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Individual and Collective Intention Recognition Combined with Evolution Prospection

Luís Moniz Pereira and Han The Anh

CENTRIA–UNL

Cognitive Science, Computational Logic, and Connectionism Summer School, 29 Aug-12 Sept, 2010 International Center for Computational Logic Technische Universität Dresden

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Outline

1 Evolution Prospection

Evolution Prospection Agents Main Constructs

2 Intention Recognition

Individual Intention Recognition Collective Intention Recognition

3 Home Ambient Intelligence

Proactive Support Security and Emergency

4 Anytime Intention Recognition

Model Construction Anytime Intention Recognition Algorithms

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Introduction

We explore a coherent combination of two jointly implemented logic programming based systems to address several Ambient Intelligence (AmI) issues in home environment, such as proactive support, security and emergency

Evolution Prospection: implements several kinds of well-studied preferences and useful environment-triggering constructs for decision making. Intention Recognition: performs in two stages, using Bayesian Networks and a Planner.

We present a novel method for collective intention recognition for multiple users’ domains. We illustrate the methods with examples in the elder care domain.

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Motivation Example: Elder Care

Single User Domain: e.g. an elder staying alone. In order to provide contextually appropriate help for the elder, the assisting system needs to be able to

Observe the elder’s actions; Recognize his/her intentions; Provide suggestions or help for achieving the recognized intentions

Multiple Users Domain: e.g. a couple of elders staying alone.

Need to perform collective intention recognition.

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Evolution Prospection Agents - EPA

Agents can prospectively look ahead into their hypothetical futures, in order to determine the best one to follow. EPA was implemented in ABDUAL, a XSB-Prolog abduction system that allows computing abductive solutions to given queries. Main constructs of EPA:

Active goals Abducibles Local preferences

Accompanying papers

IDT-09: http://centria.fct.unl.pt/~lmp/publications/online-papers/IDT-evolution.pdf KES-IDT-09: http://centria.fct.unl.pt/~lmp/publications/online-papers/KES-IDT09-evolution.pdf 5 / 113 L.M. Pereira and T. A. Han Intention Recognition

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Active Goal

Definition At each cycle, the agent has a set of active goals to be satisfied

• n_observe(AG) ← Body

"on observing Body trigger goal AG" An active goal may be triggered by events, previous commitments, or history-related information.

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Modelling reactive rules to provide reactive behaviors:

• n_observe(do(actions)) ← events_expression, preconditions

On detecting certain events, if certain preconditions are true, then certain actions should be executed. However, with additional background knowledge representing different kinds of information from different sources such as embedded network sensors, users’ preferences, etc., the EPA system can deliberate on which actions to perform rather than simply react to observations, thereby making more rational decisions.

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Abducibles

Each program has a set of abducibles, providing hypotheses for hypothetical solutions to a given query. Abducible A can be assumed if it is expected in the given situation and there is no expectation to the contrary: consider(A) ← expect(A), not expect_not(A), A expect(A) ← B1 expect_not(A) ← B2 Counter expectation rules supplement expectation rules by representing defeasible conditions. These rules encode pros and cons of user towards some choice, as represented by an abducible, and may enable other rules, e.g. the rule for an active goal.

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Local Preferences

Definition (A priori or pre preferences) Preferences over abducibles a ⊳ b ← Body "Prefer abducible a to abducible b" Definition (A posteriori or post preferences) Preferences over abductive solutions Ai ≪ Aj ← holds_given(Li, Ai), holds_given(Lj, Aj) "Ai is preferred to Aj if Li and Lj are true consequences of Ai and Aj, respectively"

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An Example

Example (Intention Recognition (IR) – Solving Intrusion) IR system recognized an intention of intrusion at night. The system must either warn the elders who are sleeping, automatically call the nearest police, or activate the embedded burglary alarm. If the elders are sleeping and ill, they do not expect to be warned, but prefer other solutions. Due to potential disturbance, the elders prefer simply to activate the burglary system instead of calling the police, as long as no weapon is detected and there is a single intruder.

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EPA Encoding

Three abducibles: call_police, warn_persons, activate_alarm Example

1 on_observe(solve_intrusion) ← night, intrud_int_detected. 2 solve_intrusion ← call_police.

solve_intrusion ← warn_persons. solve_intrusion ← activate_alarm.

3 expect(call_police).

expect(warn_persons). expect(activate_alarm).

4 expect_not(warn_persons) ← ill, sleeping. 5 activate_alarms ⊳ call_police ← no_weapon_det, indiv. 6 call_police ⊳ activate_alarms ← weapon_detected. 11 / 113 L.M. Pereira and T. A. Han Intention Recognition

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Suppose it is night-time and an intrusion intention is recognized: active goal solve_intrusion is triggered. Three abductive solutions: [call_police], [warn_persons], and [activate_alarm], since all the abducibles are expected and there is no expectation to their contrary. Suppose IR detects that the elders are sleeping and known to be ill: the elders do not expect to be warned, thus ruling out [warn_persons]. And if no weapon is detected, and only single intruder, the a priori preference in line 5 is triggered, which defeats solution call_police. Hence, the only solution is to activate the burglary alarm. However, if weapons were detected, the preference in line 6 is triggered and defeats [activate_alarm]. The only solution is to call the police.

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Example (cont.): A Posteriori Preference

Example

7 Ai ≪ Aj ← holds_given(not_annoying, Ai),

holds_given(annoying, Aj)

8 annoying ← call_police.

not_annoying ← activate_alarm. Suppose no weapon is detected, and more than one intruder. Then, no a priori preference is triggered: two abductive solutions [call_police] and [activate_alarm]. The a posteriori preference in line 7 is triggered. So the abductive solution leading to not_annoying is preferred to that leading to annoying. Thus [call_police] is ruled out, and [activate_alarm] is the only solution.

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1 Evolution Prospection

Evolution Prospection Agents Main Constructs

2 Intention Recognition

Individual Intention Recognition Collective Intention Recognition

3 Home Ambient Intelligence

Proactive Support Security and Emergency

4 Anytime Intention Recognition

Model Construction Anytime Intention Recognition Algorithms

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Approach to Intention Recognition

Intention recognition (IR): process by which an agent becomes aware of the intention of others. Main stream of IR: reducing to plan generation – generating conceivable plans achieving intentions and choosing ones matching observations. Main problem: finding initial set of possible intentions the planner has to work with.

This set should depend on situation at hand: generating plans for all possible intentions is unrealistic.

Accompanying paper

EPIA-09: http: //centria.fct.unl.pt/~lmp/publications/online-papers/EPIA09-intention-recognition.pdf 15 / 113 L.M. Pereira and T. A. Han Intention Recognition

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New Approach

We propose a new approach, using situation-sensitive Causal Bayes Nets – CBNs that change according to situation under consideration, itself subject to change, in order to:

compute likelihood of intentions conditional on observations, and then filter out the much less likely ones.

CBNs are modelled using P-log. A logical component is added to dynamically compute situation specific probabilistic information, to update into P-log program.

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Advantages of New Approach

Plan generator only needs to deal with remaining relevant intentions. From likelihood value of intentions, the recognizing agent can see which are more likely and worth addressing first – especially important in case of quick decision making. Comparing to approaches using BNs solely, combining with plan generation guides recognition process: which actions should be checked for whether they were hiddenly executed, from their side-effects, say.

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Network structure

Intentions are intermediate nodes whose ancestor nodes represent causes that give rise to them. When their prior probabilities can be specified without considering the causes, intentions are top nodes. Observed actions are children of the intentions that causally affect them. Observable effects are bottom nodes. They can be children of

• bserved action nodes, of intention nodes, or of unobserved

actions that might cause effects added as children of intention nodes. The causal relations, the conditional probability distribution (CPD) and the prior probabilities of top nodes are specified by domain

• experts. They may also be learnt automatically.

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An Example: Fox-Crow Fable

There is a crow, holding a cheese. A fox, being hungry, approaches the crow and praises her, hoping the crow will sing and the cheese will fall down near

• him. Unfortunately

for the fox, the crow is very intelligent, having the ability of intention recognition.

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Fox’s Intentions CBN

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P-log

The computation in CBNs can be automated by using P-log P-log – a declarative language based on a logic formalism for probabilistic reasoning It uses Answer Set Programming as its logical and Causal Bayes Nets as its probabilistic foundations.

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Fox’s Intentions CBN in P-log

Two sorts, bool and fox_intentions, represent boolean values and set of Fox’s intentions

bool = {t,f}. fox_intentions = {food,please,ter}.

Attributes hungry_fox, friendly_fox, praised and i state that first 3 have no domain parameter and are boolean, and last maps each Fox intention to a boolean

hungry_fox : bool. friendly_fox : bool. i : fox_intentions --> bool. praised : bool.

These attributes are randomly distributed in their full ranges

random(rh,hungry_fox,full). random(rf,friendly_fox,full). random(ri,i(I),full). random(rp,praised,full).

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Representing CPD in P-log: pa rule

Probability distribution of top nodes hungry_fox, friendly_fox

pa(rh,hungry_fox(t),d(1,2)). pa(rf,friendly_fox(t),d(1,100)).

Probability distribution of Fox having intention food, conditional on its parents (hungry_fox, friendly_fox)

pa(ri(food),i(food,t),d(8,10)) :- friendly_fox(t),hungry_fox(t). pa(ri(food),i(food,t),d(9,10)) :- friendly_fox(f),hungry_fox(t). pa(ri(food),i(food,t),d(1,100)) :- friendly_fox(t),hungry_fox(f). pa(ri(food),i(food,t),d(2,10)) :-friendly_fox(f),hungry_fox(f).

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Probability distribution of observation Praised, conditional on its parents (i(food), i(please), i(ter))

pa(rp, praised(t),d(95,100)) :- i(food, t), i(please, t). pa(rp, praised(t),d(6,10)) :- i(food, t), i(please, f). pa(rp, praised(t),d(8,10)) :- i(food, f), i(please, t). pa(rp, praised(t),d(1,100)) :- i(food, f),i(please,f),i(ter,t). pa(rp, praised(t),d(1,1000)) :- i(food,f),i(please,f),i(ter,f).

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Likelihood of intentions

Probabilities of Fox having intention Food, Territory, Please given the observation that Fox praised Crow can be found in P-log with queries ? − pr(i(food, t) | obs(praised(t)), V1). V1 = 0.9317. ? − pr(i(ter, t) | obs(praised(t)), V2). V2 = 0.8836. ? − pr(i(please, t) | obs(praised(t)), V3). V3 = 0.0900.

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Situation-sensitive CBNs

CBNs should be situation-sensitive since using a general CBN for all specific situations of a problem domain is unrealistic and most likely imprecise. We provide a way to construct these CBNs

use Logic Programming techniques, such as abduction, preferences, inductive learning, etc. to compute situation specific probabilistic information, and then update the specific information into a CBN general for the problem domain.

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Example (cont.)

Logical component of Fox’s intentions CBN

pa_rule(pa(ri(ter),i(ter,t),d(0,1)),[friendly_fox(t)]) :- territory(tree). pa_rule(pa(ri(ter),i(ter,t),d(1,100)),[friendly_fox(f)]) :- territory(tree).

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If territory is tree, in the updated CBN the pa-rules for territory pa(ri(ter),i(ter,t),d(1,10)) :- friendly_fox(t). pa(ri(ter),i(ter,t),d(9,10)) :- friendly_fox(f). are replaced with pa(ri(ter),i(ter,t),d(0,1)) :- friendly_fox(t). pa(ri(ter),i(ter,t),d(1,100)) :- friendly_fox(f). The likelihood of the intentions Food, Territory, Please are: V1 = 0.9407; V2 = 0.0099; V3 = 0.0908, respectively. Thus, most likely, the only surviving intention is food.

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Second Phase - Plan Generation

The second phase of the intention recognition system is to generate conceivable plans that can achieve the most likely intentions surviving after the first phase. In our system we deploy ASCP – an ASP based conditional planner, re-implemented in XSB Prolog.

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ASCP Syntax

ASCP uses Ac

K – action language extending language A.

Alphabet of Ac

K consists of a set of actions A and a set of

fluents F. A literal is a fluent f ∈ F or its negation ¬f . A fluent formula ϕ is a propositional formula constructed from the set of literals using operators ∧, ∨ and/or ¬.

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To describe an action theory, 5 kinds of propositions used:

1 initially(l): l holds in the initial situation. Initial situation is

described by a set of these propositions.

2 executable(a, ψ): a is executable in any situation in which ψ

holds

3 causes(a, l, φ): performing a in a situation in which φ holds

causes l to hold in the successor situation

4 if(l, ϕ): l holds in any situation in which ϕ holds. 5 determines(a, θ): values of literals in θ will be known after a

is executed.

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Example: Fox’s Plans for Food

Initial situation

initially(holds(crow,cheese)). initially(hungry(fox)).

Executability conditions

executable(eat(A,E),[holds(A,E)]) :- animal(A),edible(E). executable(sing(B),[accepted(B)]) :- bird(B). executable(praise(fox,A), []) :- animal(A). executable(grab(A,O),[holds(no,O)]) :- animal(A),obj(O).

Causal laws

causes(sing(B),holds(no,O),[holds(B,O)]):- bird(B),obj(O). causes(eat(A,E),neg(hungry(A)),[hungry(A)]) :- animal(A),edible(E). causes(grab(A,O),holds(A,O),[]) :- animal(A),obj(O).

Knowledge propositions

determines(praise(fox,B),[accepted(B),declined(B)]):-bird(B).

Goal:

goal([neg(hungry(fox))]).

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Fox’s Plans to get food

[praise(fox, crow), cases({ accepted(crow) → [sing(crow), grab(fox, cheese), eat(fox, cheese)]; declined(crow) → ⊥})] where ⊥ means no appropriate plans

i.e, first, Fox praises Crow. If Crow accepts to sing, Fox grabs the dropped cheese and eats it. Otherwise, i.e. if Crow declines to sing, nothing happens.

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Elder Care domain

In order to provide contextually appropriate help for elders, the assisting system needs to be able to

• bserve elders’ actions,

recognize their intentions, then provide suggestions achieving the recognized intentions

We focus on dealing with the last two steps. Elders’ intention recognition is employed using the previously described system. Accompanying papers

Springer Book Chapter 2010: http://centria.fct.unl.pt/~lmp/publications/online-papers/IR-EPA-CBNs-book.pdf INAP-09: http://centria.fct.unl.pt/~lmp/publications/online-papers/INAP09_Elder_Care.pdf 34 / 113 L.M. Pereira and T. A. Han Intention Recognition

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For providing suggestions to achieve recognized intentions: use our Evolution Prospection Agent (EPA) Prospectively look ahead into the future to choose the best course of evolution achieving the recognized intention, while aware of external environment and elder’s preferences Expectation rules and a priori preferences take into account physical health reports to guarantee only contextually safe healthy choices are generated; then, information like elder’s pleasure, interests, etc. are met by post preferences. Advance in handling and easiness of expressing preferences in EPA enable to closely take into account the elders’ preferences – to increase degree of acceptance of elders w.r.t. technological help – an important issue of Elder Care domain.

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An Example

An elder stays alone in his apartment. The intention recognition system observes that he is looking for something in the living room. In order to assist him, the system needs to figure out what he intends to find. The possible things are:

something to read (Book); something to drink (Drink); TV remote control (Rem); the light switch (Switch).

The states of the light and TV are observed.

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Elder’s Intentions CBN in P-log

Two sorts, bool and elder_intentions, represent boolean values and set of the Elder’s intentions

bool = {t,f}. elder_intentions = {book,drink,rem,switch}.

Attributes thsty (thirsty), lr (like reading), lw (like watching), tv (tv on), lt (light on) have no domain parameter and get boolean values

thsty:bool. lr:bool. lw:bool. tv:bool. lt:bool.

Attribute i maps each Elder’s intention to a boolean value

i:elder_intentions --> bool.

These attributes are randomly distributed in their full ranges

random(rth,thsty,full). random(rlr,lr,full). random(rlw,lw,full). random(rtv,tv,full). random(rl, lt, full). random(ri, i(I), full).

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Representing CPD in P-log: pa rule

Probability distribution of top nodes

pa(rth,thsty(t),d(1,2)). pa(rlr,lr(t),d(8,10)). pa(rlw,lw(t),d(7,10)). pa(rtv,tv(t),d(1,2)). pa(rl,lt(t),d(1,2)).

Probability distribution of the Elder having intention book, conditional on its parents (all top nodes)

pa(ri(b),i(b,t),d(0,1)) :-lt(f). pa(ri(b),i(b,t),d(0,1)) :-lt(t),tv(t). pa(ri(b),i(b,t),d(3,5)) :-lt(t),tv(f),lr(t),lw(t),thsty(t). pa(ri(b),i(b,t),d(13,20)):-lt(t),tv(f),lr(t),lw(t),thsty(f). pa(ri(b),i(b,t),d(7,10)) :-lt(t),tv(f),lr(t),lw(f),thsty(t). pa(ri(b),i(b,t),d(4,5)) :-lt(t),tv(f),lr(t),lw(f),thsty(f). pa(ri(b),i(b,t),d(1,10)) :-lt(t),tv(f),lr(f),lw(t). pa(ri(b),i(b,t),d(4,10)) :-lt(t),tv(f),lr(f),lw(f).

Similarly for other intentions.

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Probability distribution of observation look, conditional on its parents (i(b), i(dr), i(rem), i(sw))

pa(rla,look(t),d(99,100)):-i(b,t),i(dr,t),i(rem,t). pa(rla,look(t),d(7,10)) :-i(b,t) i(dr,t),i(rem,f). pa(rla,look(t),d(9,10)) :-i(b,t),i(dr,f),i(rem,t). pa(rla,look(t),d(6,10)) :-i(b,t),i(dr,f),i(rem,f). pa(rla,look(t),d(6,10)) :-i(b,f),i(dr,t),i(rem,t). pa(rla,look(t),d(3,10)) :-i(b,f),i(dr,t), i(rem,f). pa(rla,look(t),d(4,10)) :-i(b,f),i(dr,f),i(rem,t). pa(rla,look(t),d(1,10)) :-i(b,f),i(dr,f),i(rem,f),i(sw,t). pa(rla,look(t),d(1,100)) :-i(b,f),i(dr,f),i(rem,f),i(sw,f).

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Likelihood of intentions

Probabilities of the Elder having intention book, drink, remote control, switch given the observation that he is looking for something and the states of light and TV (on or off), can be found in P-log with queries ? − pr(i(b, t) | (obs(tv(S1)) & obs(lt(S2)) & obs(look(t))), V1). ? − pr(i(dr, t) | (obs(tv(S1)) & obs(lt(S2)) & obs(look(t))), V2). ? − pr(i(rem, t) | (obs(tv(S1)) & obs(lt(S2)) & obs(look(t))), V3). ? − pr(i(sw, t) | (obs(tv(S1))&obs(lt(S2)) & obs(look(t))), V4). where S1, S2 are boolean values (t or f ) instantiated during execution, depending on the states of the light and TV.

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If the light is off (S2 = f ), then V1 = V2 = V3 = 0, V4 = 1.0, regardless of the state of the TV. If the light is on and TV is off (S1 = t, S2 = f ), then V1 = 0.7521, V2 = 0.5465, V3 = 0.5036, V4 = 0.0101. If both light and TV are on (S1 = t, S2 = t), then V1 = 0, V2 = 0.6263, V3 = 0.9279, V4 = 0.0102. Thus, if one observes the light is off, definitely the elder is looking for the light switch. Otherwise, if one observes the light is on, no matter TV is on or off, the first three intentions book, drink, remote control still under consideration in next phase. The intention of looking for the light switch is very unlikely compared with the others, thus being ruled out. When there is light, one goes directly to the light switch if the intention is to turn it off, without having to look for it.

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Situation-sensitive CBNs

The CBN may vary depending on some observed factors, e.g. the time of day, the current temperature, etc. We design a logical component for the CBN to deal with those factors:

pa_rule(pa(rlk,lr(t), d(0,1)),[]):-time(T), T>0, T<5,!. pa_rule(pa(rlk,lr(t), d(1,10)),[]):-time(T), T>=5, T<8,!. pa_rule(pa(rlw,lw(t), d(9,10)),[]):- time(T), schedule(T,football),!. pa_rule(pa(rlw,lw(t), d(1,10)),[]):-time(T), (T>23; T<5),!. pa_rule(pa(rth,thsty(t), d(7,10)),[]):-temp(T), T>30,!. pa_rule(pa(rlk,lr(t), d(1,10)),[]):-temp(TM), TM >30,!. pa_rule(pa(rlw,lw(t), d(3,10)),[]):-temp(TM), TM>30,!.

If body of rule true, corresponding P-log pa-rule is formed and updated into the CBN.

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Collective Intention

Collective intention of a group of agents is not mere summation of individual intentions (Philosophy, AI):

It involves a sense of acting and willing something together, cooperatively. It requires some “glue" supplement amongst the agents’ activity.

“Glue" amongst agents, e.g. mutual beliefs and reactions:

Agents hold reciprocal expectations. Thus, expectation reciprocal actions should be observed, and reactions to any lack thereof.

Accompanying paper

AAAI-FS-PAA-10: http://centria.fct.unl.pt/~lmp/publications/online-papers/PAA10_collective.pdf 44 / 113 L.M. Pereira and T. A. Han Intention Recognition

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Bottom-up Approach

Tuomela: collective intention = individual intentions + mutual

• beliefs. Briefly, agents A and B intend to do some task X

cooperatively if these “glue" conditions for A—symmetrically for B—hold: (a) A intends to do his part of X (b) A believes B will do his part of X (c) A believes B believes A will do his part of X Kanno et al.: bottom-up approach to collective IR:

constituents’ individual intentions and beliefs are inferred first; then, collective intention is inferred by checking for consistency amongst the inferred mental components.

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Bottom-up Method: Disadvantage

Disadvantage: combinatorial explosion problem of possible combinations of individual intentions and beliefs for forming collective intentions.

Given the situation at hand, each agent may have several conceivable intentions and beliefs, but not many of the combinations partake of conceivable collective intentions.

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Collective Intention Recognition

We propose a top-down method for collective intention recognition, based on Searle’s account Searle’s Account Collective intention is non-summative, but remains individualistic. Having the presuppositions of mutual awarenesses or beliefs, posit the existence of a virtual plural agent that holds the collective intention This virtual plural agent is ascribed all the activities of the group of agents.

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Collective Intention Recognition Method Collective IR is thus reduced to individual IR plus checking if there are actions reflecting mutual expectations amongst agents:

1 First step: From the observations (actions or their effects in

the environment of all agents in the group) infer intentions, as if these observations are of the plural agent;

2 Second step: Figure out which of the recognized intentions is

a genuine collective intention, by checking for actions reflecting the mutual expectations between the agents, while ignoring any irrelevant actions for each considered intention.

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Example

Example (Elder Care Couple) A couple of elderly people are alone in their apartment. The IR system observes they are both now in the kitchen. The man looking for something, the woman holding a kettle. In order to assist them, the system needs to figure out what they intend to do, cooperatively or separately. Possible collective intentions: “making a drink" or “cooking". The first step can be done by any existent individual IR

• method. We employ our previous system for this purpose,

which relies on a Bayesian Network connecting possible intentions to actions and circumstances.

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CBN: Couple’s Collective Intentions

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Computing Likelihood with P-log

The probabilities of the elders having the collective intentions of cooking (cook) and making a drink (mD), given the man is looking for something and the woman holding a kettle, are computed with these P-log queries, respectively: ? − pr(i(cook, t) | (obs(look(t) & obs(holdKettle)), V1). ? − pr( i(mD, t) | (obs(look(t) & obs(holdKettle)), V2). Their results are: V1 = 0.478; V2 = 0.667. Meaning the collective intention of making a drink is more likely and should be examined first for confirmation. However, it is still necessary to look at the other collective intention as it is not much less likely.

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Situation-Sensitive CBN

Example (Elder Care, cont’d) In this scenario, the Casual Bayesian Network (CBN) may vary depending on observed factors, e.g. time of day, of elders’ last drink

• r last meal.

We design a logical component for the CBN to deal with these factors:

pa_rule(pa(hg(t),d_(0,1)),[]):- time(T), eat(T1), T-T1<1. pa_rule(pa(hg(t),d_(9,10)),[]):- time(T), last_eating(T1),T-T1>3. pa_rule(pa(thty(t),d_(1,10)),[]):- time(T), drink(T1), T1-T<1. pa_rule(pa(thty(t),d_(9,10)),[]):- time(T), last_drink(T1),T1-T>3.

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Suppose current time is 18: time(18) last time elders ate was half an hour ago: last_eating(17.5) they have not had any drink for 3 hours, e.g.: last_drink(14) These three facts are asserted. Hence, these two pa_rule/2 literals become true, and are updated into the general CBN:

pa_rule(pa(hg(t),d_(0,1)),[]). pa_rule(pa(thty(t),d_(9,10)),[]).

Now the result is: V1 = 0.0206; V2 = 0.9993. This time around,

• nly the collective intention of making a drink should be sought for

confirmation in the second stage, as the one of cooking is quite unlikely.

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Second Step: Confirming Collective Intention

This step aims to confirm whether the recognized intention is a genuine collective intention of the group. Let {a1, ...., an} be the set of agents and A the plural agent standing for them (i.e. having all their presuppositions and actions). Suppose W is an intention of A, recognized in the first step. P = [p1, ..., pk] is a plan achieving W. We determine an assigned subplan to each agent towards achieving the collective intention of the whole group, by looking at each agent’s actions.

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Grouping

Recall that P = [p1, ..., pk] is a plan for achieving W. Let si be the first action of agent ai in P. Determine indices di such that pdi = si. Group the agents with same first action, i.e. with same index

• di. They are doing the same task or at least some part of it

together. Suppose we obtain m groups g1, ..., gm: gt is responsible for the subplan [pjt+1, ..., pjt+1] where 1 ≤ t ≤ m and 0 = j1 < j2 < ... < jm+1 = k. Grouping is unique for a given set of agents and a given plan.

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Checking Expectation Actions

To check if there are expectation actions reflecting agents’ mutual expectations, consider 2 cases: mutual expectations between agents in a group, and between agents in consecutive groups

The number of interactions amongst the agents that need to be observed (e.g. by an activity recognition system) is considerably reduced. The number of possible expectation actions between two particular agents is reduced and becomes better specified.

One group may interact (produces or consumes results) with more than another group, but that is not considered here.

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Inside Group

When some agents are assigned or intended to do something together, they expect from each other to do some task.

“complain" is an expectation action when a group member “deviates" from the task without an “inform" action. Example: two people intend to walk together, but if one changes direction without informing the other, he will be subject to complaint. This distinguishes it from the situation

• f walking in step by coincidence..

When agents work cooperatively, we usually observe a “help"

• action. This type of action is often preceded by a “complain"

action. The term “deviation" should be specified for concrete application domains. E.g., in the spatial domain, an agent is said to deviate if he is not in his assigned position or does not move in the right direction, or does so at wrong speed.

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Between Consecutive Groups

When agents are in consecutive groups: agents from the second would expect some result from the first, who in turn expect the agents from the second to use their result. If the agents from one of the groups “deviated" from their task

• r did not finish it as assigned, the agents from the other

group would “complain". Let resultt be the assigned result of group gt. Usually, this result comes from the last action, pjt+1, of the group subplan. Assume a given set of possible actions reflecting expectation of resultt of action pjt+1, denoted by expect_resultt, and a given set of possible actions reflecting expectation of using resultt, denoted by expect_uset.

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Then, we say these two groups of agents are working towards achieving a collective intention either if agents in gt+1 have some action belonging to expect_resultt, or agents in gt have some action belonging to expect_uset, or there occur “complain" actions from one of the groups. As long as the collective intentionality inside each group is confirmed, a single expectation action observed between the two groups is enough to conclude that they have a collective intention. Usually, it is useful to identify one of the result-delivery and result-receiver agents.

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Advantages of Grouping Method: 1

Reduces number of interactions needing to be observed Set of n agents: number of interactions to be observed n(n−1)

2

. Grouping method: m groups of nj (1 ≤ nj ≤ n, 1 ≤ j ≤ m) agents, where m

j=1 nj = n.

The number of interactions to be observed (

m

• j=1

nj(nj − 1) 2 ) + m − 1 = n(n − 1) 2 − S where S = (

1≤j<k≤m njnk) − m + 1 is the number of

interactions not needing observation, compared with the case without grouping. If groups are equally divided, we reduce approximately m times the number of interactions to be observed.

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Advantages of Grouping Method: 2

Focuses on smaller groups, with smaller sets of possible expectation actions Divides big set of agents into smaller groups: more easily

• bserved, e.g. by an activity recognition system.

Specifies which are the expectation actions requiring recognition between a particular pair of agents: Are they in the same group? Are they in consecutive groups? Or else? Without grouping, a bigger set of possible expectation actions needs to be considered for any pair.

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Elder Care: General Case

In order to provide appropriate assistance: When observing only individual activity (e.g. when the other is absent from the apartment): recognize individual intentions. When observing both elders’ activity: recognize whether there is collective intentionality.

If no collective intentionality is detected, the system then should then perform individual IR.

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An Example: Couple of Elder

Example (Confirming Collective Intention) Suppose “making a drink" was the collective intention recognized in the first step. We check if it is a genuine one. Plan achieving the intention: [take kettle, fill up with water, boil water, look for tea or coffee, put into water]

woman’s subplan: [take kettle, fill up with water, boil water] man’s subplan: [look for tea or coffee, put into water].

Woman’s assigned result: provide boiled water. Man’s expectation: boiled water. He may have “expect result" actions, e.g. ask whether the water is ready or get the water from the woman. Or, if after a while the woman had not boiled water or was doing something else, the man would complain.

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If such “expect result" or “‘complain" action occurs, we conclude that they really have a collective intention of making a drink. Otherwise, e.g. the man does not show any expectation for water the woman has actually boiled, then we conclude that there was not a genuine collective intention, even if later he might use the boiled water for his own purpose. We emphasize that there are necessary actions showing mutual expectations of results and usage of results when agents have indeed a collective intention towards achieving some task.

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Now suppose the system has found there is no collective intention amongst the elders, and the man keeps looking for something. To assist him, the system should then figure out what is his individual intention. Example (Man’s intentions) Suppose the possibilities are: book, drink, remote control, cook, make drink. We have shown elsewhere how to proceed with individual intention recognition in our setting.

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Related Work

We are not the first to suggest the use of the plural agent concept: most works in multi-agent plan recognition rely on the assumption that the plan is carried out by a single entity (team, group, troop). However, none of them has addressed the necessary cognitive underpinnings amongst the constituents (mutual beliefs or mutual awarenesses) in order to confirm the existence of a collective intention. That is however the main concern of the philosophical community regarding collective intentionality.

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Consequently, extant work is restricted to considering only sets

• f agents with an initially assigned collective intention

(football team, army troop). In Elder Care domain concerning multiple users: elders’ actions may accidentally form a plausible plan for a conceivable intention, but each of them is following his/her own intention. Our method supplements existing ones that deal with arbitrary sets of agents; without an initially assigned collective task or intention.

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1 Evolution Prospection

Evolution Prospection Agents Main Constructs

2 Intention Recognition

Individual Intention Recognition Collective Intention Recognition

3 Home Ambient Intelligence

Proactive Support Security and Emergency

4 Anytime Intention Recognition

Model Construction Anytime Intention Recognition Algorithms

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Proactively Providing Support

AmI key feature: system should take initiative to help. In order to provide contextually appropriate help, e.g. for elders, the assisting system needs being able to:

1

• bserve the elders’ actions,

2

recognize his/her intentions, or their collective intention,

3

provide suggestions or help for achieving the recognized intentions (Evolution Prospection Agents—EPA).

Accompanying paper

AITAmI-10: http://centria.fct.unl.pt/~lmp/publications/online-papers/AITAmI10.pdf 69 / 113 L.M. Pereira and T. A. Han Intention Recognition

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Example (Elder Intentions) An elder stays alone in his apartment. One day, the Burglary Alarm is ringing. IR system observes that he is looking for something. In order to assist him, it needs to figure out what he intends to find. Possible things are: Alarm button (AlarmB) Contact Device (ContDev) Defensible Weapons (Weapon) Light switch (Switch)

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Bayesian Network for Intention Recognition

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Security for Home AmI

1 Security in terms of Burglary Alarm systems 2 Security in terms of health and well-being of residents 72 / 113 L.M. Pereira and T. A. Han Intention Recognition

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Security: Burglary Alarm System

Burglary Alarm technology has been based on sensing and recognizing very last actions of an intrusion plan (e.g. “breaking the door"). It may be too late to provide appropriate protection. Need to guess in advance possibility of intrusion from very first

• bserved actions of potential intruders.

This information can be sent to the carer to get prepared (e.g. turn on the light to scare off burglars). Two-stage Intention Recognition:

From the observed actions the CBN computes likelihood of conceivable intentions If worrisome enough, carer should be informed of potential intrusion.

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Security: Health and Well-being

AmI systems need to be able to prevent hazardous situations, which usually come from dangerous ideas or intentions (e.g. take bath when drunk, drink alcohol while not permitted, or even commit suicide) of the assisted persons, especially those with mental impairments. To this end, guessing their intentions from the very first relevant behaviors is indispensable for takeing timely actions. Our IR system: CBN computes how likely there is a dangerous intention, and carers should be informed in case it is likely enough, in order to get prepared.

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Emergency Handling

Wide range of emergency situations: recognizing intrusion intention; dangerous intentions of assisted persons; detecting fire; unconsciousness or unusualness in regular activities (e.g. sleep for too long), etc. Emergency handling in EPA

Active goal rule for each emergency situation. For each goal: a list of possible actions, represented by abducibles, are available to form solutions. Users preferences:

a priori prefs. for preferring amongst available actions; a posteriori prefs. for comparing solutions taking into account their consequences and utility; evolution-level preferences to compare further consequences.

Expectation and counter expectations rules: encode pros and cons of users towards available actions.

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Outline

1 Evolution Prospection

Evolution Prospection Agents Main Constructs

2 Intention Recognition

Individual Intention Recognition Collective Intention Recognition

3 Home Ambient Intelligence

Proactive Support Security and Emergency

4 Anytime Intention Recognition

Model Construction Anytime Intention Recognition Algorithms

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Figure: On the left: Heinze’s tri-level decompositional model of intentional behavior of the intending agent: Intentional level; Activity level; and State level. On the right: intention recognition is the reversal

• f this process.

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Introduction

We present an intention recognition (IR) model that is situation-dependent

One aspect of intention is future-directedness: actions for the intention may be executed at a far distance in time. During that period the world may be changing, and an intention may be changed or abandoned. IR model is reconfigurable to take into account changes.

incremental and anytime

Bayesian Network (BN) is used as IR model. IR model is incrementally constructed as more actions are

• bserved.

An anytime BN inference algorithm is deployed to design an anytime IR algorithm.

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Summary

We present an anytime algorithm for incremental intention recognition in a changing world. The algorithm is performed by dynamically constructing the intention recognition model on top of a prior domain knowledge base. The model is occasionally reconfigured by situating itself in the changing world and removing newly found out irrelevant intentions. Reconfigurable Bayesian networks are employed to produce the intention recognition model. Accompanying paper

AAAI-FS-PAA-10: http://centria.fct.unl.pt/~lmp/publications/online-papers/PAA10_incremental.pdf 79 / 113 L.M. Pereira and T. A. Han Intention Recognition

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Intention Recognition Model Construction

The IR model is dynamically reconstructed as more actions are

• bserved.

The IR model is occasionally introspected (e.g. to remove found out irrelevant intentions) and reconfigured w.r.t. to the situation at hand. E.g., in the Elder Care domain, different elders have distinct conditions and habits to be taken into account to recognize their intentions. Also, place, time of day, temperature, etc. need to be considered. In the next slides we formalize the IR model based on the proposed structure, and provide some operators for incremental construction of the IR model.

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Definition (Bayes Networks) A Bayes Network is a pair a dag whose nodes represent variables and missing edges encode conditional independencies between the variables, and an associated probability distribution satisfying the Causal Markov Assumption (CMA), that is, variables are independent

• f their non-effects conditional on their direct own causes.

Definition (Causal Bayes Network) A BN is causal if its associated probability distribution satisfies the condition specifying that if a node X of its dag is actively caused to be in a given state x (an operation written as do(x), e.g. in P-log syntax), then the probability density function changes to the one of the network obtained by cutting the links from X’s parents to X, and setting X to the caused value x.

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Bayesian Network for Intention Recognition

Definition (Intention Recognition BN – IRBN) BN for intention recognition: triple W = {Cs, Is, As}, pa, PW Cs, Is and As are sets of cause, intention and action nodes (binary random variables). pa maps a node to the set of its parent nodes:

pa(C) = ∅ ∀C ∈ Cs (causes are top node) pa(I) ⊆ Cs ∀I ∈ Is (intention connect to its causes) pa(A) = Is ∀A ∈ As (all intentions connect to each action).

CPD tables are given by probability distribution PW

PW (X|pa(X)) defines the probability of X conditional on pa(X) in W, for all X ∈ VW , where VW = Cs ∪ Is ∪ As.

Furthermore, it is required that Cs =

I∈Is pa(I), i.e. there is no

isolated cause node in W.

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Unit Bayesian Net for Intention Recognition

Our Intention Recognition method is performed by incrementally constructing an IRBN on top of a knowledge base of Unit IRBNs, defined as follows: Definition (Unit IRBN) The Bayesian Network for intention recognition for an action A, denoted by irBN(A), is an IRBN where its set of intentions refers to a single action A. We denote by PA the probability distribution in irBN(A).

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Knowledge Base

We stipulate a reasonable set of assumptions for a domain knowledge base. Definition (Knowledge Base) The domain knowledge base KB consists of a set of actions AS and a set of unit IRBNs for every action in AS, satisfying that An intention I has the same set of parents (causes) and CPD table in all the unit IRBNs that it belongs to.

C(I) denotes the set of parents of I. PKB(I|C(I)) defines CPD table of I.

A cause C has the same prior probability distribution in all the unit IRBNs that it belongs to, denoted by PKB(C).

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Operators for Handling IRBNs

We present some operators for handling IRBNs and CPD tables Project of CPD table to a set of variables Combine IRBNs Remove a set of intentions from an IRBN Situate a IRBN in a situation

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Project of CPD Table

This operator is useful later to reconfigure the model when new actions or intentions need to be removed from the model. Definition (Project of CPD Table) Let T be a CPD table defining P(X|V ): probability of random variable X conditional on set of random variables V. Project of T on strict subset V ′ of V (V ′ ⊂ V ) according to assignment u to variables in U = V \ V ′ is a CPD table defining P(X|V ′, U = u) (the part of T for V corresponding to U = u) If variables of V are binary: proj(T, V′) denotes the project of T on V ′ according the assignment of all variables in U to false.

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Combination of Unit IRBNs

Definition (Combination of Unit IRBNs) Let O = {A1, ..., An} ⊆ AS (n ≥ 0) be a set of actions, and irBN(Ai) = {Csi, Isi, {Ai}}, pai, PAi be the unit IRBN for action Ai (1 ≤ i ≤ n). The IRBN of O is irBN(O) = {CsO, IsO, O}, paO, PO}, where ∀C ∈ CsO, ∀I ∈ IsO, 1 ≤ i ≤ n IsO = n

i=1 Isi

CsO =

I∈IsO C(I)

paO(C) = ∅ paO(I) = C(I) paO(Ai) = IsO PO(C) = PKB(C) PO(I|paO(I)) = PKB(I|C(I)) PO(Ai|paO(Ai)) is defined by proj(T, IsO), where T is the CPD table for Ai in irBN(Ai).

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Combination of IRBNs

Definition (Combination of IRBNs) Let W1 = {Cs1, Is1, As1}, pa1, P1, W2 = {Cs2, Is2, As2}, pa2, P2 be two IRBNs. The combination of W1 and W2 is comb(W1, W2) = {Cs, Is, As}, pa, PW , where ∀C ∈ Cs, ∀I ∈ Is, ∀A ∈ As As = As1 ∪ As2 Is = Is1 ∩ Is2 Cs =

I∈Is C(I)

pa(C) = ∅ pa(I) = C(I) pa(A) = Is PW (C) = PKB(C) PW (I|pa(I)) = PKB(I|C(I)) PW (A|pa(A)) is defined by the CPD table proj(T, Is), where T is the CPD table for A in irBN(A).

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Remove Intentions from IRBN

If some intentions are found out to be irrelevant, e.g. when their likelihoods are very small, they are removed from the model. Definition (Remove Intentions from IRBN) Let W = {Cs, Is, As}, pa, PW be an IRBN; R ⊂ Is. The result of removing the intentions in R from W is the IRBN remove(W, R) = {CsR, IsR, AsR}, paR, PR where ∀C ∈ CsR, ∀I ∈ IsR, ∀A ∈ AsR AsR = As IsR = Is \ R CsR =

I∈IsR C(I)

paR(C) = ∅ paR(I) = C(I) paR(A) = IsR PR(C) = PKB(C) PR(I|paR(I)) = PKB(I|C(I)) PR(A|paR(A)) is defined by the CPD table proj(T, IsR), where T is the CPD table for A in W.

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Situate an IRBN

In a given situation, an IRBN is re-situated by recomputing the prior probabilities of the top nodes. Definition (Situate IRBN) Let W = {Cs, Is, As}, pa, PW be an IRBN. We say W is situated into situation SIT if prior probabilities of top nodes of W, i.e. PW (C) (C ∈ Cs), are recomputed according to SIT. The result is an IRBN: situate(W, SIT) = {Cs, Is, As}, pa, PS PS(C) (C ∈ Cs) are the new prior probabilities of top nodes, resulting from re-computation according to SIT. PS(X|pa(X)) = PW (X|pa(X)) ∀X ∈ Is ∪ As.

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Algorithm 1.

KB: domain knowledge base; AS: set of actions in KB. Repeat these steps until one intention remains in the IRBN or time limit is reached; in the latter case, the most likely intention in the previous cycle is the final result.

1 Let O ⊆ AS: set current observed actions, W: current IRBN.

Let W ′ = comb(W, irBN(O)). If O is the set of initially

• bserved actions: W ′ = irBN(O).

2 If there is an (expert defined) “salient" intention in W ′, situate

it according to the situation at hand curSIT: situate(W′, curSIT) = W ′′; otherwise, the IRBN remains the same: W ′′ = W ′.

3 Compute the likelihood of each intention in W ′′, conditional

• n the set of current observed actions in W ′′. Remove

intentions much less likely than the others.

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Algorithm 1: Some Remarks

If an observed action makes the set of conceivable intentions empty, the action is considered irrelevant and discarded. At any cycle, if the likelihood of all intentions are very small: the sought for intention is abandoned.

The causes and actions do not support or force the intending agent to keep pursuing his/her intention anymore.

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Anytime Intention Recognition

Definition (Anytime Algorithm) An algorithm is anytime if it can produce a solution in a given time T and the quality of solutions improves with time after T. In Alg. 1, the standard BN inference algorithm is applied. If the time limit is reached within the first cycle, it cannot provide any IR decision. Otherwise, using an anytime BN inference algorithm provides an anytime IR algorithm.

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Revising Knowledge Base Representation

Currently, all conceivable intentions that may give rise to some action, regardless of the situation in which it is observed, figure in Unit IRBN for that action.

This set is usually quite big.

The set of conceivable intentions for an action should depend

• n the situation the action is observed.

We propose two methods for revising the KB representation to tackle this problem

Conditional Unit IRBNs. Situation-sensitive Intentions.

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Conditional Unit IRBNs

We revise the current representation by providing a set of Unit IRBNs for each action, and a criterion for choosing the appropriate one for the situation at hand.

Each Unit IRBN is accompanied by a precondition. The criterion must satisfy that in each situation only one Unit IRBN is chosen.

Situation is encoded as a logic program. Difficulties: design a BN for each situation is costly; moreover, it is difficult to specify all situations from the beginning.

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Situation-sensitive Intentions

Whether an intention may give rise to a particular action should be situation-dependent. Common sense reasoning can be employed for this purpose. We combine LP techniques for common sense reasoning.

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KB is accompanied by a LP PKB to help decide which intentions are conceivable in a given situation. AS = {A1, ..., AN}: set of actions of KB. BNs = {W1, ..., WN}: set of unit IRBNs of KB, where Wi = {Csi, Isi, Asi}, pai, Pi (1 ≤ i ≤ N). Intention I is conceivable w.r.t. action A if it is expected in the given situation and there is no expectation to the contrary. Thus, for 1 ≤ i ≤ N and I ∈ Isi, PKB contains rule: conceivable(I) ← Ai, expect(I), not expect_not(I) Furthermore, for each I ∈ N

i=1 Isi, PKB contains two rules:

expect(I) ← Cond1. expect_not(I) ← Cond2.

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Suppose action Ai (1 ≤ i ≤ N) is observed. The current situation is encoded by logic program SIT. To compute set of conceivable intentions that may give rise to Ai, we use XSB Prolog built-in findall/3 predicate to detect all true conceivable/1 atoms of program PKB ∪ SIT ∪ {Ai ←}. Suppose O is the obtained set of conceivable intentions. The IRBN obtained by removing the other intentions from Wi, i.e. remove(Wi, Isi \ O), is used for intention recognition. If Ai is not given in SIT, but is allowed as an abducible, then an observation action can be triggered to discover whether hypothetical Ai took place, and SIT is updated accordingly. If explicit negation is available in SIT, known absence of actions may be recorded and duly taken into account.

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An Example

Example (Elder Care) An elderly person stays alone in his apartment. An IR system is set up to support his activities in the living room. Once, at night, the system observes that the elder is looking around for something (look). KB of the system has a Unit IRBN for this action. For illustration, consider a small set of conceivable intentions: Is = {book, water, weapon, lightSwitch}. The accompanying logic program PKB contains the following rules, for each concrete action and conceivable I ∈ Is: conceivable(I) ← look, expect(I), not expect_not(I).

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Example (cont.)

Suppose in PKB the expectation and counter-expectation rules for these intentions are

• 1. expect(book).

expect_not(book) ← light_off . expect_not(book) ← burglar_alarm_ring.

• 2. expect(water).

expect_not(water) ← light_off . expect_not(water) ← burglar_alarm_ring.

• 3. expect(weapon) ← burglar_alarm_ring.

expect_not(weapon) ← light_off . expect_not(weapon) ← no_weapon_availabe.

• 4. expect(lightSwitch).

expect_not(lightSwitch) ← light_on, tv_on.

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The rules in part 1 say the intention of looking for a book is always expected except when the light is off or the burglar alarm is ringing. If at the moment light is off (SIT = {light_off ←}), then conceivable(light_switch) is the only true conceivable/1 atom of program PKB ∪ SIT ∪ {look ←}. In this case, as there is just one conceivable intention, we can conclude immediately. Now suppose light is on, the tv is not on, and the burglar alarm is not ringing. There are three conceivable intentions: book, water and lightSwitch. Then we need to remove the intention weapon from the unit IRBN. If light is on, tv is not on, and the burglar alarm is ringing, the conceivable intentions are: weapon and lightSwitch.

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Complexity Assessment

Let MI, MC be the maximal number of intentions and maximal number of causes in a unit IRBN of KB, respectively, and NA be the number of actions in KB (i.e. |AS|). Let M = MI + MC + NA. The complexity will be evaluated in terms of M. proj(T, V′): 2|V ′| (create new CPD from original table). In real implementation, it can be done by setting a pointer to

• riginal table: constant time.

comb and remove: linear time (O(M)). situate: top-down querying procedure of XSB-Prolog (polynomial time).

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Summary – 1

We have shown a novel approach to Intention Recognition by combing situation-sensitive CBNs and a plan generator Such CBNs are dynamically reconfigured using LP techniques to compute likelihood of intentions w.r.t. a situation. The planner just needs to work with likely intentions.

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Summary – 2

We have shown a system for assisting elderly people based on Intention Recognition and Evolution Prospection The recognizer figures out intentions of elders based on

• bserved actions or effects of their actions on environment.

The prospector, being aware of external environment, of elders’ preferences and their scheduled events, provides contextually appropriate suggestions for achieving the recognized intention. And shown how their combination is useful to tackle some issues of AmI in home environment Providing proactive support Security and Emergency issues

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Summary – 3

We have presented and formalized an anytime, situation-dependent intention recognition model

It is based on incrementally constructing a Bayesian Network model as more actions are observed. LP techniques are used to situate the model according to different situations. Some methods for revising KB representation are proposed, which enable to reduce the size of the IRBN model.

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Summary – 4

We set forth a top-down approach to general collective intention recognition

It is based on Searle’s account of collective intentionality. It starts with the assumption that a collective intention is had by a virtual plural agent. This intention can be inferred using any existent individual IR methods. The inferred intention undergoes a confirmation process: Check whether there are actions reflecting mutual expectations amongst the agents. If not, fall back on individual intention recognition.

We have shown how to tackle the issue with the case of multiple elderly users.

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Future work – 1

Implement interplay between CBN and planner

Feedback from the real observed intentions may change the probabilistic relations in CBN. New observed actions by planner may rule out intentions; just change the CBN as a result of observations.

Employ advanced LP semantics for evolving program to give more flexibility in updating CBNs with new information. Implement meta-explanation about evolution prospection

Elder care assisting system should explain to elders the whys and wherefores of suggestions made.

Keep found abductive solutions or evolutions labeled by preferences used (in a partial order) instead of exhibiting only the most favorable ones.

This would allow for final preference change by the elder.

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Future Work – 2

Come up with more a easily maintained KB representation. Explore how to generalize the model to account for the case when the observed agent displays multiple intentions simultaneously. Explain the issue of changing and abandoning intentions within

• ur framework.

Apply to Ambient Intelligence domain: anytime intention recognition is very desirable therein.

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Thank you!

QUESTIONS?

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Appendix: Causal Bayes Nets - A Recap

Definition (Directed Acyclic Graph) A directed acyclic graph, also called a dag, is a directed graph with no directed cycles; that is, for any node v, there is no nonempty directed path that starts and ends on v. Definition Let G be a dag that represents causal relations between its nodes. For two nodes A and B of G If there is an edge from A to B, A is called parent of B and B is a child of A. Set of all parents of A: parents(A). If A has no parents – top node. If A has no children – bottom node. If A is neither top nor bottom – intermediate node. If value of A is observed – evidence node.

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Definition (Bayes Networks) A Bayes Network is a pair a dag whose nodes represent variables and missing edges encode conditional independencies between the variables, and an associated probability distribution satisfying Causal Markov Assumption (saying that variables are independent of their non-effects conditional on their direct causes) Definition (Causal Bayes Network) A BN is causal if its associated probability distribution satisfies the condition specifying that if a node X of its dag is actively caused to be in a given state x (an operation written as do(x), e.g. in P-log syntax), then the probability density function changes to the one of the network obtained by cutting the links from X’s parents to X, and setting X to the caused value x.

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Conditional Probability Distribution (CPD) of BN Associated with each intermediate node is a specification of the distribution of its variable, conditioned on its parents, i.e. P(A|parents(A)) is specified. For a top node, the unconditional distribution of the variable is specified. Joint distribution of all node values can be determined as the product of conditional probabilities of the value of each node on its parents P(X1, ..., XN) =

N

• i=1

P(Xi|parents(Xi)) where V = {Xi|1 ≤ i ≤ N} is the set of nodes of the dag.

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Let O = {O1, ..., Om} ⊂ V be a set of evidence nodes. Conditional probability of variable X given the observed value of evidence nodes can be determined using conditional probability formula P(X|O1, O2, ...., Om) = P(X, O) P(O) = P(X, O1, ..., Om) P(O1, ..., Om) (1) where the numerator and denominator are computed by summing the joint probabilities over all absent variables w.r.t. V .

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