Quantitative Verification and Strategy Synthesis for BDI Agents

Abstract

Belief-Desire-Intention (BDI) agents feature probabilistic outcomes, e.g. the chance an agent tries but fails to open a door, and non-deterministic choices: what plan/intention to execute next? We want to reason about agents under both probabilities and non-determinism to determine, for example, probabilities of mission success and the strategies used to maximise this. We define a Markov Decision Process describing the semantics of the Conceptual Agent Notation (Can) agent language that supports non-deterministic event, plan, and intention selection, as well as probabilistic action outcomes. The model is derived through an encoding to Milner’s Bigraphs and executed using the BigraphER tool. We show, using probabilistic model checkers PRISM and Storm, how to reason about agents including: probabilistic and reward-based properties, strategy synthesis, and multi-objective analysis. This analysis provides verification and optimisation of BDI agent design and implementation.

A Appendix

The language used in the plan-body in Can is defined by the grammar:

$\pm b \ | \ act \ | \ e \ | \ P _{1}; P _{2} \ | \ P _{1} \triangleright P _{2} \ | \ P _{1}\parallel P _{2} \ | \ goal (\varphi _{s}, P , \varphi _{f})$

where $\pm b$ stands for belief addition and deletion, act a primitive agent action, and e is a sub-event (i.e. internal event). Actions act take the form $act = \psi \leftarrow \langle \phi ^{+}, \phi ^{-} \rangle$, where $\psi$ is the pre-condition, and $\phi ^{+}$ and $\phi ^{-}$ are the addition and deletion sets (resp.) of belief atoms, i.e. a belief base $\mathcal {B}$ is revised to be $(\mathcal {B} \setminus \phi ^{-}) \cup \phi ^{+}$ when the action executes. To execute a sub-event, a plan (corresponding to that event) is selected and the plan-body added in place of the event. In this way we allow plans to be nested (similar to sub-routine calls in other languages). In addition, there are composite programs $P _{1}; P _{2}$ for sequence, $P _{1} \rhd P _{2}$ that executes $P _{2}$ in the case that $P _{1}$ fails, and $P _{1}\parallel P _{2}$ for interleaved concurrency. Finally, a declarative goal program $goal (\varphi _{s}, P , \varphi _{f})$ expresses that the declarative goal $\varphi _{s}$ should be achieved through program P, failing if $\varphi _{f}$ becomes true, and retrying as long as neither $\varphi _{s}$ nor $\varphi _{f}$ is true (see in [19] for details).

Figure 7 gives the complete set of semantic rules for evolving an intention. For example, act handles the execution of an action, when the pre-condition $\psi$ is met, resulting in a belief state update. Rule event replaces an event with the set of relevant plans, while rule select chooses an applicable plan from a set of relevant plans while retaining un-selected plans as backups. With these backup plans, the rules for failure recovery $\rhd _{;}$, $\rhd _{\top }$, and $\rhd _{\bot }$ enable new plans to be selected if the current plan fails (e.g. due to environment changes). Rules ;  and $;_{\top }$ allow executing plan-bodies in sequence, while rules $\Vert _{1}$, $\Vert _{2}$, and $\Vert _{\top }$ specify how to execute (interleaved) concurrent programs (within an intention). Rules $G_s$ and $G_f$ deal with declarative goals when either the success condition $\varphi _s$ or the failure condition $\varphi _f$ become true. Rule $G_{init}$ initialises persistence by setting the program in the declarative goal to be $P \rhd P$, i.e. if P fails try P again, and rule $G_;$ takes care of performing a single step on an already initialised program. Finally, the derivation rule $G_{\rhd }$ re-starts the original program if the current program has finished or got blocked (when neither $\varphi _s$ nor $\varphi _f$ is true).

Fig. 7.

Complete intention-level Can semantics.