A Formal Theory of Plan Recognition and its Implementation

ion [2] [3] PrepareMeal(*I1) Kautz Plan Recognition 09/09/97 page 22 Abstraction [3] [4] End(*I1) Although the recognized plan is not fully specified, enough is known to allow the observer to make predictions about future actions of the agent. For example, the observer can predict that the agent will boil water:ion [3] [4] End(*I1) Although the recognized plan is not fully specified, enough is known to allow the observer to make predictions about future actions of the agent. For example, the observer can predict that the agent will boil water: Decomposition [2] [5] Boil( step3(*I1) ) ∧ After(time(Obs1), time(step3(*I1)) ) The observer may choose to make further inferences to refine the hypothesis. The single formula “MakePastaDish(*I1)” above does not summarize all the information gained by plan recognition. The actual set of conclusions is always infinite, since it includes all formulas that are entailed by the hierarchy, the observations, and the assumptions. (The algorithms discussed later in this chapter perform a limited number of inferences and generate a finite set of conclusions.) Several inference steps are required to reach the conclusion that the agent must be making spaghetti rather than fettucini. Given Knowledge [6] ∀x . ¬MakeAlfredoSauce(x) Exhaustiveness Assumption [2] [7] MakeSpaghettiMarinara(*I1) ∨ MakeSpaghettiPesto(*I1) ∨ MakeFettuciniAlfredo(*I1) Decomposition & Universal Instantiation [8] MakeFettuciniAlfredo(*I1) ⊃ MakeAlfredoSauce(step2(*I1)) Modus Tollens [6,8] [9] ¬MakeFettuciniAlfredo(*I1) Disjunction Elimination [7,9] [10] MakeSpaghettiMarinara(*I1) ∨ MakeSpaghettiPesto(*I1) Decomposition & Universal Instantiation [11] MakeSpaghettiMarinara(*I1) ⊃ MakeSpaghetti(step1(*I1)) [12] MakeSpaghettiPesto(*I1) ⊃ MakeSpaghetti(step1(*I1)) Kautz Plan Recognition 09/09/97 page 23 Reasoning by Cases [10,11,12] [13] MakeSpaghetti(step1(*I1)) Suppose that the second observation is that the agent is making marinara sauce. The minimal cardinality assumption allows the observer to intersect the possible explanations for the first observation with those for the second, in order to reach the conclusion that the agent is making spaghetti marinara. Second Observation [14] MakeMarinara(Obs2) Component/Use Assumption [14] & Existential Instantiation [15] MakeSpaghettiMarinara(*I2) ∨ MakeChickenMarinara(*I2) Abstraction [15] [16] MakePastaDish(*I2) ∨ MakeMeatDish(*I2)ion [15] [16] MakePastaDish(*I2) ∨ MakeMeatDish(*I2) Abstraction [16] [17] PrepareMeal(*I2)ion [16] [17] PrepareMeal(*I2) Abstraction [17] [18] End(*I2)ion [17] [18] End(*I2) Minimality Assumption [19] ∀ x,y . End(x) ∧ End(y) ⊃ x=y Universal Instantiation & Modus Ponens [4,17,19] [20] *I1 = *I2 Substitution of Equals [2,30] [21] MakePastaDish(*I2) Disjointness Assumption [22] ∀ x . ¬MakePastaDish(x) ∨ ¬MakeMeatDish(x) Disjunction Elimination [21,22] [23] ¬MakeMeatDish(*I2) Abstraction & Existential Instantiation [24] MakeChickenMarinara(*I2) ⊃ MakeMeatDish(*I2)ion & Existential Instantiation [24] MakeChickenMarinara(*I2) ⊃ MakeMeatDish(*I2) Kautz Plan Recognition 09/09/97 page 24 Modus Tollens [23,24] [25] ¬MakeChickenMarinara(*I2) Disjunction Elimination [15,25] [26] MakeSpaghettiMarinara(*I2) 3.6. Circumscription and Plan Recognition Earlier we discussed the relation of circumscription to plan recognition in informal terms. Now we will make that relation precise, and in so doing, develop a model theory for part of the plan recognition framework. Circumscription is a syntactic transformation of a set of sentences representing an agent’s knowledge. Let S[π] be a set of formulas containing a list of predicates π. The expression S[σ] is the set of formulas obtained by rewriting S with each member of π replaced by the corresponding member of σ. The expression σ ≤ π abbreviates the formula stating that the extension of each predicate in σ is a subset of the extension of the corresponding predicate in π; that is (∀x . σ1(x) ⊃ π1(x)) ∧ ... ∧ (∀x . σn(x) ⊃ πn(x)) where each x is a list of variables of the proper arity to serve as arguments to each σi. The circumscription of π relative to S, written Circum(S,π), is the second-order formula (∧S) ∧ ∀ σ . [(∧S[σ]) ∧ σ ≤ π] ⊃ π ≤ σ Circumscription has a simple and elegant model theory. Suppose M1 and M2 are models of S which are identical except that the extension in M2 of one or more of the predicates in π is a proper subset of the extensions of those predicates in M1. This is denoted by the expression M1 >> M2 (where the expression is relative to the appropriate S and π). We say that M1 is minimal in π relative to S if there is no such M2. The circumscription Circum(S,π) is true in all models of S that are minimal in the π [Etherington 1986]. Therefore to prove that some set of formulas S ∪ T entails Circum(S,π) it suffices to show that every model of S ∪ T is minimal in π relative to S. Kautz Plan Recognition 09/09/97 page 25 The converse does not always hold, because the notion of a minimal model is powerful enough to capture such structures as the standard model of arithmetic, which cannot be axiomatized [Davis 1980]. In the present work, however, we are only concerned with cases where the set of minimal models can be described by a finite set of first-order formulas. The following assertion about the completeness of circumscription appears to be true, although we have not uncovered a proof: Supposition: If there is a finite set of first-order formulas T such that the set of models of S ∪ T is identical to the set of models minimal in π relative to S, then that set of models is also identical to the set of models of Circum(S,π). Another way of saying this is that circumscription is complete when the minimal-model semantics is finitely axiomatizable. Given this supposition, to prove that Circum(S,π) entails some set of formulas S ∪ T it suffices to show that T holds in every model minimal in π relative to S.