Probabilistic reasoning about epistemic action narratives

Abstract

We propose the action language EPEC – Epistemic Probabilistic Event Calculus – that supports probabilistic, epistemic reasoning about narratives of action occurrences and environmentally triggered events, and in particular facilitates reasoning about future belief-conditioned actions and their consequences in domains that include both perfect and imperfect sensing actions. To provide a declarative semantics for sensing and belief conditioned actions in a probabilistic, narrative setting we introduce the novel concept of an epistemic reduct. We then formally compare our language with two established frameworks for probabilistic reasoning about action – the action language PAL by Baral et al., and the extension of the situation calculus to reason about noisy sensors and effectors by Bacchus et al. In both cases we prove a correspondence with EPEC for a class of domains representable in both frameworks.

Introduction

The action language EPEC – Epistemic Probabilistic Event Calculus – described in this article combines probabilistic reasoning, epistemic reasoning, reasoning about the general effects of actions, and reasoning about particular narratives of action occurrences (i.e. events along an explicitly represented time-line). Each of these topics has its own venerable history of AI research, and notable work has already been done in some sub-combinations. For example, the work of Moore, Scherl, Levesque, Belle, Bacchus and others concerns epistemic (and sometimes also probabilistic) reasoning about actions (see e.g. [1], [2], [3] [4]), the work of Ma et al. [5] facilitates epistemic reasoning about narratives of events, and the work of Baral, Skarlatidis, Artikis and others focuses on probabilistic reasoning about action narratives (see e.g. [6], [7]). (See Section 4.3 for a full discussion of related work.) However, to our knowledge little or no previous research has been undertaken towards a full integration of all four of these topics, and we aim to demonstrate the benefits of such an integration in this paper. EPEC is related to PEC (Probabilistic Event Calculus) [8], an earlier probabilistic framework for narrative reasoning that did not contain any epistemic features. The utility of our EPEC framework is partly illustrated by the following (imaginary) example medical scenario.

Example 1.1 Epectisis

A doctor is 95% certain that a patient has made skin contact with her, and 80% certain that this patient is suffering from epectisis, a rare disease caused by a bacterial infection. With such contact, epectisis is typically passed on 75% of the time. A course of epecillin is known to eliminate the disease 99% of the time if taken before symptoms manifest themselves, although with a 15% risk of side effects. A blood test gives a pre-symptom indication of the disease, but with a 10% false positive and a 5% false negative error rate. The doctor decides that she will undertake the blood test, and if after this she still has a more than 50% belief that she is infected she will take a course of epecillin. She reasons that, assuming that she did not have epectitis prior to contact with her patient, in this way she will eventually be at least 93.1% sure that she does not have the disease, while giving herself only a 2.875% chance of suffering epecillin's occasional side effects. (A probability tree diagram of this domain is given in Appendix B.)

The scenario above has a number of interesting features, all of which can be represented in our EPEC framework (see Appendix C for the corresponding EPEC domain description ${\mathcal{D}}_e$ and example entailments). First, it includes a narrative, in this case containing a single probable past event – the doctor is 95% certain that she had skin contact with a patient. In deciding a course of action, the doctor appends two future events to this narrative – performing a blood test and (conditionally) taking epecillin. This reflects an abductive view of plan specification commonplace in the context of event-calculus-like frameworks (see e.g. [9]). Second, some of the causal information about actions is probabilistic – in general, contact has a 75% probability of causing infection, and taking epecillin has a 15% probability of causing a side effect. Third, one of the actions mentioned – performing a blood test – is a sensing action in that it has an effect on the doctor's (probabilistic) knowledge, so that the doctor's plan has an epistemic dimension. Moreover, the sensing is imperfect, with the possibility of false positives and negatives. Fourth, the doctor's plan includes a conditional action, conditioned on a future belief state – if after performing the blood test she has a strong belief that she is infected then she will take the medicine. We view this as a key feature of epistemic planning, in that sensing actions and actions conditioned on (revised) beliefs resulting directly or indirectly from sensing outcomes must go hand-in-hand, or there is little point in including sensing actions within a plan. A principal advantage of EPEC is that it allows for explicit representation and probabilistic reasoning about future belief-conditioned actions and their consequences in domains that include both perfect and imperfect sensing actions. This is a key contribution of our work.

The medical domain is one of a number of application areas where we envisage EPEC having a useful role. Other domains in which A.I. and knowledge-based applications have to reason with imperfect sensor input include cognitive and mobile robotics, and physical monitoring systems. Not all of the representational features of EPEC are illustrated by the example above. Descriptions of more complex domains may for example include non-Boolean-valued properties (fluents), concurrent actions, conflicting concurrent noisy sensory inputs, and, importantly, events (probabilistically) triggered by the environment under certain conditions. This latter feature in turn allows EPEC to be used to model domains involving decay and analogous dynamic behaviours.

To reflect the high degree of non-determinism inherent in probabilistic models, as well as the epistemic nature of EPEC domains, EPEC's semantics uses a structure of “possible worlds”, each with its own timeline and overall probability attached. The semantics is developed in two stages. First, the “non-epistemic” case is considered, where all action occurrences are considered to be executed by the environment and are independent from the agent's belief state. Second, the semantics is generalised to the epistemic case, where possible worlds are considered to have two components – a timeline of actual environmental conditions and events, together with history of the agent's sensory experience and decisions regarding its own actions. We show how the notion of an epistemic reduct can be used to model this second case in terms of the first, while taking into account the agent's sensory input, changing belief state and associated decision making process as time progresses.

In order to progress understanding of the space of formalisms available for probabilistic reasoning about actions, we conclude with an investigation of the relation of EPEC to two established frameworks in this area of research. These are the action language PAL developed by Baral, Tran and Tuan [6], and the extension of the situation calculus to reason about noisy sensors and effectors by Bacchus, Halpern and Levesque [3]. In both cases we provide a general translation procedure of a class of domains written in these languages into EPEC, and prove that probabilistic entailment is preserved under the translations.

In summary, the main contribution of this paper is the formulation of an action language, EPEC (Epistemic Probabilistic Event Calculus), and associated semantics that supports probabilistic, epistemic reasoning about narratives of (potentially simultaneous) action occurrences and environmentally triggered events, and in particular facilitates reasoning about future belief-conditioned actions and their consequences in domains that include both perfect and imperfect sensing actions, and potentially simultaneous and/or conflicting sensory inputs. Additional contributions are formal comparisons with two established frameworks for probabilistic reasoning about actions by provably correct translations into EPEC.

The paper is organised as follows. Section 2 gives some general background about the topic areas related to this research. Section 3 describes the syntax, semantics and key properties of EPEC. Section 4 describes the translations into EPEC of domains written in the frameworks in [6] and [3], and follows with a wider discussion of related work. In Section 5 we briefly comment on ongoing experiments implementing EPEC. Section 6 concludes the paper with a final summary and remarks about possible future directions of research.