational model. We show that our approach is a conservative extension of Nielson's approach for reasoning about the run-…
This work introduces a notion of approximate probabilistic trace equivalence for labelled Markov chains, and relates this new concept to the known notion of approximate probabilistic bisimulation. In particular this work shows that the latter notion induces a tight upper bound on the approximation between finite-horizon traces, as expressed by a total variation distance. As such, this work extends corresponding results for exact notions and analogous results for non-probabilistic models. This bound can be employed to relate the closeness in satisfaction probabilities over bounded linear-time p…
This paper introduces a new constraint domain for reasoning about data with uncertainty. It extends convex modeling with the notion of p-box to gain additional quantifiable information on the data whereabouts. Unlike existing approaches, the p-box envelops an unknown probability instead of approximating its representation. The p-box bounds are uniform cumulative distribution functions (cdf) in order to employ linear computations in the probabilistic domain. The reasoning by means of p-box cdf-intervals is an interval computation which is exerted on the real domain then it is projected onto the…
We study weakest precondition reasoning about the (co)variance of outcomes and the variance of run-times of probabilistic programs with conditioning. For outcomes, we show that approximating (co)variances is computationally more difficult than approximating expected values. In particular, we prove that computing both lower and upper bounds for (co)variances is $Σ^{0}_{2}$-complete. As a consequence, neither lower nor upper bounds are computably enumerable. We therefore present invariant-based techniques that do enable enumeration of both upper and lower bounds, once appropriate invariants are …
We present quantitative separation logic ($\mathsf{QSL}$). In contrast to classical separation logic, $\mathsf{QSL}$ employs quantities which evaluate to real numbers instead of predicates which evaluate to Boolean values. The connectives of classical separation logic, separating conjunction and separating implication, are lifted from predicates to quantities. This extension is conservative: Both connectives are backward compatible to their classical analogs and obey the same laws, e.g. modus ponens, adjointness, etc. Furthermore, we develop a weakest precondition calculus for quantitative rea…
University of California, Berkeley（加州大学伯克利分校）；Stanford University（斯坦福大学）
Qualitative and quantitative approaches to reasoning about uncertainty can lead to different logical systems for formalizing such reasoning, even when the language for expressing uncertainty is the same. In the case of reasoning about relative likelihood, with statements of the form $φ\succsimψ$ expressing that $φ$ is at least as likely as $ψ$, a standard qualitative approach using preordered preferential structures yields a dramatically different logical system than a quantitative approach using probability measures. In fact, the standard preferential approach validates principles of reasonin…
Probabilistic logic reasoning is a central component of such cognitive architectures as OpenCog. However, as an integrative architecture, OpenCog facilitates cognitive synergy via hybridization of different inference methods. In this paper, we introduce a differentiable version of Probabilistic Logic networks, which rules operate over tensor truth values in such a way that a chain of reasoning steps constructs a computation graph over tensors that accepts truth values of premises from the knowledge base as input and produces truth values of conclusions as output. This allows for both learning …
We present a framework to formally describe probabilistic system behavior and symbolically reason about it. In particular we aim at reasoning about possible failures and fault tolerance. We regard systems which are composed of different units: sensors, computational parts and actuators. Considering worst-case failure behavior of system components, our framework is most suited to derive reliability guarantees for composed systems. The behavior of system components is modeled using monad like constructs that serve as an abstract representation for system behavior. We introduce rules to reason ab…
Standard approaches to probabilistic reasoning require that one possesses an explicit model of the distribution in question. But, the empirical learning of models of probability distributions from partial observations is a problem for which efficient algorithms are generally not known. In this work we consider the use of bounded-degree fragments of the "sum-of-squares" logic as a probability logic. Prior work has shown that we can decide refutability for such fragments in polynomial-time. We propose to use such fragments to answer queries about whether a given probability distribution satisfie…
Free University of Bozen-Bolzano（博尔扎诺-博岑自由大学）；University of Osnabrück（奥斯纳布吕克大学）
We present a probabilistic extension of the description logic $\mathcal{ALC}$ for reasoning about statistical knowledge. We consider conditional statements over proportions of the domain and are interested in the probabilistic-logical consequences of these proportions. After introducing some general reasoning problems and analyzing their properties, we present first algorithms and complexity results for reasoning in some fragments of Statistical $\mathcal{ALC}$.
Effectively combining logic reasoning and probabilistic inference has been a long-standing goal of machine learning: the former has the ability to generalize with small training data, while the latter provides a principled framework for dealing with noisy data. However, existing methods for combining the best of both worlds are typically computationally intensive. In this paper, we focus on Markov Logic Networks and explore the use of graph neural networks (GNNs) for representing probabilistic logic inference. It is revealed from our analysis that the representation power of GNN alone is not e…
This thesis explores proofs by coupling from the perspective of formal verification. Long employed in probability theory and theoretical computer science, these proofs construct couplings between the output distributions of two probabilistic processes. Couplings can imply various guarantees comparing two runs of a probabilistic computation. We first show that proofs in the program logic pRHL describe couplings. We formalize couplings that establish various probabilistic properties, including distribution equivalence, convergence, and stochastic domination. Then we give a proofs-as-programs int…
Institute for Research in Fundamental Sciences (IPM)（基础科学研究所（IPM））；The University of Quebec at Trois-Rivieres（魁北克大学三河分校）
In this paper, we introduce a fundamental framework to create a bridge between Probability Theory and Fuzzy Logic. Indeed, our theory formulates a random experiment of selecting crisp elements with the criterion of having a certain fuzzy attribute. To do so, we associate some specific crisp random variables to the random experiment. Then, several formulas are presented, which make it easier to compute different conditional probabilities and expected values of these random variables. Also, we provide measure theoretical basis for our probabilistic fuzzy logic framework. Note that in our theory,…
Robots assisting humans in complex domains have to represent knowledge and reason at both the sensorimotor level and the social level. The architecture described in this paper couples the non-monotonic logical reasoning capabilities of a declarative language with probabilistic belief revision, enabling robots to represent and reason with qualitative and quantitative descriptions of knowledge and degrees of belief. Specifically, incomplete domain knowledge, including information that holds in all but a few exceptional situations, is represented as a Answer Set Prolog (ASP) program. The answer s…
We propose a hybrid probabilistic process calculus for modelling and reasoning on cyber-physical systems (CPSs). The dynamics of the calculus is expressed in terms of a probabilistic labelled transition system in the SOS style of Plotkin. This is used to define a bisimulation-based probabilistic behavioural semantics which supports compositional reasonings. For a more careful comparison between CPSs, we provide two compositional probabilistic metrics to formalise the notion of behavioural distance between systems, also in the case of bounded computations. Finally, we provide a non-trivial case…
