A Logical Account of Prioritized Goals and their Dynamics

Most previous logical accounts of goals do not deal with prioritized goals and goal dynamics properly. Many are restricted to achievement goals. In this paper, we develop a logical account of goal change that addresses these deficiencies. In our account, we do not drop lower priority goals permanently when they become inconsistent with other goals and the agent’s knowledge; rather, we make such goals inactive. We ensure that the agent’s chosen goals /intentions are consistent with each other and the agent’s knowledge. When the world changes, the agent recomputes her chosen goals and some inactive goals may become active again. This ensures that our agent maximizes her utility. We prove that the proposed account has desirable properties. Introduction There has been much work on modeling agent’s mental states, beliefs, goals, and intentions, and how they interact and lead to rational decisions about action. As well, there has been a lot of work on modeling belief change. But the dynamics of motivational attitudes has received much less attention. Most formal models of goal and goal change (Cohen and Levesque 1990; Rao and George ff 1991; Konolige and Pollack 1993; Shapiro et al. 1995) assume that all goals are equally important and many only deal with achievement goals. Moreover, most of these frameworks do not guarantee that an agent’s goals will properly evolve when an action /event occurs, e.g. when the agent’s beliefs /knowledge changes or a goal is adopted or dropped (one exception to this is the model of prioritized goals in (Shapiro and Brewka 2007)). Dealing with these issues is important for developing e ff ctive models of rational agency. It is also important for work on BDI agent programming languages, where handling declarative goals is an active research topic. In this paper, we present a formal model of prioritized goals and their dynamics that addresses some of these issues. In our framework, an agent can have multiple goals at di fferent priority levels, possibly inconsistent with each other. We define intentions as the maximal set of highest priority goals that is consistent given the agent’s knowledge. Our model of goals supports the specification of general temporally extended goals, not just achievement goals. We start with a (possibly inconsistent) initial set of prioritized goalsor desires that are totally ordered according to priority, and specify how these goals evolve when actions/events occur and the agent’s knowledge changes. We define the agent’s chosen goalsor intentions in terms of this goal hierarchy. Our agents maximize their utility; they will abandon a chosen goal φ if an opportunity to commit to a higher priority but inconsistent withφ goal arises. To this end, we keep all prioritized goals in the goal base unless they are explicitly dropped. At every step, we compute an optimal set of chosen goals given the hierarchy of prioritized goals, preferring higher priority goals such that chosen goals are consistent with each other and with the agent’s knowledge. Thus at any given time, some goals in the hierarchy are active, i.e. chosen, while others are inactive. Some of these inactive goals may later become active, e.g. if a higher priority active goal that is currently blocking an inactive goal becomes impossible. Our formalization of prioritized goals ensures that the agent always tries to maximize her utility, and as such displays an idealized form of rationality. In the fifth section, we discuss how this relates to Bratman’s (1987) theory of practical reasoning. We use an action theory based on the situation calculus along with our formalization of paths in the situation calculus as our base formalism. In the next section, we outline our base framework. In the third section, we formalizepathsin the situation calculus to support modeling goals. In the fourth section, we present our model of prioritized goals. In the fifth and sixth section, we present our formalization of goal dynamics and discuss some of its properties. Then in the last section, we summarize our results, discuss previous work in this area, and point to possible future work. Action and Knowledge Our base framework for modeling goal change is the situation calculus as formalized in (Reiter 2001). In this framework, a possible state of the domain is represented by a situation. There is a set of initial situations corresponding to the ways the agent believes the domain might be initially, i.e. situations in which no actions have yet occurred. Init( s) means thats is an initial situation. The actual initial state is represented by a special constant S0. There is a distinguished binary function symbol do wheredo(a, s) denotes the successor situation to s resulting from performing the actiona. Relations (and functions) whose truth values vary from situation to situation, are called relational (functional, resp.) fluents, and are denoted by predicate (function, resp.) symbols taking a situation term as their last argument. There is a special predicate Poss( a, s) used to state that action a is executable in situations. Our framework uses a theory Dbasic that includes the following set of axioms: 1 (1) action precondition axioms, one per actiona characterizing Poss( a, s), (2) successor state axioms (SSA), one per fluent, that succinctly encode both effect and frame axioms and specify exactly when the fluent changes (Reiter 2001), (3) initial state axioms describing what is true initially including the mental states of the agents, (4) unique name axioms for actions, and (5) domainindependent foundational axioms describing the structure of situations (Levesquet al. 1998). Following (Scherl and Levesque 2003), we model knowledge using a possible worlds account adapted to the situation calculus. K(s′, s) is used to denote that in situation s, the agent thinks that she could be in situation s′. UsingK, the knowledge of an agent is defined as: 2 Know(Φ, s) def = ∀s′. K(s′, s) ⊃ Φ(s′), i.e. the agent knows Φ in s if Φ holds in all of herK-accessible situations in s. K is constrained to be reflexive, transitive, and Euclidean in the initial situation to capture the fact that agents’ knowledge is true, and that agents have positive and negative introspection. As shown in (Scherl and Levesque 2003), these constraints then continue to hold after any sequence of actions since they are preserved by the successor state axiom for K. We also assume that all actions are public, i.e. whenever an action (including exogenous events) occurs, the agent learns that it has happened. Note that, we work with knowledge rather than belief. Although much of our formalization should extend to the latter, we leave this for future work. Paths in the Situation Calculus To support modeling temporally extended goals, we introduce a new sort of paths, with (possibly sub/super-scripted) variablesp ranging over paths. A path is essentially an infinite sequence of situations, where each situation along the path can be reached by performing some ex cutableaction in the preceding situation. We introduce a predicate OnPath(p, s), meaning that the situation s is on pathp. Also, Starts(p, s) means that s is the starting situation of path p. A pathp starts withs iff s is the earliest situation on p:3 Axiom 1 Starts(p, s) ≡ OnPath(p, s) ∧ ∀s′. OnPath(p, s′) ⊃ s≤ s′. In the standard situation calculus, paths are implicitly there, and a path can be viewed as a pair ( s, F) consisting of a situations representing the starting situation of the path, and a functionF from situations to actions (called Action Selection Functions(ASF) or strategies in (Shapiro et al. 1995)), such that from the starting situation s, F defines an infinite sequence of situations by specifying an action for every situation starting froms. Thus, one way of axiomatizing paths 1We will be quantifying over formulae, and thus assume Dbasic includes axioms for encoding of formulae as first order terms, as in (Shapiroet al. 2007). 2Φ is a state formula that can contain a situation variable, now, in the place of situation terms. We often suppress now when the intent is clear from the context. 3In the following, s < s′ means thats′ can be reached from s by performing a sequence of executable actions. s ≤ s′ is an abbreviation fors< s′ ∨ s = s′. is by making them correspond to such pairs ( s, F): Axiom 2 ∀p. Starts(p, s) ⊃ (∃F. Executable( F, s) ∧ ∀s′. OnPath(p, s′) ≡ OnPathASF( F, s, s′)), ∀F, s. Executable( F, s) ⊃ ∃p. Starts(p, s) ∧ ∀s′. OnPathASF( F, s, s′) ≡ OnPath(p, s′). This says that for every path there is an executable ASF that produces exactly the sequence of situations on the path from its starting situation. Also, for every executable ASF and situation, there is a path that corresponds to the sequence of situations produced by the ASF starting from that situation. OnPathASF( F, s, s′) def = s≤ s′ ∧ ∀a, s∗. s< do(a, s∗) ≤ s′ ⊃ F(s∗) = a, Executable( F, s) def = ∀s′. OnPathASF( F, s, s′) ⊃ Poss( F(s′), s′). Here, OnPathASF( F, s, s′) means that the situation sequence defined by ( s, F) includes the situations′. Also, the situation sequence encoded by a strategy F and a starting situation s is executable i ff for all situationss′ on this sequence, the action selected by F in s′ is executable ins′. We will use both state and path formulae. A state formula Φ(s) is a formula that has a free situation variable s in it, whereas a path formula φ(p) is one that has a free path variablep. State formulae are used in the context of knowledge while path formulae are used in that of goals. Note that, by incorporating infinite paths in our framework, we can evaluate goals over these and handle arbitrary temporally extended goals; thus, unlike some other situation calculus based accounts where goal formulae are evaluated w.r.t. finite paths (e.g. (Shapiro and Brewka 2007)), we can handle for example unbounded maintenance goals. We next define some useful constructs. A state formula Φ eventually holdsover