Journal of Articial Intelligence Research Submitted published

A Temporal Description

for Reasoning ab out Actions and Plans

Alessandro Artale artaleirstitcit

ITCIRST Cognitive and Communication Technologies Division

I Povo TN Italy

Enrico Franconi franconicsmanacuk

Department of University of Manchester

Manchester M PL UK

Abstract

A class of intervalbased temp oral languages for uniformly representing and reasoning

ab out actions and plans is presented Actions are represented by describing what is true

while the action itself is o ccurring and plans are constructed by temp orally relating actions

and world states The temp oral languages are members of the family of Description

which are characterized by high expressivity combined with go o d computational prop erties

The subsumption problem for a class of temp oral Description Logics is investigated and

sound and complete decision pro cedures are given The basic language TLF is considered

rst it is the comp osition of a temp oral logic TL able to express interval temp oral

networks together with the nontemp oral logic F a Feature It is

proven that subsumption in this language is an NPcomplete problem Then it is shown

how to reason with the more expressive languages T LU FU and TLALC F The former

adds disjunction b oth at the temp oral and nontemp oral sides of the language the latter

extends the nontemp oral side with setvalued features ie roles and a prop ositionally

complete language

Introduction

The representation of temp oral knowledge has received considerable attention in the Ar

ticial Intelligence community in an attempt to extend existing knowledge representation

systems to deal with actions and change At the same many logicbased formalisms

were developed and analyzed by logicians and philosophers for the same purp oses In this

class of logical formalisms prop erties such as expressive p ower and computability have b een

studied as regards typical problems involving events and actions

This pap er analyzes from a theoretical p oint of view the logical and computational

prop erties of a knowledge representation system that allows us to deal with time actions

and plans in a uniform way The most common approaches to mo del actions are based

on the notion of state change eg the formal mo dels based on the original situation

calculus McCarthy Hayes Sandewall Shoham or the Stripslike planning

systems Fikes Nilsson Lifschitz in which actions are generally considered

instantaneous and dened as functions from one state to another by means of pre and

p ostconditions Here an explicit notion of time is introduced in the mo deling language

and actions are dened as occurring over time intervals following the Allen prop osal Allen

c

AI Access Foundation and Morgan Kaufmann Publishers All rights reserved

Artale Franconi

In this formalism an action is represented by describing the time course of the

world while the action o ccurs Concurrent or overlapping actions are allowed eects of

overlapping actions can b e dierent from the sum of their individual eects eects may

not directly follow the action but more complex temp oral relations may hold For instance

consider the motion of a p ointer on a screen driven by a mouse the p ointer moves b ecause

of the movement of the device on the pad there is a causeeect relation but the two

events are contemporary in the commonsense notion of the word

A class of interval temp oral logics is studied based on Description Logics and inspired by

the works of Schmiedel and of Weida and Litman In this class of formalisms

a state describ es a collection of prop erties of the world holding at a certain time Actions are

represented through temp oral constraints on world states which p ertain to the action itself

Plans are built by temp orally relating actions and states To represent the temp oral dimen

sion classical Description Logics are extended with temp oral constructors thus a uniform

representation for states actions and plans is provided Furthermore the distinction made

by Description Logics b etween the terminological and assertional asp ects of the knowledge

allows us to describ e actions and plans b oth at an abstract level actionplan types and

at an instance level individual actions and plans In this environment the subsumption

calculus is the main inference to ol for managing collections of action and plan types Action

and plan types can b e organized in a subsumptionbased taxonomy which plays the role

of an actionplan library to b e used for the tasks known in the literature as plan retrieval

and individual plan recognition Kautz A renement of the plan recognition no

tion is prop osed by splitting it into the dierent tasks of plan description classication

involving a plan type and specic plan recognition with respect to a plan description

involving an individual plan According to the latter reasoning task the system is able to

recognize which type of actionplan has taken place at a certain time interval given a set

of observations of the world

Advantages of using Description Logics are their high expressivity combined with de

sirable computational prop erties such as decidability soundness and completeness of de

duction pro cedures Buchheit Donini Schaerf Schaerf Donini Lenzerini

Nardi Schaerf Donini Lenzerini Nardi Nutt The main purp ose of this

work is to investigate a class of decidable temp oral Description Logics and to provide com

plete algorithms for computing subsumption To this aim we start with TLF a language

b eing the comp osition of a temp oral logic TL able to express interval temp oral networks

together with the nontemp oral Description Logic F a Feature Description Logic Smolka

It turns out that subsumption for TLF is an NPcomplete problem Then we show

how to reason with more expressive languages T LU FU which adds disjunction b oth at

the temp oral and nontemp oral sides of the language and TLALC F which extends the

nontemp oral side with setvalued features ie roles and a prop ositionally complete De

scription Logic Hollunder Nutt In b oth cases we show that reasoning is decidable

and we supply sound and complete pro cedures for computing subsumption

The pap er is organized as follows After introducing the main features of Description

Logics in Section Section organizes the intuitions underlying our prop osal The technical

bases are introduced by briey overviewing the temp oral extensions of Description Logics

relevant for this approach together with the interrelationships with the interval temp oral

mo dal logic sp ecically intended for time and action representation and reasoning The

A Temporal Description Logic for Reasoning about Actions and Plans

basic feature temp oral language TLF is introduced in Section The language syntax is

rst describ ed in Section together with a worked out example illustrating the informal

meaning of temp oral expressions Section reveals the mo del theoretic semantics of TLF

together with a formal denition of the subsumption and instance recognition problems

Section shows that the temp oral language is suitable for action and plan representation

and reasoning the well known cooking domain and blocks world domain are taken into

consideration The sound and complete calculus for the feature temp oral language TLF

is presented in details in Section A pro of that subsumption for TLF is an NPcomplete

problem is included The calculus for TLF forms the basic reasoning pro cedure that

can b e adapted to deal with logics having an extended prop ositional part An algorithm

for checking subsumption in presence of disjunction T LU FU is devised in Section

while in Section the nontemp oral part of the language is extended with roles and

full prop ositional calculus TLALC F In b oth cases the subsumption problem is still

decidable Op erators for homogeneity and p ersistence are presented in Section for an

adequate representation of world states In particular a p ossible solution to the frame

problem ie the problem to compute what remains unchanged by an action is suggested

Section surveys the whole sp ectrum of extensions of Description Logics for representing

and reasoning with time and action This Section is concluded by a comparison with State

Change based approaches by briey illustrating the eort made in the situation calculus

area to temp orally extend this class of formalisms Section concludes the pap er

Description Logics

Description Logics are formalisms designed for a logical reconstruction of representa

tion to ols such as frames semantic networks objectoriented and semantic data mo dels

see Calvanese Lenzerini Nardi for a survey Nowadays Description Logics

are also considered the most imp ortant unifying formalism for the many ob jectcentered

representation languages used in areas other than Knowledge Representation Imp ortant

characteristics of Description Logics are high expressivity together with decidability which

guarantee the existence of reasoning algorithms that always terminate with the correct

answers

This Section gives a brief introduction to a basic Description Logic which will serve as

the basic representation language for our prop osal As for the formal apparatus the formal

ism introduced by SchmidtSchau Smolka and further elab orated by Donini

Hollunder Lenzerini Spaccamela Nardi Nutt Donini et al Buchheit

et al De Giacomo Lenzerini is followed in this way Description

Logics are considered as a structured fragment of predicate logic ALC SchmidtSchau

Smolka is the minimal Description Logic including full negation and disjunction

ie prop ositional calculus and it is a notational variant of the prop ositional mo dal logic

K Halp ern Moses Schild

(m)

The basic types of a Description Logic are concepts roles features and individuals A

concept is a description gathering the common prop erties among a collection of individuals

from a logical p oint of view it is a unary predicate ranging over the domain of individu

Description Logics have b een also called FrameBased Description Languages Term Subsumption Lan

guages Terminological Logics Taxonomic Logics Concept Languages or KLOnelike languages

Artale Franconi

C D A j atomic concept

j top

j b ottom

C j complement

C u D j conjunction

C t D j disjunction

P C j universal quantier

P C j existential quantier

p C j selection

p q j agreement

p q j disagreement

p undenedness

p q f j atomic feature

p q path

Figure Syntax rules for the ALC F Description Logic

als Prop erties are represented either by means of roles which are interpreted as binary

relations asso ciating to individuals of a given class values for that prop erty or by means

of features which are interpreted as functions asso ciating to individuals of a given class

a single value for that prop erty The language ALC F extending ALC with features ie

functional roles is considered By the syntax rules of Figure ALC F concepts denoted by

the letters C and D are built out of atomic concepts denoted by the letter A atomic roles

denoted by the letter P and atomic features denoted by the letter f The syntax rules

are expressed following the tradition of Description Logics Baader B urckert Heinsohn

Hollunder M uller Neb el Nutt Protlich

The meaning of concept expressions is dened as sets of individuals as for unary pred

icates and the meaning of roles as sets of pairs of individuals as for binary predicates

I I I

Formally an interpretation is a pair I consisting of a set of individuals the

I

domain of I and a function the interpretation function of I mapping every concept

I I I

to a subset of every role to a subset of every feature to a partial function

I I I I I

from to and every individual into a dierent element of ie a b if a b

Unique Name Assumption such that the equations of the left column in Figure are

satised

The ALC F semantics identies concept expressions as fragments of rstorder predicate

logic Since the interpretation I assigns to every atomic concept role or feature a unary or

I

binary functional relation over resp ectively one can think of atomic concepts roles

and features as unary and binary functional predicates This can b e seen as follows an

atomic concept A an atomic role P and an atomic feature f are mapp ed resp ectively to

the op en formulas F P and F with F satisfying the functionality axiom

A

f f

y z F x y F x z y z ie F is a functional relation

f f f

The rightmost column of Figure gives the transformational semantics of ALC F ex

pressions in terms of FOL wellformed formul while the left column gives the standard

extensional semantics As far as the transformational semantics is concerned a concept C

a role P and a path p corresp ond to the FOL op en formulae F F and F

C P p

A Temporal Description Logic for Reasoning about Actions and Plans

I I

true

I

false

I I I

C n C F

C

I I I

C u D C D F F

C D

I I I

C t D C D F F

C D

I I I I

P C fa j ba b P b C g x F x F x

P C

I I I I

P C fa j ba b P b C g xF x F x

P C

I I I I

p C fa dom p j p a C g x F x F x

p C

I I I I I

p q fa dom p dom q j p a q ag (x F x F x)

p q

I I I I I

p q fa dom p dom q j p a q ag (x y F x F y )

p q

(x y F x F y x y )

p q

I I I

p n dom p x F x

p

I I I

p q p q x F x F x

p q

Figure The extensional and transformational semantics in ALC F

resp ectively It is worth noting that the extensional semantics of the left column gives also

an interpretation for the formulas of the right column so that the following prop osition

holds

Prop osition Concepts vs fol formul Let C be an ALC F concept expression

Then the transformational semantics of Figure maps C into a logically equivalent rst

order formula

A terminology or TBox is a nite set of terminological axioms For an atomic concept A

called dened concept and a p ossibly complex concept C a terminological axiom is of the

form A C An atomic concept not app earing on the lefthand side of any terminological

axiom is called a primitive concept Acyclic simple TBoxes only are considered a dened

concept may app ear at most once as the lefthand side of an axiom and no terminological

cycles are allowed ie no dened concept may o ccur neither directly nor indirectly

within its own denition Neb el An interpretation I satises A C if and only if

I I

A C

As an example consider the unary relation ie a concept denoting the class of happy

fathers dened using the atomic predicates Man Doctor Rich Famous concepts and

CHILD FRIEND roles

HappyFather Man u CHILD u CHILDDoctor u FRIENDRich t Famous

ie the men whose children are do ctors having some rich or famous friend

An ABox is a nite set of assertional axioms ie predications on individual ob jects Let

O b e the alphab et of symbols denoting individuals an assertion is an axiom of the form

C a R a b or pa b where a and b denote individuals in O C a is satised by an

I I I I I

interpretation I i a C P a b is satised by I i a b P and pa b is satised

I I I

by I i p a b

Artale Franconi

A know ledge base is a nite set of terminological and assertional axioms An interpre

tation I is a model of a knowledge base i every axiom of is satised by I logically

I I

implies A v C written j A v C if A C for every mo del of we say that A is

subsumed by C in The reasoning problem of checking whether A is subsumed by C in

I I

is called subsumption checking logically implies C a written j C a if a C

for every mo del of we say that a is an instance of C in The reasoning problem of

checking whether a is an instance of C in is called instance recognition

An acyclic simple TBox can b e transformed into an expanded TBox having the same

mo dels where no dened concept makes use in its denition of any other dened concept

In this way the interpretation of a dened concept in an expanded TBox do es not dep end

on any other dened concept It is easy to see that A is subsumed by C in an acyclic simple

TBox if and only if the expansion of A with resp ect to is subsumed by the expansion of

C with resp ect to in the empty TBox The expansion pro cedure recursively substitutes

every dened concept o ccurring in a denition with its dening expression such a pro cedure

may generate a TBox exp onential in size but it has b een proved Neb el that it works

in p olynomial time under reasonable restrictions The following interchangeably refers either

to reasoning with resp ect to a TBox or to reasoning involving expanded concepts with an

empty TBox In particular while devising the subsumption calculus for the logics considered

here it is always assumed that all dened concepts have b een expanded

Towards a Temporal Description Logics

Schmiedel prop osed to extend Description Logics with an intervalbased temp oral

logic The temp oral variant of the Description Logic is equipp ed with a mo deltheoretic

semantics The underlying Description Logic is F LE N R Donini et al it diers

from ALC F in that it do es not contain the and concepts it do es not have neither

negation nor disjunction and it has cardinality restrictions and conjunction over roles

The new temp oral termforming op erators are the temp oral qualier at the existential and

universal temp oral quantiers sometime and alltime The qualier op erator sp ecies the

time at which a concept holds The temp oral quantiers introduce the temp oral variables

constrained by means of temp oral relationships based on Allens interval algebra extended

with metric constraints to deal with durations absolute and granularities of intervals

To give an example of this temp oral Description Logic the concept of Mortal can b e dened

by

Mortal LivingBeing u (sometimex after x NOW at x LivingBeing)

with the meaning of a LivingBeing at the reference interval NOW who will not b e alive

at an interval x sometime after the reference interval NOW Schmiedel do es not prop ose

any algorithm for computing subsumption but gives some preliminary hints Actually

Schmiedels logic is argued to b e undecidable Bettini sacricing the main b enet

of Description Logics the p ossibility of having decidable inference techniques

Schmiedels temp oral Description Logic when closed under complementation contains

as a prop er fragment the temp oral logic HS prop osed by Halp ern and Shoham

The logic HS is a prop ositional mo dal logic which extends prop ositional logic with mo dal

formul of the kind hR i and R where R is a basic Allens temp oral relation and hi

A Temporal Description Logic for Reasoning about Actions and Plans

and are the p ossibility and necessity mo dal op erators For example the mo dal formula

LivingBeing hafter i LivingBeing corresp onds to the ab ovementioned Mortal concept

Unfortunately the HS logic is shown to b e undecidable at least for most interesting classes

of temp oral structures One gets decidability only in very restricted cases such as when

the set of temporal models considered is a nite collection of structures each consisting of

a nite set of natural numbers Halp ern Shoham

Weida and Litman prop ose TRex a lo ose hybrid integration b etween

Description Logics and constraint networks Plans are dened as collections of steps together

with temp oral constraints b etween their duration Each step is asso ciated with an action

type represented by a generic concept in KRep a nontemp oral Description Logic Thus

a plan is seen as a plan network a temp oral constraint network whose no des corresp onding

to time intervals are lab eled with action types and are asso ciated with the steps of the plan

itself As an example of plan in TRex they show the plan of preparing spaghetti marinara

(defplan AssembleSpaghettiMarinara

step BoilSpaghetti

step MakeMarinara

step PutTogetherSM

step b efore meets step

step b efore meets step)

This is a plan comp osed by three actions ie b oiling spaghetti preparing marinara sauce

and assembling all things at the end Temporal constraints b etween the steps establish

the temp oral order in doing the corresp onding actions A structural plan subsumption

algorithm is dened characterized in terms of graph matching and based on two separate

notions of subsumption pure terminological subsumption b etween action types lab eling

the no des and pure temp oral subsumption b etween interval relationships lab eling the arcs

The plan library is used to guide plan recognition Weida in a way similar to that

prop osed by Kautz Even if this work has strong motivations no formal semantics

is provided for the language and the reasoning problems

Starting from the assumption that an action has a duration in time our prop osal con

siders an intervalbased mo dal temp oral logic in the spirit of Halp ern and Shoham

and reduces the expressivity of Schmiedel in the direction of Weida Litman

While Schmiedels work lacks computational machinery and Halp ern and Shohams

logic is undecidable here an expressive decidable logic is obtained providing sound and

complete reasoning algorithms Dierently from TRex which uses two dierent languages

to represent actions and plans a non temp oral Description Logic for describing actions

and a second language to comp ose plans by adding temp oral information here an exten

sion of a Description Logic is chosen in which time op erators are available directly as term

constructors This view implies an integration of a temp oral domain in the semantic struc

ture where terms themselves are interpreted giving the formal way b oth for a wellfounded

notion of subsumption and for proving soundness and completeness of the corresp onding

pro cedure As an example of the formalism the plan for preparing spaghetti marinara

introduced ab ove is represented as follows

Artale Franconi

AssembleSpaghettiMarinara 3y z w y b efore meets w z b efore meets w

BoilSpaghettiy u

MakeMarinaraz u

PutTogetherSMw

Moreover it is p ossible to build temporal structured actions as opp osed to the atomic

actions prop osed in TRex describing how the world state changes b ecause of the o ccur

rence of an action in fact our language allows for feature representation in order to relate

actions to states of the world see Section This kind of expressivity is not captured

by TRex since it uses a nontemp oral Description Logic to represent actions The main

application of TRex is plan recognition according to the ideas of Kautz a Closed

World Assumption CWA Weida is made assuming that the plan library is com

plete and an observed plan will b e fully accounted for by a single plan CWA is not relied

on here following the Op en World Semantics characterizing Description Logics Weaker

but monotonic deductions are allowed in the plan recognition pro cess However their pro

cedures for recognizing a necessary optional or impossible individual plan with resp ect to

a plan description is still applicable if the plan library is given a closed world semantics

The Feature Temporal Language TLF

The feature temp oral language TLF is the basic logic considered here This language is

comp osed of the temp oral Logic TL able to express interval temp oral networks and the

nontemp oral Feature Description Logic F Note that each logic of the family of Temporal

Description Logics introduced in this pap er is identied by a comp osed string in which

the rst part refers to the temp oral part of the language while the other one refers to the

nontemp oral part

Syntax

Basic types of the language are concepts individuals temporal variables and intervals

Concepts can describ e entities of the world states and events Temporal variables denote

intervals b ound by temp oral constraints by means of which abstract temp oral patterns in

the form of constraint networks are expressed Concepts resp individuals can b e sp ecied

to hold at a certain temp oral variable resp interval In this way action types resp

individual actions can b e represented in a uniform way by temp orally related concepts

resp individuals

For the basic temp oral interval relations the Allen notation Allen Figure is

used b efore b meets m during d overlaps o starts s nishes f equal after

a metby mi contains di overlappedby oi startedby si nishedby Concept

expressions denoted by C D are syntactically built out of atomic concepts denoted by A

atomic features denoted by f atomic parametric features denoted by g and temporal

variables denoted by X Y Temporal concepts C D are distinguished from nontemp oral

concepts E F following the syntax rules of Figure Names for atomic features and

atomic parametric features are from the same alphab et of symbols the symbol is not

intended as op erator but only as dierentiating the two syntactic types

A Temporal Description Logic for Reasoning about Actions and Plans

Relation A bbr Inverse i j

beforei j b a

meetsi j m mi

overlapsi j o oi

startsi j s si

duringi j d di

f inishesi j f

Figure The Allens interval relationships

Temporal variables are introduced by the temp oral existential quantier 3 excluding

the sp ecial temp oral variable usually called NOW and intended as the reference interval

Tc must b e declared within the same temp oral Variables app earing in temp oral constraints

quantier with the exception of the sp ecial variable Temporal variables app earing in the

right hand side of an op erator are called bindable Concepts must not include unbound

aka free bindable variables Informally a bindable variable is said to b e bound in a

concept if it is declared at the nearest temp oral quantier in the b o dy of which it o ccurs

this avoid the usual formal inductive denition of a b ound variable Moreover in chained

constructs of the form C Y X Y X non bindable variables ie the ones on

the left hand side of an op erator cannot app ear more than once Note that since

Description Logics are a fragment of FOL with one free variable the ab ove mentioned

restrictions force the temp oral side of the language to have only one free temp oral variable

ie the reference time

As usual terminological axioms for building simple acyclic TLF TBoxes are allowed

While using in a concept expression a name referring to a dened concept it is p ossible to

use the substitutive qualier construct to imp ose a coreference with a variable app earing

in the denition asso ciated to the dened concept The statement C Y X constrains the

variable Y which should app ear in the denition of the dened concept C to corefer with X

see Section for an example A drawback in the use of this op erator is the requirement

to know the internal syntactical form of the dened concept namely the names of its

temp oral variables

Let O and OT b e two alphab ets of symbols denoting individuals and temporal intervals

resp ectively An assertion ie a predication on temp orally qualied individual entities

is a statement of one of the forms C i a pi a b g a b R i j where C is a concept p

is a feature g is a parametric feature R is a temp oral relation a and b denote individuals

in O i and j denote temp oral intervals in OT

Artale Franconi

TL C D E j nontemp oral concept

C u D j conjunction

C X j qualier

C Y X j substitutive qualier

3X Tc C existential quantier

Tc X R Y j temp oral constraint

X R j

R Y

Tc Tc j Tc Tc

R S R S j disjunction

s j mi j f j Allens relations

X Y x j y j z j temp oral variables

X X j X X

F E F A j atomic concept

j top

E u F j conjunction

p q j agreement

p E selection

p q f j atomic feature

g j atomic parametric feature

p q path

Figure Syntax rules for the interval Description Logic TLF

A clarifying Example

Let us now informally see the intended meaning of the terms of the language TLF for the

formal details see Section Concept expressions are interpreted over pairs of temporal

intervals and individuals hi ai meaning that the individual a is in the extension of the con

cept at the interval i If a concept is intended to describ e an action then its interpretation

can b e seen as the set of individual actions of that type o ccurring at some interval

Within a concept expression the sp ecial variable denotes the current interval of

evaluation in the case of actions it is thought that it refers to the temp oral interval

at which the action itself occurs The temp oral existential quantier introduces interval

variables related to each other and p ossibly to the variable in a way dened by the set of

temporal constraints To evaluate a concept at an interval X dierent from the current one

it is necessary to temp orally qualify it at X written C X in this way every o ccurrence of

A Temporal Description Logic for Reasoning about Actions and Plans

Basic StackBLOCK

 -

OnTableBLOCK OnBlockBLOCK

 -  -

x y

Figure Temporal dep endencies in the denition of the BasicStack action

embedded within the concept expression C is interpreted as the X variable The informal

meaning of a concept with a temp oral existential quantication can b e understo o d with the

following examples in the action domain

BasicStack 3x y x m m y ( BLOCK OnTablex u BLOCK OnBlocky )

Figure shows the temp oral dep endencies of the intervals in which the concept BasicStack

holds BasicStack denotes according to the denition a terminological axiom any

action o ccurring at some interval involving a BLOCK that was once OnTable and then

OnBlock The interval could b e understo o d as the o ccurring time of the action type b eing

dened referring to it within the denition is an explicit way to temp orally relate states

and actions o ccurring in the world with resp ect to the o ccurrence of the action itself The

temp oral constraints x m and m y state that the interval denoted by x should meet

the interval denoted by the o ccurrence interval of the action type BasicStack and

that should meet y The parametric feature BLOCK plays the role of formal parameter of

the action mapping any individual action of type BasicStack to the blo ck to b e stacked

indep endently from time Please note that whereas the existence and identity of the BLOCK

of the action is time invariant it can b e qualied dierently in dierent intervals of time

eg the BLOCK is necessarily OnTable only during the interval denoted by x

Let us comment now on the introduction of explicit temp oral variables The absence of

explicit temp oral variables would weaken the temp oral structure of a concept since arbitrary

relationships b etween more than two intervals could not b e represented anymore For

example having only implicit intervals it is not p ossible to describ e the situation in which

two concept expressions say C and D hold at two meeting intervals say x y with the rst

interval starting and the second nishing the reference interval ie the temp oral pattern

x meets y x starts y nishes cannot b e represented More precisely it is not p ossible

to represent temp oral relations b etween more than two intervals if they are not derivable by

the temp oral propagation of the constraints imp osed on pairs of variables While explicit

variables go against the general thrust of Description Logics the gained expressive p ower

together with the observation that the variables are limited only to the temp oral part of

the language are the main motivations for using them However it is easy to drop them by

limiting the temp oral expressiveness as prop osed by Bettini see also Section

An assertion of the type BasicStacki a states that the individual a is an action of

the type BasicStack o ccurred at the interval i Moreover the same assertion implies that

a is related to a BLOCK say b which is of type OnTable at some interval j meeting i and

of type OnBlock at another interval l met by i

Since any concept is implicitly temp orally qualied at the sp ecial variable it is not necessary to

explicitly qualify concepts at

Artale Franconi

E

s fhu v u v i T T j u u v v g

E

j v v u ug T f fhu v u v i T

E

j u v g T mi fhu v u v i T

meaning of the other Al len temporal relations

E E E

R S R S

E E

h j X R Y X Tc i fV X T Tc hV X V Y i R g

Figure The temp oral interpretation function

BasicStacki a b BLOCKa b j l OnTablej b OnBlockl b

mj i mi l

An individual action is an ob ject in the conceptual domain asso ciated with the relevant

prop erties or states of the world aected by the individual action itself via a bunch of

features moreover temp oral relations constrain time intervals imp osing an ordering in the

change of the states of the world

Semantics

In this Section a Tarskistyle extensional semantics for the TLF language is given and a

formal denition of the subsumption and recognition reasoning tasks is devised

Assume a linear unbounded and dense temp oral structure T P where P is

a set of time p oints and is a strict partial order on P In such a structure given an

interval X and a temp oral relation R it is always p ossible to nd an interval Y such that

X R Y The assumption of linear time which means that for any two p oints t and

t such that t t the set of p oints ft j t t t g is totally ordered ts the intuition

ab out the nature of time so that the pair t t can b e thought as the closed interval of

p oints b etween t and t The interval set of a structure T is dened as the set T of all

closed intervals u v fx P j u x v u v g in T

I I

A primitive interpretation I hT i consists of a set T the interval set of

I I

the selected temp oral structure T a set the domain of I and a function the

primitive interpretation function of I which gives a meaning to atomic concepts features

and parametric features

par tial par tial

I I I I I I I I

A T f T g

Atomic parametric features are interpreted as partial functions they dier from atomic

features for b eing indep endent from time

In order to give a meaning to temp oral expressions present in generic concept expres

sions Figure denes the temporal interpretation function The temporal interpretation

E

X Tc i function dep ends only on the temp oral structure T The lab eled directed graph h

X is the set of variables representing the no des and Tc is the set of temp oral con where

straints representing the arcs is called temporal constraint network The interpretation

A Temporal Description Logic for Reasoning about Actions and Plans

I I I I

A fa j ht ai A g A

t

V tH

I I I

V tH

I I I

C u D C D

V tH V tH V tH

I I I I I I

p q fa dom p dom q j p a q ag p q

t t t t t

V tH

I I I I

p C fa dom p j p a C g

t t

V tH V tH

I I

C X C

V tH

V V X H

I I

C C Y X

V tH

V tHfY V X g

I I I E

3 g a C X Tc C fa j W W hX Tc i

V tH

W t

Hf tg

par tial

I I I I

f j a a dom f ht ai dom f f

t t

t

I

f a f t a

t

I I I

p q p q

t t t

I I

g g

t

Figure The interpretation function

of a temp oral constraint network is a set of variable assignments that satisfy the temp oral

X T asso ciating an interval value to constraints A variable assignment is a function V

a temp oral variable A temp oral constraint network is consistent if it admits a non empty

E

X Tc i used to interpret concept expressions interpretation The notation h

fx t x t g

1 1 2 2

E

denotes the subset of h X Tc i where the variable x is mapp ed to the interval value t

i i

It is now p ossible to interpret generic concept expressions Consider the equations

I

introduced in Figure An interpretation function based on a variable assignment

V tH

V an interval t and a set of constraints H fx t g over the assignments of inner

variables extends the primitive interpretation function in such a way that the equations

I

of Figure are satised Intuitively the interpretation of a concept C is the set of

V tH

entities of the domain that are of type C at the time interval t with the assignment for the

I

free temp oral variables in C given by V see C X and with the constraints for

V tH

the assignment of variables in the scop e of the outermost temp oral quantiers given by H

Note that H interprets the variable renaming due to the temp oral substitutive qualier

I

see C Y X and it takes eect during the choice of a variable assignment as the

V tH

I

X Tc C shows equation for 3

V tH

In absence of free variables in the concept expression with the exception of for

I

b eing equivalent to the notational simplication the natural interpretation function C

t

I

interpretation function C with any V such that V t and H is introduced The

V tH

I

set of interpretations fC g obtained by varying I V t with a xed H is maximal wrt set

V tH

inclusion if H ie the set of natural interpretations includes any set of interpretations

with a xed H In fact since H represents a constraint in the assignment of variables the

unconstrained set is the larger one Note that for feature interpretation only the natural

one is used since it is not admitted to temp orally qualify them

Artale Franconi

BoilSpaghetti

 -

MakeSpaghetti Boil

 -  -

x

Figure Temporal dep endencies in the denition of the BoilSpaghetti plan

I I

for every t C An interpretation I satises the terminological axiom A C i A

t t

I I

for every interpretation I D A concept C is subsumed by a concept D C v D if C

t t

I

for some t and every interval t An interpretation I is a model for a concept C if C

t

If a concept has a mo del then it is satisable otherwise it is unsatisable

Each TLF concept expression is always satisable with the proviso that the temp oral

constraints introduced by the existential quantiers are consistent This latter condition

can b e easily checked during the reduction of the concept into a normal form when the

minimal temp oral network see Section denition is computed

It is interesting to note that only the relations s f mi are really necessary b ecause it is

p ossible to express any temp oral relationship b etween two distinct intervals using only these

three relations and their transp ositions si m Halp ern Shoham The following

equivalences hold

3x x a C x 3xy y mi x mi y C x

3x x d C x 3xy y s x f y C x

3x x o C x 3xy y s x y C x

E

To assign a meaning to ABox axioms the temp oral interpretation function is extended

E

to temp oral intervals so that i is an element of T for each i OT The semantics of

I I

pi a b assertions is the following C i a is satised by an interpretation I i a C

E

i

I I I I I I

a b g a b is satised by I i g a b and R i j is is satised by I i p

E

i

E E E

satised by I i hi j i R Given a knowledge base an individual a in O is said to

b e an instance of a concept C at the interval i if j C i a

Now we are able to give a semantic denition for the reasoning task we already called

specic plan recognition with respect to a plan description This is an inference service that

computes if an individual actionplan is an instance of an actionplan type at a certain

interval ie the task known as instance recognition in the Description Logic community

Given a knowledge base an interval i an individual a and a concept C the instance

recognition problem is to test whether j C i a

Action and plan representation two examples

An action description represents how the world state may evolve in relation with the p ossible

o ccurrence of the action itself A plan is a complex action it is describ ed by means of

temp orally related world states and simpler actions The following introduces examples of

action and plan representations from two well known domains the co oking domain Kautz

A Temporal Description Logic for Reasoning about Actions and Plans

Make Marinara Put Together SM

 -  -

z w

Boil Spaghetti

 -

y

Make Spaghetti Boil

 -  -

x y

Figure Temporal dep endencies in the denition of AssembleSpaghettiMarinara

Weida Litman and the blo ck world Allen with the aim of showing

the applicability of our framework

The Co oking Domain

Let us introduce the plan BoilSpaghetti

BoilSpaghetti 3x x b MakeSpaghettix u Boil

Figure shows the temp oral dep endencies of the intervals in which the concept BoilSpa

ghetti holds The denition employs the interval to denote the o ccurrence time of the

plan itself in this way it is p ossible to describ e how dierent actions or states of the world

concurring to the denition of the plan are related to it This is why the variable is

explicitly present in the denition of BoilSpaghetti instead of a generic variable the

Boil action should take place at the same time of the plan itself while MakeSpaghetti

o ccurs b efore it

The denition of a plan can b e reused within the denition of other plans by exploiting

the full comp ositionality of the language The plan dened ab ove BoilSpaghetti is used

in the denition of AssembleSpaghettiMarinara

AssembleSpaghettiMarinara 3y z w y b w z b w

BoilSpaghettiy u

MakeMarinaraz u

PutTogetherSMw

In this case precise temp oral relations b etween the no des of two corresp onding temp oral

constraint networks are asserted eg the action PutTogetherSM takes place strictly after

the Boil action Figure Observe that the o ccurrence interval of the plan AssembleSpa

ghettiMarinara do es not app ear in the Figure b ecause it is not temp orally related with

any other interval

A plan subsuming AssembleSpaghettiMarinara is the more general plan dened b e

low PrepareSpaghetti supp osing that the action MakeSauce subsumes MakeMarinara

This means that among all the individual actions of the type PrepareSpaghetti there are

all the individual actions of type AssembleSpaghettiMarinara

PrepareSpaghetti 3 y z BoilSpaghettiy u MakeSaucez

Artale Franconi

StackOBJ OBJ

- 

ClearBlockOBJ HoldingBlockOBJ ClearBlockOBJ

 - -

v w z

ClearBlockOBJ ONOBJ OBJ

-

x y

Figure Temporal dep endencies in the denition of the Stack action

However note that BoilSpaghetti do es not subsume PrepareSpaghetti even if it is a

conjunct in the denition of the latter This could b e b etter explained observing how the

denition of PrepareSpaghetti plan is expanded

PrepareSpaghetti 3 x y z x b y MakeSpaghettix u Boily u

MakeSaucez

Then the Boil action o ccurs at the interval y which can b e dierent from the o ccurring

time of PrepareSpaghetti as the eect of binding BoilSpaghetti to the temp oral vari

able y On the contrary in the denition of BoilSpaghetti the Boil action takes place nec

essarily at the same time Subsumption b etween PrepareSpaghetti and BoilSpaghetti

fails since dierent temp oral relations b etween the actions describing the two plans and the

plans themselves are sp ecied In particular observe that the BoilSpaghetti plan denotes

a narrower class than the plan expression

3x y x b y MakeSpaghettix u Boily

which subsumes b oth PrepareSpaghetti and BoilSpaghetti itself

The Blo cks World Domain

As a further example of the expressive p ower of the temp oral language it is now shown

how to represent the Stack action in the blo cks world in a more detailed way than the

previous simple BasicStack action used as a clarifying example Thus a stacking action

involves two blo cks which should b e b oth clear at the b eginning the central part of the

action consists of grasping one blo ck at the end the blo cks are one on top of another and

the b ottom one is no longer clear Figure

Our representation b orrows from the Rat Description Logic Heinsohn Kudenko Neb el

Protlich the intuition of representing action parameters by means of partial

functions mapping from the action itself to the involved action parameter see Section In

the language these functions are called parametric features For example the action Stack

has the parameters OBJECT and OBJECT representing in some sense the ob jects that are

involved in the action indep endently from time So in the assertion OBJECTa bl ock a

bl ock a denotes the rst ob ject involved in the action a at any interval On the other hand

an assertion involving a nonparametric feature eg ONi bl ock a bl ock b do es not

imply anything ab out the truth value at intervals other than i

The concept expression which denes the Stack action makes use of temp oral qualied

concept expressions including feature selections and agreements the expression OBJECT

ClearBlockx means that the second parameter of the action should b e a ClearBlock

A Temporal Description Logic for Reasoning about Actions and Plans

at the interval denoted by x while OBJECT ON OBJECTy indicates that at the

interval y the ob ject on which OBJECT is placed is OBJECT The formal denition of the

action Stack is

Stack 3x y z v w x y mi z mi v o w f w mi v

( OBJECT ClearBlockx u OBJECT ON OBJECTy u

OBJECT ClearBlockv u OBJECT HoldingBlockw u

OBJECT ClearBlockz )

The ab ove dened concept do es not state which prop erties are the prerequisites for the

stacking action or which prop erties must b e true whenever the action succeeds What this

action intuitively states is that OBJECT will b e on OBJECT in a situation where b oth

ob jects are clear at the start of the action Note that the world states describ ed at the

intervals denoted by v w z are the result of an action of grasping a previously clear blo ck

3x w z x o w f w mi xz mi Grasp

( OBJECT ClearBlockx u OBJECT HoldingBlockw u

OBJECT ClearBlockz )

The Stack action can b e redened by making use of the Grasp action

Stack 3x y u v x y mi u f v o

( OBJECT ClearBlockx u OBJECT ON OBJECTy u

Graspxv u)

The temp oral substitutive qualier Graspxv renames within the dened Grasp action

the variable x to v and it is a way of making coreference b etween two temp oral variables

while the temp oral constraints p eculiar to the renamed variable x are inherited by the

substituting interval v Furthermore the eect of temp orally qualifying the grasping action

at u is that the variable asso ciated to the grasping action referring to the o ccurrence

time of the action itself is b ound to the interval denoted by u Because of this binding on

the o ccurrence time of the grasping action the variable in the grasping action and the

variable in the stacking action denote dierent time intervals so that the grasping action

o ccurs at an interval nishing the o ccurrence time of the stacking action

Now it is shown how from a series of outside observations action recognition can b e

p erformed ie the task called specic plan recognition with respect to a plan description

The following ABox describ es a situation in which blo cks can b e clear grasped andor on

each other and in which a generic individual action a is taking place at time interval i

a

having the blo cks blocka and blockb as its parameters

OBJECTa bl ock a OBJECTa bl ock b

oi i ClearBlocki bl ock a i i ClearBlocki bl ock b

a a

mii i f i i HoldingBlocki bl ock a

a

mii i ClearBlocki bl ock a mii i ONi bl ock a bl ock b

a a

The system deduces that in the context of a knowledge base comp osed by the ab ove

ABox and the denition of the Stack concept in the TBox the individual action a is of

type Stack at the time interval i ie j Stacki a

a a

Artale Franconi

C X u D X C u D X

C X X C X

1 2 1

C X u D X C X u D X

1 2 1 2

C u 3 X Tc D 3X Tc C u D

if C do esnt contain free variables

X Tc (C u 3

1

3 X ) 3 C u D X Y Tc D Y X Y X X Y Tc Tc

+ 2 1 1 p q 1 2+

[Y X ][Y X ]

p q

1 1

[X ]

if D do esnt contain existential temp oral quantiers

p q C p q C

p C u D p C u p D

p q q p q p q

1 2 1 2

X Y returns the union of the two sets of variables X and Y where each Prescriptions

[Y X ][Y X ]

p q

1 1

Y o ccurring o ccurrence of Y Y is substituted by X X resp ectively while all the other elements of

1 p 1 q

in X are renamed with fresh new identiers Z is intended to b e the expression Z where the same

+

substitution or renaming has taken place The condition on the last rule forces application to start from the

last nested existential temp oral qualied concept

Figure Rewrite rules to transform an arbitrary concept into an existential concept

The Calculus for TL F

This Section presents a calculus for deciding subsumption b etween temp oral concepts in the

Description Logic TLF The calculus is based on the idea of separating the inference on

the temp oral part from the inference on the Description Logic part This is achieved by rst

lo oking for a normal form of concepts Concept subsumption in the temp oral language is

then reduced to concept subsumption b etween nontemp oral concepts and to subsumption

b etween temp oral constraint networks

Normal Form

Every TLF concept expression can b e reduced to an equivalent existential concept of the

X Tc Q u Q X u u Q X where each Q is a nontemp oral concept ie it form 3

n n

is an element of the language F A concept in existential form can b e seen as a conceptual

temporal constraint network ie a lab eled directed graph hX Tc QX i where arcs are

lab eled with a set of arbitrary temp oral relationships representing their disjunction

no des are lab eled with nontemp oral concepts and for each no de X the temp oral relation

X X is implicitly true Moreover since the normalized concepts do not contain free

variables or substitutive qualiers in the following the natural interpretation function see

Section is used

Prop osition Equivalence of EF Every concept C can be reduced in linear time

into an equivalent existential concept ef C by exhaustively applying the set of rewrite

rules of Figure

A Temporal Description Logic for Reasoning about Actions and Plans

Pro cedure Coveringh X Tci y

mid

result

Z fz X j z y Tcg

s Z do

X Tc i obtained by deleting the temp oral relation b etween if j s j and the graph h

the no de y and each of the no des in s is inconsistent

then mid mid fsg

s mid do

if t mid t s

then result result fsg

return result

Figure Pro cedure which computes a covering

Note that ef C makes explicit all the p ossible chains of features by reducing each non

temp oral concept Q to a conjunction of atomic concepts feature selections restricted to

atomic concepts and feature agreements ie each Q is a feature term expression Smolka

The normalization pro ceeds by discovering all the p ossibles interactions b etween no des

with the intention of making explicit all the implicit information A crucial temp oral in

teraction o ccurs when a no de is always coincident with a set of no des in every p ossible

interpretation of the temp oral network

X Tc i let y X and Denition Covering Given a temporal constraint network h

E

Z fz z z g X with p and y Z Z is a Covering for y if V hX Tc i

p

W Z W is not a covering for y If Z V y fV z V z V z g and for each

p

then y is called uncovered otherwise y is said covered by Z

X Tc i in Prop osition Covering pro cedure Given a temporal constraint network h

X then the procedure minimal form see eg van Beek Manchak and a node y

described in Figure returns al l the possible coverings for y with size

The idea b ehind the covering is that whenever a set of no des fz z z g is a covering

p

for y the disjunctive concept expression Q t t Q should b e conjunctively added to

z z

p

1

the concept expression Q Actually since in TLF concept disjunction is not allowed it will

y

b e sucient to add to the no de y the Least Commom Subsumer lcs of Q t t Q

z z

p

1

as dened b elow

Denition lcs Let Q Q Q C be F concept expressions Then the concept

n

Q lcsfQ Q g is such that Q v Q Q v Q and there is no C such that

n n

Q v C Q v C C < Q

n

Given a concept in existential form the temp oral completion of the constraint network is

computed as describ ed b elow

Denition Completed existential form The temporal completion of a concept in

existential form the Completed Existential Form CEF is obtained by sequentially ap

plying the fol lowing steps

Artale Franconi

closure The transitive closure of the Al len temporal relations in the conceptual

temporal constraint network is computed obtaining a minimal temporal network see

eg van Beek Manchak

collapsing For each equality temporal constraint collapse the equal nodes by

applying the fol lowing rewrite rule

X n fx g Tc Q if x x and x 3

i j j

j

x x x x

j i j i

3X Tc x x Q

i j

3 X n fx g Tc Q if x x and x

i j j

i

x x

i i

Then apply exhaustively the rst rule of Figure

X let compute the covering fZ Z g fol lowing the covering For each y

n

procedure showed by proposition Whenever the covering is not empty translate

Q Q applying the fol lowing rewrite rule Q Q u lcsfQ Z g where

i y y y in i i

m

1

X Tc QX i fz z g and Q z h

i i i i

m

1

j j

parameter introduction New information is added to each node because of the pres

ence of parameters as the fol lowing rules show The ; symbol is intended so that

each time the concept expression in the left hand side appears in some node of the

temporal constraint network possibly conjoined with other concepts then the right

hand side represents the concept expression that must be conjunctively added to al l the

other nodes square brackets point out optional parts the letters f f and g g

possibly with subscripts denote atomic parametric features while p and q stand for

generic features

g g f p C ; g g

1 n 1 n

g g f p g q ; g g

1 n 1 n

g g f f ; g g f f

1 n 1 m 1 n 1 m

g g g p f f f q ; g g u

1 n 1 m 1 n

f f

1 m

Prop osition Equivalence of CEF Every concept in existential form can be re

duced into an equivalent completed existential concept

Both the covering and the parameter introduction steps can b e computed indep endently

after the collapsing step and then conjoining the resulting concept expressions Observe

that to obtain a completed existential concept the steps of the normalization pro cedure

require linear time with the exception of the computation of the transitive closure of the

temp oral relations and the covering step Both these steps involve NPcomplete temp oral

constraint problems van Beek Cohen However it is p ossible to devise reasonable

subsets of Allens algebra for which the problem is p olynomial Renz Neb el The

most relevant prop erties of a concept in CEF is that all the admissible interval temp oral

relations are explicit and the concept expression in each no de is no more renable without

changing the overall concept meaning this is stated by the following prop osition

X Tc QX i be a conceptual tem Prop osition No de indep endence of CEF Let h

poral constraint network in its completed form CEF then for al l Q Q and for al l

A Temporal Description Logic for Reasoning about Actions and Plans

F concept expressions C such that C w Q there exists an interpretation I such that

I I

h h for some interval t X Tc Q u C X i X Tc QX i

t t

Proof The prop osition states that the information in each no de of the CEF is indep endent

I I

X Tc Q u C X i hX Tc QX i if from the information in the other no des In fact h

t t

the concept expression in one no de implies new information in some other no de Two cases

can b e distinguished

i Covered Nodes Both the collapsing rule and the covering rule provide to restrict a

covered no de with the most sp ecic F concept expression Indeed the collapsing rule

provides collapsing two contemporary no des conjoining the concept expressions of each of

them On the other hand the covering rule adds to the covered no de the most sp ecic

F concept expression that subsumes the disjunctive concept expression that is implicitly

true at the covered no de Note that thanks to the Closure rule all the p ossible equal

temp oral relations are made explicit So these two normalization rules cover all the p ossible

cases of temp oral interactions b etween no des

ii No coincident nodes Every timeinvariant information should spread over all the no des

Both parametric features and the concept have a timeinvariant semantics the only time

invariant concept expressions are g g g g f f with

n n m

n m or an arbitrary conjunction of these terms The parameter introduction rule

captures all the p ossible syntactical cases of completion concerning timeinvariant concept

expressions By induction on the syntax it can b e proven that adding to a no de any other

concept expression changes the overall interpretation 2

The last normalization pro cedure eliminates no des with redundant information This

nal normalization step ends up with the concept in the essential graph form that will b e

the normal form used for checking concept subsumption

Denition Essential graph The subgraph of the CEF of a conceptual temporal con

X Tc QX i obtained by deleting the nodes labeled only with time straint network T h

invariant concept expressions with the exception of the node is called essential graph

of T ess T

Prop osition Equivalence of essential graph Every concept in completed existen

tial form can be reduced in linear time into an equivalent essential graph form

Theorem Equivalence of normal form Every concept expression can be reduced

into an equivalent essential graph form If a polynomial fragment of Al lens algebra is

adopted the reduction takes polynomial time

As an example the normal form is shown ie the essential graph of the previously

introduced Stack action see Section

Stack 3x y v w z x y mi z mi w f v o y mi xz mi xw f x

v o d s xz s si y w m y v b y w m z v b z w mi v

( OBJECT ClearBlock u OBJECT x u

OBJECT ON OBJECTy u

OBJECT ClearBlock u OBJECT v u

OBJECT HoldBlock u OBJECT w u

OBJECT ClearBlock u OBJECT z )

Artale Franconi

In this example the essential graph is also the CEF of Stack since there are no redundant

no des

Computing Subsumption

A concept subsumes another one just in case every p ossible instance of the second is also an

instance of the rst for every time interval Thanks to the normal form concept subsump

tion in the temp oral language is reduced to concept subsumption b etween nontemp oral

concepts and to subsumption b etween temp oral constraint networks A similar general pro

cedure was rst presented in Weida Litman where the language for nontemp oral

concepts is less expressive it do es not include features or parametric features

To compute subsumption b etween nontemp oral concepts which may p ossibly include

lcs concepts we refer to Cohen Borgida Hirsh In the following we will write

w for subsumption b etween nontemp oral F concepts taking into account lcs concepts

F

Denition Variable mapping A variable mapping M is a total function M

X X such that M We write MX to intend fM X j X X g and M Tc

Tcg to intend fM X R MY j X R Y

Denition Temporal constraint subsumption A temporal constraint X R Y

is said to subsume a temporal constraint X R Y under a generic variable mapping M

E E

written X R Y w X R Y if M X X MY Y and R R

M

for every temporal interpretation E

Prop osition TC subsumption algorithm X R Y w X R Y if and

M

only if MX X MY Y and the disjuncts in R are a superset of the disjuncts

in R

Proof Follows from the observation that the temp oral relations are mutually disjoint

and their union covers the whole interval pairs space 2

Denition Temporal constraint network subsumption A temporal constraint

X Tc i subsumes a temporal constraint network hX Tc i under a variable map network h

E E

X X written hX Tc i w hX Tc i if hMX M Tc i hX Tc i ping M

M

for every temporal interpretation E

Prop osition TCN subsumption algorithm h X Tc i w hX Tc i i after

M

X X computing the temporal transitive closure there exists a variable mapping M

w Y R X exist X X which satisfy X Y Y such that for al l X

m n

j ij i j i

M

X R Y

m mn n

Proof Since from denition X R Y w X R Y implies that

m mn n

i ij j

M

E E

R R for every E then from the denition of interpretation of a temp oral

mn

ij

constraint network it is easy to see that each assignment of variables V in the interpretation

X Tc i is also an assignment in the interpretation of hM X M Tc i of h

Supp ose that one is not able to nd such a mapping then by hypothesis for each

p ossible variable mapping there exists some i j such that R is not a sup erset of R

mn

ij

A Temporal Description Logic for Reasoning about Actions and Plans

Since by assumption the temp oral constraint networks are minimal the temp oral relation

R cannot b e further restricted So for each variable mapping and each temp oral inter

mn

E

pretation E we can build an assignment V such that hV X V X i R while

m n mn

E

hV X V X i R Now we can extend the assignment V in such a way that

i j ij

E E

X Tc i while V hM X M Tc i This contradicts the assumption that V h

X Tc i w hX Tc i 2 h

M

Denition Smapping An smapping from a conceptual temporal constraint net

work h X Tc QX i to a conceptual temporal constraint network hX Tc QX i is a

variable mapping S X X such that the nontemporal concept labeling each node

X subsumes the nontemporal concept labeling the corresponding node in S X and in

h X Tc i w hX Tc i

S

The algorithm for checking subsumption b etween temp oral concept expressions reduces the

subsumer and the subsumee in essential graph form then it lo oks for an smapping b e

tween the essential graphs by exhaustive search To prove the completeness of the overall

subsumption pro cedure it will b e showed that the introduction of lcss preserves the sub

sumption A mo deltheoretic characterization of the lcs will b e given for showing this

prop erty Lets start to build an Herbrand mo del for an F concept Let C x denote the

rst order formula corresp onding to a concept C see prop osition while the function

ality of features can b e expressed with a set of formul F By syntax induction it easy to

show that C x is an existentially quantied formula with one free variable Moreover the

matrices of such formula is a conjunction of p ositive predicates F fC xg is logically

equivalent to F fC xg where the functionality axioms allow to map every subformula

V

y F x y into y F x y Then C x is such that all the existential quantiers in

f f

y

C x which come from the rst order conversion of features are replaced by quantiers

Now F fC ag where a is a constant substituting the free variable x and C a is

obtained by skolemizing the quantied variables is a set of denite Horn clauses

Denition Herbrand mo del Let C be an F concept expression Then we dene

its Minimal Herbrand Mo del H as the Minimal Herbrand Mo del of the above mentioned

C

set of denite Horn clauses F fC ag

Lemma F concept subsumption Let C D be F concept expressions and H H

C D

their minimal Herbrand models obtained by skolemizing the rst order set F fC a D ag

Then C v D i H H

D C

Proof C v D i F fC xg j D x i F fC xg j D x where C and D

are obtained by applying the functionality axioms to the set fC x D xg ie uni

fying the variables in the functional predicates and then replacing all the existential

quantiers by quantiers Now C x and D x are obtained by skolemizing the

quantied variables in the following way let C x y y x y y and let

n n

D x y y z z x y y z z with k n then skolemize

m m

k k

the formula y y z z x y y x y y z z and

n m n m

k

let x indicate its skolemized form Then C x x and D x x Now

since every existential quantication in C x D x was of type then the thesis is true

Artale Franconi

i F fC ag j D a where a is a constant substituting the free variable x see van

Dalen Now as showed by lemma b oth C a and D a have minimal Her

brand mo dels H H that verify the lemma hypothesis Then F fC ag j D a i

C D

H H 2

D C

We are now able to give a mo deltheoretic characterization of the lcs that will b e crucial

to prove the subsumptionpreserving prop erty

Lemma lcs mo del prop erty Let Q Q be F concept expressions and H

n Q

1

H their minimal Herbrand models obtained by skolemizing the rst order set F

Q

n

H a Q ag Then Q lcsfQ Q g i H H fQ

Q n Q Q

n n

1

Proof First of all let show that H is the minimal Herbrand mo del of a concept Q in

Q

the language F Every H can b e seen as a ro oted directed acyclic graph where no des

Q

i

are lab elled with p ossible empty set of atomic concepts and arcs with atomic features

while equality constraints b etween no des corresp ond to features agreement Whithout loss

of generality let us consider the case where H H H It is sucient to show

Q Q Q

1 2

that H is a ro oted directed acyclic graph Let a b e the ro ot of H H then will b e

Q Q Q

1 2

proved by induction that if F a a H where F is the rst order translation of a

i i i Q i

feature a a are obtained as a result of the skolemization pro cess and a a then

i i

fF a a F a a g H The case i is trivial Let i Now F a a

i i i Q i i i

H i F a a H H But a is uniquely dened by the skolem function

Q i i i Q Q i

1 2

are newly generated for each feature F by the where the function symbols f f

i F F

i i1

a H H skolemization pro cess Then F a a H H i F a f

i Q Q i i i Q Q i i F

1 2 1 2

i1

a H H Then the thesis is true by induction i F a f

i Q Q i i F

1 2

i1

Let us now prove the direction Supp ose by absurd that there is an F concept C

such that Q v C Q v C C < Q Then Q v C i H H and Q v C i

C Q

1

H H But then H H H ie H H Then Q v C which contradicts the

C Q C Q Q C Q

2 1 2

hypothesis

The direction can b e proved with analogous considerations 2

Prop osition lcs subsumptionpreserving prop erty Let A B C D be F con

cepts then A u B t C v D i A u lcsfB C g v D

Proof A u B t C v D i A u B v D and A u C v D Now A u B v D i F

fA a B ag j D a i H H j D a i H H H For the same reasons

A B D A B

A u C v D i H H H But then H H H and H H H ie

D A C D A B D A C

H H H H ie H H H But H H H i

D A B C D A D A

lcsfB C g lcsfB C g

A u lcsfB C g v D 2

The following theorem provides a sound and complete pro cedure to compute subsump

tion The completeness pro of takes into account that the temp oral structure is dense and

unbounded This allows us to introduce any new no de to a conceptual temp oral constraint

network without changing its meaning Remember that for each of these redundant no des

timeinvariant information holds

Theorem TLF concept subsumption A concept C subsumes a concept C i

there exists an smapping from the essential graph of C to the essential graph of C

A Temporal Description Logic for Reasoning about Actions and Plans

X Tc QX i b e the essential graph of C and T hX Tc QX i Proof Let T h

b e the essential graph of C

Soundness Follows from the fact that the essential graph form is logically equiv

alent to the starting concept and from the soundness of the pro cedures for computing

b oth the TCN subsumption prop osition and the subsumption b etween nontemp oral

concepts Cohen et al

Completeness Supp ose that such an smapping do es not exist Two main cases

can b e distinguished

i There is not a mapping M such that h X Tc i w hX Tc i By adding redundant

M

i X Tc QX h no des to T an equivalent conceptual temp oral constraint network T

may b e obtained Let us consider such an extended network in a way that there exists a

variable mapping M such that h X Tc i w X Tc i Now for all p ossible M there h

M

is a no de X X such that M X X with X X Now Q w Q since X

F

i i j j i j j

X neither can have a covering otherwise the hypothesis cannot coincide with other no des in

that the mapping M do es not exist would b e contradicted Then from prop osition Q

j

is not since T is an essential graph Then although is in a timeinvariant no de whereas Q

i

the construction of M allows for the existence of a unique V for b oth networks follows

from prop osition it is p ossible to build an instance of T that is not an instance of T

ii For each p ossible mapping M such that h X Tc i w hX Tc i there will b e always

M

Now the concept ex w Q and Q X such that MX and X two no des X

F

j i j i j i

pression Q cannot b e rened lo oking for a subsumption relationship with Q by adding

j i

to it an F concept since from prop osition this would change the overall interpretation

On the other hand the lcs introduction which would substitute the more sp ecic con

cept disjunction implicitly presents b ecause of a no de covering is a subsumptioninvariant

concept substitution as showed by lemma

Both cases contradict the assumption that T subsumes T 2

Complexity of Subsumption

Now it is shown that checking subsumption b etween TLF concept expressions in the es

sential graph form is an NPcomplete problem Therefore a p olynomial reduction from the

NPcomplete problem of deciding whether a graph contains an isomorphic subgraph is pre

sented It is then shown that the subsumption computation as prop osed in theorem

can b e done by a nondeterministic algorithm that takes p olynomial time in the size of the

concepts involved First of all let us consider the complexity of computing subsumption

b etween nontemp oral concepts

Lemma F subsumpion complexity Let C D be F concept expressions that can

contain lcss Then checking whether C v D takes polynomial time

F

Proof See Cohen et al 2

Here the problem of subgraph isomorphism is briey recalled Given two graphs G

V E and G V E G contains a subgraph isomorphic to G if there exists a

Since subsumption is computed with resp ect to a xed evaluation time V maps the dierent o ccurrences

of to the same interval this justies the choice that M

Artale Franconi

subset of the vertices V V and a subset of the edges E E such that j V jj V j

j E jj E j and there exists a onetoone function f V V satisfying fu v g E i

ff u f v g E

Given a graph G V E with V fv v g asso ciate a temp oral concept expression

n

C 3v v v b a v Av u u Av where A is an atomic concept

n i j n

and fv v g E This transformation allows us to prove that the problem of subgraph

i j

isomorphism can b e reduced to the subsumption of temp oral concepts

Prop osition Given two graphs G and G G contains a subgraph isomorphic to G

i C w C where C and C are the corresponding temporal concepts expressions

Proof A temp oral network with edges lab eled only with the b efore after relation is always

consistent minimal and nondirected Gerevini Schubert Then each temp oral

concept is in the essential graph form Now the pro of easily follows since every time G is

an isomorphic subgraph of G the onetoone function f is also an smapping from C to

C and it is true that C w C On the other hand the smapping that gives rise to the

subsumption is also the onetoone isomorphism from G to G 2

Theorem NPhardness Concept subsumption between TLF concept expressions

in normal form is an NPhard problem

Proof Follows from prop osition and the reduction b eing clearly p olynomial 2

Now the NPcompleteness is proven

Theorem NPcompleteness Concept subsumption between TLF concept expres

sions in normal form is an NPcomplete problem

Proof To prove NPcompleteness it is necessary to show that the prop osed calculus can

b e solved by a nondeterministic algorithm that takes p olynomial time Now given two

X j N and j X j N temp oral concepts T and T in their essential graph form let j

N

1

Then to check whether T w T the algorithm guesses one of the N variable mapping

from T to T and veries whether it is an smapping to o This last step can b e done in

deterministic p olynomial time since given a mapping M it is p ossible to determine whether

X Tc i w hX Tc i by checking at most N N edges lo oking for subsumption h

M

b etween the corresp onding temp oral relations solved by a set inclusion pro cedure while

the N nontemp oral concept subsumptions can b e computed in p olynomial time 2

Extending the Prop ositional Part of the Language

The prop ositional part of the temp oral language can b e extended to have a more p owerful

but still decidable Description Logic It is p ossible either to add full disjunction b oth at

the temp oral and nontemp oral levels T LU FU or to have a prop ositionally complete

language at the nontemp oral level only TLALC F

Please note that in these languages it is not p ossible to express full negation and

in particular the negation of the existential temp oral quantier This is crucial and it

If v b a v then v b a v to o

i j j i

A Temporal Description Logic for Reasoning about Actions and Plans

C t D X C X t D X

p C t D p C t p D

C t C u D C u D t C u D

1 2 1 2

3 X Tc C t D 3X Tc C t 3X Tc D

Figure Rewrite rules for computing the disjunctive form

makes the dierence with other logicbased approaches Schmiedel Bettini

Halp ern Shoham The dual of 3 ie the universal temp oral quantier 2 makes

the satisability problem and the subsumption for prop ositionally complete languages

undecidable in the most interesting temp oral structures Halp ern Shoham Venema

Bettini For the representation of actions and plans in the context of plan

recognition the universal temp oral quantier is not strictly necessary This limitation makes

these languages decidable with nice computational prop erties and capable of supp orting

other kinds of useful extensions The examples shown throughout the pap er may serve as a

partial validation of the claim Section prop oses the introduction of a limited universal

temp oral quantication that maintains decidability of subsumption

Disjunctive Concepts T LU FU

The language T LU FU adds to the basic language TLF the disjunction op erator with

the usual semantics b oth at the temp oral and nontemp oral levels

C D TL j C t D T LU

E F F j E t F FU

Before showing how to mo dify the calculus to check subsumption let us b egin with a

clarifying example The gain in expressivity allows us to describ e the alternative realizations

that a given plan may have Let us consider a scenario with a rob ot moving in an empty

ro om that can move only either horizontally or vertically Lets call RectMove that which

involves a simple sequence of the two basic moving actions Then to describ e a RectMove

plan we can make use of the disjunction op erator

RectMove 3x y m xx m y (HorMovex u VerMovey ) t

3x y m xx m y (VerMovex u HorMovey )

The Calculus for T LU FU

Normal Form

In computing subsumption a normal form for concepts is needed The normalization pro

cedure is similar to that rep orted in Section Let us start by reducing each concept

expression into an equivalent disjunctive concept of the form

X Tc X Tc

G ) t t (3 G ) t Q t t Q (3

n n n m

Artale Franconi

where G are conjunctions of concepts of the form Q X and each Q do es not contain

i i i

k k

neither temp oral information nor disjunctions ie it is an element of the language F

Prop osition Equivalence of disjunctive form Every concept C can be reduced

into an equivalent disjunctive form df C by exhaustively applying the set of rewrite rules

of Figure in addition to the rules introduced in Figure

It is now p ossible to compute the completed disjunctive normal form cdnf C Each dis

junct of such normal form has some interesting prop erties which are crucial for the pro of

of the theorem on concept subsumption temp oral constraints are always explicit ie

any two intervals are related by a basic temp oral relation there is no disjunction either

implicit or explicit neither in the conceptual part nor in the temp oral part ie it is a

TLF concept the information in each no de is indep endent of the information in the other

no des and it do es not contain timeinvariant ie redundant no des

Denition Completed disjunctive normal form Given a concept in disjunctive

form the completed disjunctive normal form is obtained by applying the fol lowing rewrite

rules to each disjunct

Temporal completion The rules of denition are applied to each disjunct with

the exclusion of the covering step which is replaced by the tintroduction step If a

disjunct is unsatisable ie the temporal constraint network associated with it is

inconsistent then eliminate it

Essential form The rules of denition are applied to each disjunct

t introduction Reduce to concepts containing only basic temporal relationships

X X R S X TcC 3X X R X Tc C t 3X X S X Tc C 3

Prop osition Equivalence of CDNF Every concept expression can be reduced into

an equivalent completed disjunctive normal concept

Subsumption

The theorem reduces subsumption b etween CDNF concepts into subsumption of disjun

ctionfree concepts such that the results of theorem can b e applied The following

theorem gives a terminating sound and complete subsumption calculus for T LU FU

Theorem T LU FU concept subsumption Let C C t t C and D D t

m

t D be T LU FU concepts in CDNF Then C v D if and only if ij C v D

n i j

Proof Since it is easy to show that C t t C v D i iC v D we need only to prove the

n i

restricted thesis C v D t t D i C v D C v D Every concept expression in

i n i i n

CDNF corresp onds to an existential quantied formula with two free variables Moreover

the matrices of such formul are conjunctions of p ositive predicates Let us denote the

formula corresp onding to a concept C as C t x Now the restricted thesis holds i it is

a b Now let H the minimal Herbrand mo del a b D true that F fC a bg j D

B

i

a b a b D a b i H j D a b D a bg j D a bg Then F fC of F fC

B

i i

a b is valid in H if and only if either a b D Since we are talking of a single mo del D

B

a b is valid in H This proves the theorem 2 a b or D D

B

The pro of of this theorem comes from an idea of Werner Nutt

A Temporal Description Logic for Reasoning about Actions and Plans

As a consequence of the theorems the following complexity result holds

Theorem T LU FU subsumption complexity Concept subsumption between T LU FU

concept expressions in normal form is an NPcomplete problem

A Prop ositionally Complete Language TLALC F

TLALC F uses the prop ositionally complete Description Logic ALC F Hollunder Nutt

for nontemp oral concepts by changing the syntax rules for TLF in the following

way

E F FU j j E j p q j p j P E j P E ALC F

The interpretation functions are extended to take into account roles

I I I

P T

I I I I

P j a b ha bi P ht a bi P P

t t

t

As seen in Section ALC F adds to F full negation thus introducing disagreement p q

and undenedness p for features and role quantication P E P E

As an example of the expressive p ower gained let us rene the description of the world

states involved in the Stack action see Section Supp ose that a blo ck is describ ed by

saying that it has LATERALSIDEs role and BOTTOM and TOPSIDEs features Then the

prop erty of b eing clear could b e represented as follows

ClearBlock Block u LATERALSIDEClear u TOPSIDE HASABOVE

which says that in order to b e clear each LATERALSIDE has to b e clear and nothing has

to b e over the TOPSIDE Now the situation in which a blo ck involved in a Stack action is

on top of another one is reformulated with the following concept expression

OBJECT TOPSIDE HASABOVE OBJECT

Furthermore given the ab ove denition of ClearBlocks it can b e derived that

OBJECT TOPSIDE HASABOVE OBJECT v OBJECT ClearBlock

ie an ob ject having another ob ject on top of it is no more a clear ob ject

In TLALC F it is p ossible to describ e states with some form of incomplete knowledge

by exploiting the disjunction among nontemp oral concepts For example let us say that

the agent of an action can b e either a human b eing or a machine AGENTPerson t Robot

The Calculus for TLALC F

This Section presents a calculus for deciding subsumption b etween temp oral concepts in

the Description Logic TLALC F Again the calculus is based on the idea of separating the

inference on the temp oral part from the inference on the Description Logic part v

ALC F

and adopting standard pro cedures developed in the two areas

Normal Form

Once more the subsumption calculus is based on a normalization pro cedure The rst

X Tc Q u step reduces a concept expression into an equivalent existential form 3

Q X u u Q X by applying the rewrite rules of Figure augmented with the

n n

Artale Franconi

C u D C t D

C t D C u D

C C

P C P C

P C P C

f C f t f C

p C f t f q C if p f q

p q p t q t p q

p q p t q t p q

f p f t f p

Note By f we denote b oth an atomic feature and an atomic parametric feature

Figure Rewrite rules to transform an arbitrary concept into a simple concept

rule p q q p q p q Each Q is a nontemp oral concept ie it is an element

of the language ALC F

In the following normalization step there will b e a need to verify concept satisability

for nontemp oral concept expressions An ALC F concept E is unsatisable i E v

ALC F

Algorithms for checking satisability and subsumption of concepts terms in ALC F are well

known Hollunder Nutt

Denition Completed existential form The temp oral completion of a concept

in existential form the Completed Existential Form CEF is obtained by sequentially

applying the fol lowing steps

closure collapsing covering As reported in denition As for the covering

translate the concept expression Q applying the rewrite rule Q Q u Q t

y y y in i

1

t Q

i

m

parameter introduction This requires two phases

Each Q is translated in disjunctive normal form First the simple form is ob

tained by transforming each Q fol lowing the rewrite rules reported in Figure

The disjunctive normal form is then obtained by rewriting each Q which is now

in simple form using the fol lowing rules which correspond to the rst order

rules for computing the disjunctive normal form of logical formul

C t C u D C u D t C u D

p C t D p C t p D

A simple concept contains only complements of the form A where A is a primitive concept and no

subconcepts of the form p where p is not an atomic parametric feature this corresp onds to a rst

order logical formula in negation normal form

A Temporal Description Logic for Reasoning about Actions and Plans

g g f p C g g

1 n 1 n

g g f p g q g g

1 n 1 n

g g f f g g f f

1 n 1 m 1 n 1 m

g g g p f f f q g g u

1 n 1 m 1 n

f f

1 m

g g f f g g f f

1 n 1 m 1 n 1 m

g g

g g g g g g

1 n n+1 1 n n+1

g g f p g q g g

1 n 1 n

g g g p f f f q g g u

1 n 1 m 1 n

f f

1 m

Figure Rewrite rules that compute the parameter introduction step

For each Q E t t E on compute its timeinvariant part let us indicate

j j j

n

1

this particular concept expression as Q This gives Q by computing for each

j j

v then If E in Q its timeinvariant information E disjunct E

j j j j

ALC F

i i i

as showed in Figure while Otherwise rewrite every conjunct in E E

j j

i i

the conjuncts not considered there are rewrote to Now unless there is an

Q E t t E E must be conjunctively added to al l the other nodes

j j j j

n

1

i

Prop osition Equivalence of CEF Every concept in existential form can be re

duced into an equivalent completed existential concept

As for the TLF case b oth covering and parameter introduction can b e computed inde

p endently As a consequence of the ab ove normalization phase the prop osition no de

indep endence is now true for TLALC F concepts in CEF Observe that to obtain a CEF

concept the steps of the normalization pro cedure require the computation of the transitive

closure of the temp oral relations which is an NPcomplete problem van Beek Co

hen and the computation of ALC F subsumption which is a PSPACEcomplete

problem Hollunder Nutt

Before the presentation of the last normalization phase which will eliminate redundant

no des it is now p ossible to check whether a concept expression is satisable

X Tc QX i Prop osition Concept satisability A TLALC F concept in CEF h

is satisable with the proviso that the temporal constraints are satisable if and only if the

nontemporal concepts labeling each node in X are satisable Checking satisability of a

TLALC F concept in CEF is a PSPACEcomplete problem

Proof Is a direct consequence of the no de indep endence established by prop osition

which is true also for TLALC F concepts in CEF 2

The normalization pro cedure now go es on by rewriting unsatisable concepts to and

then computing the essential graph form for satisable concepts This last phase is more

Artale Franconi

complex than for the other temp oral languages considered in this pap er essentially b ecause

ALC F can express the concept by means of a concept expression eg A t A

From this consideration it follows that in TLALC F a redundant no de can b e derived from

a complex concept expression eg b oth A t A and g A t g A are redundant no des

The key idea is that all the timeinvariant information is present in the no de thanks to

the CEF Thus it is needed only to extract this information from the no de by computing

the disjunctive normal form of Q applying the translation and then testing whether

Q v Q for a given no de x

i i

ALC F

Denition Essential graph The subgraph of the CEF of a TLALC F conceptual

X Tc QX i obtained by deleting the nodes x such that temporal constraint network T h

i

Q v Q with the exception of the node is called essential graph of T ess T

i

ALC F

Prop osition Equivalence of essential graph Every CEF concept can be reduced

into an equivalent essential graph form and obviously every concept can be reduced into

an equivalent essential graph form

Subsumption

The overall normalization pro cedure reduces the subsumption problem in TLALC F to the

subsumption b etween ALC F concepts

Theorem TLALC F concept subsumption A concept C subsumes a concept

C if and only if there exists an smapping from the essential graph of C to the essential

graph of C

The ab ove theorem gives a sound and complete algorithm for computing subsumption b e

tween TLALC F concepts the pro of is the same as the one for theorem The sub

sumption problem is now PSPACEhard since satisability and subsumption for ALC F

concepts were proven to b e PSPACEcomplete Hollunder Nutt

Extending the Expressivity for States

The following suggests how to extend the basic language to cop e with imp ortant issues in

the representation of states i Homogeneity allows us to consider prop erties of the world

p eculiar to states which remain true in each subinterval of the interval in which they hold

ii Persistence guarantees that a state holding as an eect of an action continues to hold

unless there is no evidence of its falsity at some time An approach to the frame problem is

then presented showing a p ossible solution to one of the most infamous problems in AI

literature The following subsections shall b e interested more in semantically characterizing

actions and states than on computational prop erties The extensions prop osed now to the

temp oral languages are for having a full edged Description Logic for time and action

Homogeneity

In the temp oral literature homogeneity characterizes the temp oral b ehavior of world states

when a state holds over an interval of time t it also holds over subintervals of t Thus if

A Temporal Description Logic for Reasoning about Actions and Plans

SimpleStackBLOCK

 -

rOnTableBLOCK rOnBlockBLOCK

 -  -

x y

Figure Temporal dep endencies in the denition of the SimpleStack action

a blo ck is on the table for a whole day one can conclude that it is also on the table in the

morning On the other hand actions are not necessarily homogeneous In the linguistic

literature a dierence is made b etween activity and performance verbs The distinction

comes out in the fact that activity verbs do have subevents that are denoted by the same

verb whereas p erformance verbs do not Generally activity verbs represent ongoing events

for example to eat and to run and can b e describ ed as homogeneous predicates whereas

p erformance verbs represent events with a well dened granularity in time such as to prepare

spaghetti Performance verbs are an example of antihomogeneous events if they o ccur over

an interval of time t then they do not o ccur over a subinterval of t as they would not yet

b e completed

The language is extended by introducing the Homogeneity op erator

C D rC homogeneous concept

The semantics of homogeneous concepts is easily given in terms of the semantics of the

temp oral universal quantier rC 2x x s d f C x This means that rC

is an homogeneous concept if and only if when it holds at an interval it remains true at

each subinterval In particular 2x universally qualies the temp oral variable x while the

temp oral constraint x s d f imp oses that x is a generic interval contained in

Moreover it is always true that rC v C ie rC is a more sp ecic concept than C

Let us consider as an example a more accurate denition of the BasicStack action

see Section

SimpleStack 3x y x m m y ( BLOCK rOnTablex u

BLOCK rOnBlocky )

Figure shows the temp oral dep endencies of the intervals in which the SimpleStack

holds The dierence with the BasicStack action is the use of the homogeneity op erator

In fact since the predicates OnTable and OnBlock denote states their homogeneity should

b e explicitly declared The assertion SimpleStacki a says that a is an individual action

of type SimpleStack o ccurred at interval i Moreover the same assertion implies that a

is related to a BLOCK say b which is of type OnTable at some interval j meeting i and

at all intervals included in j while it is of type OnBlock at another interval l met by i

and at all intervals included in l

SimpleStacki a b BLOCKa b j l mj i mi l

l s d f j s d f l l

OnTable b OnBlockl b

Artale Franconi

Instant StackBLOCK

 -

rOnTableBLOCK rOnBlockBLOCK

 - -

z y

Figure Temporal dep endencies in the denition of the InstantStack action

Note that the SimpleStack action subsumes the InstantStack action whose temp oral

dep endencies are depicted in Figure

InstantStack 3z y f z m y (BLOCK rOnTablez u

BLOCK rOnBlocky )

Subsumption holds b ecause the class of intervals obtained by homogeneity of the state

OnTable as dened in the SimpleStack action including x and all its subintervals is a

subset of the class of intervals over which the blo ck is known to b e on the table according

to the denition of InstantStack this latter class includes all the subintervals of z

If the InstantStack action had b een dened without the r op erator then it would not

sp ecialize any more the SimpleStack action In fact according to such a weaker denition

of InstantStack sp ecifying that the ob ject is on the table at z do es not imply that the

ob ject is on the table at subintervals of z in particular it is not p ossible to deduce any

more that the ob ject is on the table at x and its subintervals as sp ecied in the denition of

SimpleStack action Moreover the weak InstantStack action type would not sp ecialize

the weak SimpleStack action type ie BasicStack to o Thus homogeneity helps

us to dene states and actions in a more accurate way such that imp ortant inferences are

captured

As seen ab ove the denition of homogeneity makes use of universal temp oral quan

tication Remember that subsumption in a prop ositionally complete Description Logic

with b oth existential and universal temp oral quantication is undecidable and it is still an

op en problem if it b ecomes decidable in absence of negation Bettini The homo

geneity op erator is a restricted form of universal quantication An even more restricted

form interests us here where the concept C in rC do es not contain any other temp oral

op erator called simple homogeneous concept The expressiveness of the resulting logic is

enough for example to correctly represent the homogeneous nature of states In Artale

Bettini Franconi an algorithm to compute subsumption in TLF augmented with

the homogeneity op erator is prop osed Even if a formal pro of is still not available go o d

arguments are discussed to conjecture its completeness This would also prove decidability

of this logic and of the corresp onding mo dal logics

Persistence

This Section shows how our framework can b e successfully extended in a general way to

cop e with inertial prop erties In the basic temp oral language a prop erty holding say as

a p ostcondition of an action at a certain interval is not guaranteed to hold anymore at

other included or subsequent intervals This is the reason why we prop ose an extended

A Temporal Description Logic for Reasoning about Actions and Plans

LoadGUN LoadedGUN FireGUNTARGET LoadedGUN

 

- -

x z



LoadedGUN DeadTARGET

-

x y



or

Figure Denitions of the actions Load and Fire

formalism in which states can b e represented as homogeneous and p ersistent concepts

As a motivation for introducing the p ossibility of representing p ersistent prop erties in the

language this Section considers how to solve the frame problem and in particular the

famous example of the Yale Turkey Sho oting Scenario Sandewall Allen Ferguson

formerly known as the Yale Shooting Problem

An inertia op erator is introduced here Intuitively C is currently true if it was

true at a preceding interval say i and there is no evidence of the falsity of C at any

interval b etween the current one and i Thus the prop erty of an individual of b eing of type

C persists over time unless a contradiction arises

The formalization of the inertia op erator makes use of the epistemic op erator K Donini

Lenzerini Nardi Schaerf Nutt in which KC denotes the set of individuals known

to b e instances of the concept C

Denition Inertia C j a i

i star ti star tj C i a

h star th endi endh endj KC h a

where start and end are two functions giving resp ectively the starting and the ending p oint

of an interval conditions on endp oints are simpler and more readable than their equivalents

on interval relations KC h a means that it is not known that a is not of type C at

interval h Furthermore the following relation holds a j C j a C j a ie C

subsumes C The ab ove denition can b e captured by a temp oral language equipp ed with

the epistemic op erator K and the homogeneity op erator r

C C t 3x y x b m o di x s si y y C x u rKC y

Two action types are dened Load with the parameter GUN and Fire with the

parameters GUN and TARGET Figure

Load 3x m x GUN Loadedx

Fire 3x y z f x m y m z

GUN Loadedx t TARGET Deady u GUN Loadedz

The action Load describ es loading a gun The action Fire describ es ring the gun against

a target eects of ring are that the gun b ecomes unloaded and either the target is dead

An epistemic interpretation is a pair I W in which I is an interpretation and W is a set of interpreta

T

J W I W

C tions such that KC

J W

Artale Franconi

Loadgun Loadedgun Firegunfred Loadedgun

 -  -

j j i i

1 1

Loadedgun Deadfred

-

j j

2 0

Figure Actions instances in the Yale Sho oting Problem

or the gun was not loaded p ossibly by inertia b efore ring The Yale Sho oting Problem

considers the situation describ ed by the following set of assertions ABox

Loadi load action GUNload action gun aj i Firej re action

GUNre action gun TARGETre action fred

ie at the b eginning the gun is loaded then the action of ring the gun against the target

f r ed is p erformed According to the semantics of the language logical consequences of the

knowledge base are

j i mi i Loadedi gun

j j mj j Loadedj gun

j j f j j Loadedj gun

j j mj j Deadj fred

ie see also Figure i the Load action makes the gun loaded ii the Fire action

makes the gun unloaded at the end iii since there is no evidence to the contrary the gun

is still loaded at j by inertia iv since the gun is not unloaded at j the target f r ed must

b e dead

Since the inertia op erator is useful to describ e the b ehavior of properties which are

characterized as homogeneous concepts a simple way of representing p ersistence in the

context of homogeneous concepts is prop osed

Prop osition Let P be a property ie P rP is an homogeneous concept and

a know ledge base such that j P j a P j a is true in ie j P j a

if and only if two intervals i k exist such that j (star ti star tj P i a) and

fsi k f j k P k ag is satisable

Proof The entailment test veries the rst part of the denition of inertia while the

satisability test veries that b etween the interval at which the system knows that the

individual a b elongs to P i and the interval at which P a is deduced by inertia j

do es not exist an interval h at which the system knows that P a is false Indeed such

interval h would b e related to the interval k by the relation in and since it is supp osed

that P is homogeneous the knowledge base with P h a P k a in h k would b e

inconsistent 2

The deduction P j a P j a can b e obtained as a particular case of the ab ove stated

prop osition

A Temporal Description Logic for Reasoning about Actions and Plans

Related Works

The original formalism devised by Allen forms in its very basis the foundation for

our work It is a predicate logic in which interval temp oral networks can b e introduced

prop erties can b e asserted to hold over intervals and events can b e said to occur at inter

vals His approach is very general but it suers from problems related to the semantic

formalization of the predicates hold and occur Blackburn Moreover computa

tional prop erties of the formalism are not analyzed The study of this latter asp ect was on

the contrary our main concern

In the Description Logic literature other approaches for representing and reasoning with

time and action were prop osed In the b eginning the approaches based on an explicit notion

of time are surveyed and then the Stripslike approaches are considered This Section ends

by illustrating some of the approaches devoted to temp orally extend the situation calculus

Bettini suggests a variablefree extension with b oth existential and universal

temp oral quantication He gives undecidability results for a class of temp oral languages

resorting to the undecidability results of Halp ern and Shohams temp oral logic and in

vestigates approximated reasoning algorithms Basically he extends the ALC N description

logics with the existential and universal temp oral quantiers but unlike our formalism

explicit interval variables are not allowed The temp oral quantication makes use of a set

of temp oral constraints on two implicit intervals the reference interval and the current one

In this framework the concept of Mortal can b e dened as

Mortal LivingBeing u 3after not LivingBeing

Schild prop oses the embedding of pointbased tense op erators in a prop ositionally

closed Description Logic He proved that satisability in ALC T the p ointbased temp oral

extension of ALC interpreted on a linear unbounded and discrete temp oral structure is

PSPACEcomplete His ideas were applied by Fischer Neuwirth in the Back

system Note that a p ointbased temp oral ontology is unable to express all the variety of

relations b etween intervals

Baader and Laux integrate mo dal op erators for time and b elief in a terminological

system lo oking for an adequate semantics for the resulting combined language The ma jor

p oint in this pap er is the p ossibility of using mo dal op erators not only inside concept

expressions but also in front of concept denitions and assertions The following example

shows the notion of Happyfather where dierent mo dalities interact

BELJOHN Happyfather MARRIEDTOWoman u BELJOHNPretty u

hfutureiCHILDGraduate

In this case it is Johns b elief that a Happyfather is someone married to a woman b elieved

to b e pretty by John and whose children will b e graduates sometime in the future The

semantics has a Kripkestyle each mo dal op erator is interpreted as an accessibility relation

on a set of p ossible worlds while the domain of ob jects is split into p ossible dierent

domain ob jects each one dep ending on a given world This latter choice captures the case of

dierent denitions for the same concept such as BELJOHNA B and BELPETERA

C since the two formul are evaluated in dierent worlds The main restriction is that

all the mo dal op erators do not satisfy any sp ecic axioms for b elief or time On the other

hand the language is provided with a complete and terminating algorithm that should

Artale Franconi

serve as the authors prop ose as a basis for satisability algorithms for more complex

languages

There are Description Logics intended to represent and reasoning ab out actions following

the Strips tradition Heinsohn Kudenko Neb el and Protlich describ e the Rat

system used in the Wip pro ject at the German Research Center for AI DFKI They use a

Description Logic to represent b oth the world states and atomic actions A second formalism

is added to comp ose actions in plans and to reason ab out simple temp oral relationships No

explicit temp oral constraints can b e expressed in the language Rat actions are dened by

the change of the world state they cause and they are instantaneous as in the Stripslike

systems while plans are linear sequences of actions The most imp ortant service oered

by Rat is the simulated execution of part of a plan checking if a given plan is feasible

and if so computing the global pre and p ostconditions The feasibility test is similar

to the usual consistency check for a concept description they temporally project the pre

and p ostconditions of individual actions comp osing the plan resp ectively backward and

forward If this do es not lead to an inconsistent initial nal or intermediate state the plan

is feasible and the global pre and p ostconditions are determined as a side eect

Devanbu and Litman describ e the Clasp system a planbased knowledge

representation system extending the notion of subsumption and classication to plans to

build an ecient information retrieval system In particular Clasp was used to repre

sent planlike knowledge in the domain of telephone switching software by extending the

use of the software information system lassie Devanbu Brachman Selfridge Ballard

Clasp is designed for representing and reasoning ab out large collections of plan

descriptions using a language able to express temp oral conditional and lo oping op erators

Following the Strips tradition plan descriptions are built starting from states and actions

b oth represented by using the Classic Brachman McGuiness PatelSchneider Resnick

Borgida terminological language Since plans constructing op erators corresp ond

to regular expressions algorithms for subsumption integrate work in automata theory with

work in concept subsumption The temp oral expressive p ower of this system can capture

to sequences disjunction and iterations of actions and each action is instantaneous Fur

thermore state descriptions are restricted to a simple conjunction of primitive Classic

concepts Like Rat Clasp checks if an instantiated plan is well formed ie the sp ecied

sequence of individual actions are able to transform the given initial state into the goal state

by using the Strips rules

We end up by rep orting on the eorts made by researchers in the situation calculus

eld to overcome the strict sequential p ersp ective inherent to this framework Recent works

enrich the original framework to represent prop erties and actions having dierent truth

values dep ending not only on the situation but also on time The work of Reiter

moving from the results showed by Pinto and by Ternovskaia provides a

new axiomatization of the situation calculus able to capture concurrent actions prop erties

with continuous changes and natural exogenous actions those under natures control The

notion of uent which mo dels prop erties of the world and situation are maintained Each

action is instantaneous and resp onsible for changing the actual situation to the subsequent

one Concurrent actions are simply sets of instantaneous actions that must b e coherent

ie the actions collection must b e non empty and all the actions o ccur at the same time

Pinto and Reiter introduce the time dimension essentially to capture b oth

A Temporal Description Logic for Reasoning about Actions and Plans

the o ccurrence of the natural actions due to known laws of physics ie the ball b ouncing

at times prescrib ed by motions equations and the dynamic b ehavior of physical ob jects

ie the p osition of a falling ball This is realized by introducing a time argument for

each action function while prop erties of the world are divided into two dierent classes

classical uents that hold or do not hold throughout situations and continuous parameters

that may change their value during the time spanned by the given situation

More devoted to have a situation calculus with a time interval ontology is the work of

Ternovskaia In order to describ e processes ie actions extended in time she

introduces durationless actions that initiate and terminate those pro cesses As a matter of

fact pro cesses b ecome uents with instantaneous events StartFluent and FinishFluent

which resp ectively make true or false the corresp onding uent and with p ersistence

assumptions that make the uent true during the interval For example in a blo cks world

the pickingup pro cess is treated as a uent with Startpickingupx and Finishpicking

upx instantaneous actions that enable or falsify the pickingup uent

Conclusions

The main ob jective of this pap er was the design of a class of logical formalisms for uni

formly representing time actions and plans According to this framework an action has a

duration in time it can have parameters which are the ties with the temp oral evolution

of the world and it is p ossibly asso ciated over time with other actions A mo deltheoretic

semantics including b oth a temp oral and an ob ject domain was developed for giving b oth

a meaning to the language formul and a well founded denition of the various reasoning

services allowing us to prove soundness and completeness of the corresp onding algorithms

The p eculiar computational prop erties of this logic make it an eective representation and

reasoning to ol for plan recognition purp oses An action taxonomy based on subsumption

can b e set up and it can play the role of a plan library for plan retrieval tasks

This pap er contributes to exploration of the decidable realm of intervalbased temp oral

extensions of Description Logics It presented complete pro cedures for subsumption rea

soning with TLF T LU FU and TLALC F In addition the subsumption problem for

TLF was proven an NPcomplete problem The subsumption pro cedures are based on

an interpretation preserving transformation that op erates a separation b etween the tem

p oral and the nontemp oral parts of the formalism Thus the calculus can adopt distinct

standard pro cedures developed in the Description Logics community and in the temp oral

constraints community To obtain decidable languages the key idea was to restrict the tem

p oral expressivity by eliminating the universal quantication on temp oral variables While

a prop ositionally complete Description Logic with b oth existential and universal temp oral

quantication is undecidable it is still an op en problem if it b ecomes decidable in absence

of negation With the introduction of the homogeneity op erator investigation of the impact

of a restricted form of temp oral universal quantication in the language TLF was b egun

Several extensions were prop osed to the basic temp oral language With the p ossibility

to sp ecify homogeneous predicates the temp oral b ehavior of world states can b e describ ed

in a more natural way while the introduction of the nonmonotonic inertial op erator gives

rise to some forms of temp oral prediction Another extension not considered in this pap er

deals with the p ossibility of relating an action to more elementary actions decomposing

Artale Franconi

it in partially ordered steps Artale Franconi This kind of reasoning is found in

hierarchical planners like Nonlin Tate Sipe Wilkins and Forbin Dean

Firby Miller

Acknowledgements

This pap er is a substantial extension and revision of Artale Franconi The work

was partially supp orted by the Italian National Research Council CNR pro ject Ontologic

and Linguistic Tools for Conceptual Mo deling and by the Foundations of Data Warehouse

Quality DWQ Europ ean ESPRIT IV Long Term Research LTR Pro ject The

rst author wishes to acknowledge also LADSEBCNR of Padova and the University of

Firenze for having supp orted part of his work Some of the work carried on for this pap er

was done while the second author was working at ITCirst Trento This work owes a lot to

our colleagues Claudio Bettini and Alfonso Gerevini for having introduced us many years

ago to the temporal maze Sp ecial thanks to Achille C Varzi for taking time to review the

technical details of the pap er and for his insightful comments on the philosophy of events

and to Fausto Giunchiglia for useful discussions and feedback Thanks to Paolo Bresciani

Nicola Guarino Eugenia Ternovskaia and Andrea Schaerf for enlightening comments on

earlier drafts of the pap er Werner Nutt and Luciano Serani help ed us to have a deep er

insight into logic We would also like to thank Carsten Lutz for the helpful discussions we

had with him ab out temp oral representations Many anonymous referees checked out many

errors of previous versions of the pap er All the errors of the pap er are of course our own

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