Báo cáo khoa học: "A Modal Temporal Logic for Reasoning about Change" ppt

Furthermore, in cases where it is necessary for the temporal structure to be dense or continuous, there is no immediate argument against modal temporal logic in general.. That Is, one co

Eric Mays

D e p a r t m e n t of C o m p u t e r and I n f o r m a t i o n S c i e n c e

M o o r e School of E l e c t r i c a l E n g i n e e r l n g / D 2

U n i v e r s i t y of P e n n s y l v a n i a

P h i l a d e l p h i a , PA 19104

ABSTRACT

We examine several b e h a v i o r s for query

systems that become possible with the ability to

r e p r e s e n t and reason about change in data bases:

q u e r i e s about possible futures, queries about

a l t e r n a t i v e h i s t o r i e s , and offers of m o n i t o r s as

r e s p o n s e s to queries A modal temporal logic is

d e v e l o p e d for this purpose A c o m p l e t i o n axiom for

h i s t o r y is given and m o d e l l i n g s t r a t e g i e s are

g i v e n by example

I I N T R O D U C T I O N

In this paper we present a modal temporal

logic that has been developed for reasoning about

change in data bases The basic m o t i v a t i o n is as

follows A data base c o n t a i n s i n f o r m a t i o n about

the world: as the w o r l d changes, so does the data

base p r o b a b l y m a i n t a i n i n g some d e s c r i p t i o n of

what the world was like before the change took

place M o r e o v e r , if the w o r l d is c o n s t r a i n e d In

the ways it can change, so is the dat~ base We

are m o t i v a t e d by the b e n e f i t s to be gained by

being able to represent those c o n s t r a i n t s and use

them to reason about the possible states of a data

b a s e

It is g e n e r a l l y a c c e p t e d that a natural

language query s y s t e m often needs to provide more

than just the literal answer to a question For

example, [Kaplan 82I presents methods for

c o r r e c t i n g a q u e s t i o n e r l s m i s c o n c e p t i o n s (as

reflected in a query) about the contents of a data

base, as well as p r o v i d i n g a d d i t i o n a l i n f o r m a t i o n

in suvport of the literal a n s w e r to a query, By

e n r i c h i n g the data base model, K a p l a n ' s work on

c o r r e c t i n g m i s c o n c e p t i o n s was extended in [Mays

801 to d i s t i n q u i s h b e t w e e n m i s c o n c e p t i o n s about

data base structure and data base contents In

e i t h e r case, however, the model was a static one

By i n c o r p o r a t i n g a model of the data base in which

a dynamic view is allowed, answers to q u e s t i o n s

can include an offer to m o n i t o r for some c o n d i t i o n

w h i c h might p o s s i b l y occur in the future The

following is an example:

U: "Is the Kitty Hawk in N o r f o l k ? "

S: "No, shall I let you know when she is?"

IThJs work is p a r t i a l l y s u p p o r t e d by a grant

from the Natlonal Science Foundation, NSF-MCS

81-07290

But just having a dynamic view is not adequate, it

is necessary r-y ~at the d y n a m i c view c o r r e s p o n d to the possible e v o l u t i o n of the world that is modelled O t h e r w i s e , b e h a v i o r s such as the following might arise:

U: "Is New York less than 50 miles from

P h i l a d e l p h i a ? "

S: "No, shall I let you k n o w when it is?"

An offer of a m o n i t o r is said to be c o m p e t e n t only

if the c o n d i t l o n to be m o n i t o r e d can p o s s i b l y occur Thus, in the latter example the offer is not competent, while in the former it is This paper is c o n c e r n e d with d e v e l o p i n g a lo~ic for reasoning about change in data bases, and

a s s e s s i n g the impact of that c a p a b i l i t y on the

b e h a v i o r of q u e s t i o n a n s w e r i n g systems The general area of e x t e n d e d i n t e r a c t i o n in data base systems is d i s c u s s e d in [WJMM 831

As just pointed out, the a b i l i t y to represent and reason about change in data bases affects the range and quality of responses that may be produced by a query system R e a s o n i n g about prior

p o s s i b l l t y admits a class of queries d e a l i n g w i t h the future p o s s i b i l i t y of some event or state of affairs at some time in the past These q u e r i e s have the general form:

"Could it have been the case that p?"

This class of queries will be termed

c o u n t e r h i s t o r i c a l s in an attempt to draw some parallel with c o u n t e r f a c t u a l s The future

c o r r e l a t e of c o u n t e r h i s t o r i c a l s , which one might call futurities, are of the form:

"Can it be the case that p?"

i.e in the sense of:

"Might it ever be the case that p?"

The most i n t e r e s t i n g aspect of this form of

q u e s t i o n is that it admits the ability for a query system to offer a monitor as a response to a

q u e s t i o n for relevant i n f o r m a t i o n the s y s t e m may become aware of at some future time A query system can only c o m p e t e n t l y offer such m o n i t o r s when it has this ability, since o t h e r w i s e it cannot d e t e r m i n e if the m o n i t o r may ever be satisfied

II R E P R E S E N T A T I O N

We have chosen to use a modal temporal logic There are two basic r e q u i r e m e n t s which lead us toward logic and away from m e t h o d s such as Petri nets F~rst, it may be desirable to assert that some p r o p o s i t i o n is the case without n e c e s s a r i l y

may be disjunctive That is, our k n o w l e d g e of

temporal situations may be incomplete and

indefinite, and as others have argued [Moore 821

(as a recent example), m e t h o d s based on formal

logic (though u s u a l l y flrst-order) are the only

ones that have so far been capable of d e a l i n g with

p r o b l e m s of this nature

In contrast to flrst-order r e p r e s e n t a t i o n s ,

modal temporal logic makes a fundamental

d i s t i n c t i o n between v a r i a b i l i t y over time (as

e x p r e s s e d by modal temporal operators) and

v a r i a b i l i t y in a state (as e x p r e s s e d using

propositional or flrst-order languages) Modal

temporal logic also reflects the t e m p o r a l l y

indefinite structure of language in a way that is

more natural than the commaon method of using

state variables and constants in a flrst-order

logic On the side of flrst-order logic, however,

is expressive power that is not n e c e s s a r i l y

present in modal temporal logic (But, see [K a m p

68] and [GPSS 80] for c o m p a r i s o n s of the

expressive power of modal temporal logics with

flrst-order theories.)

There are several possible structures that

one could reasonably imagine over states in time

The one we have in mind is discrete, backwards

linear, and infinite in both directions We allow

branching into the future to capture the idea that

it is open, but the past is determined Due to

the nature of the intended a p p l i c a t i o n , we also

have assumed that time is discrete It should be

stressed that this d e c i s i o n Is not m o t i v a t e d by

the belief that time itself is discrete, but

rather by the data base application Furthermore,

in cases where it is necessary for the temporal

structure to be dense or continuous, there is no

immediate argument against modal temporal logic in

general (That Is, one could develop a modal

temporal logic that models a continuous structure

of time [RU 71].)

A modal temporal s t r u c t u r e is composed of a

set o P states Each state is a set of propositions

which are true of that state States are related

by an immediate p r e d e c e s s o r - s u c c e s s o r relation A

branch of time is defined by taking some possible

sequence of states accessible over this relation

from a given state The future fragment of the

logic is based on the unified b r a n c h i n g temporal

logic of [BMP 81], which introduces branches and

quantifies over them to make it possible to

describe properties on some or all futures Thls

is extended with an "until" o p e r a t o r (as in [K amp

68], [GPSS 801) and a past fragment Since the

structures are backwards linear the e x i s t e n t i a l

and universal operators are merged to form a

linear past fragment

A Syntax

Formulas are composed from the symbols,

- A set ~ o f atomic propositions

Temporal operators: AX (every next), EX (some next), AG (every always), EG (some always), AF (every e v e n t u a l l y ) , EF (some

e v e n t u a l l y ) , AU (every until), EU (some until), L ( i m m e d i a t e l y past), P (sometime past), H (always past), S (since) AU, EU, and S are binary; the others are unary

For the operators composed of two symbols, the first symbol ("A" or "E") can be thought of as q u a n t i f y i n g u n i v e r s a l l y or

e x i s t e n t i a l l y over branches in time; the second symbol as q u a n t i f y i n g over states

w i t h i n the branch Since b r a n c h i n g is not allowed into the past, past o p e r a t o r s have only one symbol

using the rules,

- If p ~ , then p is a formula

- If p and q are formulas, then (-p), (p v q) are formulas

- If m is a unary temporal o p e r a t o r and p is

a formula, then (m p) is a formula

- If m is a binary temporal o p e r a t o r and p and q are formulas, then (p m q) is a formula

Parentheses will o c c a s i o n a l l y be omitted, and &, >, 4 > used as a b b r e v i a t i o n s (In the next section: "Ax" should be read as the universal

q u a n t i f i e r over the variable x, "Ex" as the existential q u a n t i f i e r over x.)

B Semantics

A temporal structure T is a triple ( S , ~ , R) where,

- S is a set of states

-~'~:(S -+ 2 ~) is an assignment of atomic propositions to states

- R C (S x S) is an a c c e s s i b i l i t y relation on S Each state is required to have at least one successor and e x a c t l y one

p r e d e c e s s o r i.e., As (Et (sRt) & E!t (tRs))

Define b to be an s-branch

b = ( , S_l , S=So, Sl, .) such that s i R s i + 1

The relation ">" is the transitive closure of

R

The s a t i s f a c t i o n of a formula p at a state s

in a s t r u c t u r e T, <T,s> I = p, is defined as follows :

<T,s>I = p iff p G ~ s ) , for p ~

<T,s>l = -p iff not <T,s>i=p

<T,s>l = p v q Iff <T,s>J=p or <T,s>l=q

(p is true at every time of every future)

<T,s>[= AFp Iff A b E t ( t f b & t>s & <T,t>[=p)

(p is true at some time of every future)

<T,s>i = pAUq iff

AbEt(tf"b & t>s & < T , t > i = q &

A t ' ( ( t ' ~ b & s<t'<t) -9 <T,t'>l=p)))

(q is true at some time of every future and until

q is true p is true)

< T , s > I = AXp i ff At(sRt > <T,t>I=p)

(p is true at every immediate future)

<T,s>l= EGp iff E b A t ( ( t S b & t>s) -9 <T,t>l=p)

(p is true at every time of some future)

< T , s > l = EFp iff E b E t ( t f b & t>s & <T,t>{=p)

(p fs true at some time of some future)

<T,s>1 = EXp iff Et(sRt & <T,t>l=p)

(p is true at some immediate future)

<T,s>I = pEUq iff

EbEt(teb & t>s & < T , t > I = q &

A t ' ( ( t ' e b & s<t'<t) > < T , t ' > I = p ) ) )

(q is true at some time of some future and in that

future until q is true p is true)

<T,s>~= Hp iff A b A t ( ( t f b & t<s) - ~ <T,t>l=p)

(p is true at every time of the past)

< T , s > l = Pp iff A b E t ( t ~ b & t<s & <T,t>I=p)

(p is true at some time of T h e past)

<T,s>J= Lp iff A=(tRs > <T,t>l=p)

(p is true at the immediate past)

<T,s>I= pSq iff

A b E t ( t G b & t<s & < T , t > I = q &

A t ' ( ( t ' ~ b & s>t'>t) - 9 <T,t'>l=p)))

(q is true at some time of the past and since q is

true p is true)

A formula p is valid iff for every structure

T and every state s in T, <T,s> I= p

III M O D E L L I N G CHANGE IN K N O W L E D G E BASES

As noted earlier, this logic was d e v e l o p e d to

reason about change in data bases A l t h o u g h

u l t l m a t e l y the a p p l i c a t i o n requires e x t e n s i o n to a

flrst-order language to better express v a r l a b i l l t y

within a state, for now we are restricted to the

propositional case Such an e x t e n s l o n is not

wfthout problems, but should be m a n a g e a b l e

The set of p r o p o s i t i o n a l v a r i a b l e s for

m o d e l l i n g change in data bases is divided into two

classes A state p r o p o s i t i o n asserts the truth of

a s s o c i a t e s the o c c u r e n c e of an event with the state in which it occurs The idea is to impose

c o n s t r a i n t s on the o c c u r e n c e of events and then derive the a p p r o p r i a t e state d e s c r i p t i o n To be specfic, let O s l Q s n be state p r o p o s i t i o n s and

Q e l O e m be event p r o p o s ~ t l o n s If PHI is a boolean formula of state p r o p o s i t i o n s , then formulas of the form:

(PHI -9 EX Qei) are event constraints To d e r i v e state d e s c r i p t i o n s from events frame axioms are required:

(Qei -9 ((L PHIl) -9 PHI2)), where PHIl and PHI2 are b o o l e a n ~ormulas of state propositions In the blocks world, and event

c o n s t r a i n t w o u l d be that If b l o c k A was clear and block B was clear then move A onto B is a next possible event:

((cleartop(A) & c l e a r t o p ( B ) ) -9 EX m o v e ( A , B ) ) Two frame axioms are:

(move(A,B) -9 o n ( A , B ) ) and (move(A,B) > ((L on(C,D)) -9 o n ( C , D ) ) )

If the m o d e l l i n g s t r a t e g y was left as just outlined, nothing very s i g n i f i c a n t would have been

a c c o m p l i s h e d Indeed, a simpler s t r a t e g y w o u l d be hard to imagine, o t h e r than requiring that the state formulas be a c o m p l e t e d e s c r i p t i o n This can

be improved in two n o n - t r i v i a l ways The first is that the c o n d i t i o n s on the t r a n s i t i o n s m a y reference states earlier than the last one

~econdly, we may require that certain c o n d i t i o n s might or must e v e n t u a l l y happen, but'not

n e c e s s a r i l y next As m e n t i o n e d earller, these

c a p a b i l i t i e s are important c o n s i d e r a t l o n s for us

By placing b i c o n d i t i o n a l s on the event constraints, it can be d e t e r m i n e d that some

c o n d i t i o n may never arise, or from knowledge of some event a r e c o n s t r u c t i o n of the previous state may be obtained

The form of the frame axioms may be inverted using the until o p e r a t o r to o b t a i n a form that is perhaps more intuitive As s p e c i f i e d above the form of the frame axioms will yield identical previous and next state p r o p o s i t i o n s for those events that have no effect on them The s t a n d a r d example from the blocks world is that moving a

b l o c k does not alter the color of the block If there are a lot uf events llke move that don't change block color, there will be a lot of frame axioms around stating that the events don't change the block color But if there is only one event, say paint, that changes the color of the block, the "every until" (AU) o p e r a t o r can be used to state that the color of the block stays the same unti] it is painted This s t r a t e g y works best if

we m a i n t a i n a single event c o n d i t i o n for each state; i.e, no more than a single event can o c c u r

In each state For each a p p l i c a t i o n , a d e c i s i o n must be made as to how to best represent the frame axioms Of course, if the world is very

complicated, there will be a lot of c o m p l i c a t e d frame axioms I see no easy way around this problem in this logic

As p r e v i o u s l y mentioned, we assume t h a t the

past is determined (i.e backwards linear)

However this does not imply that our knowledge of

the past is complete Since in some cases we may

wish to claim complete k n o w l e d g e with respect to

one or more predicates in the past, a c o m p l e t i o n

axiom is developed for an intuitively natural

c o n c e p t i o n of history Examples of predicates for

which our knowledge might be complete are

presidential inaugurations, employees of a

company, and courses taken by someone in college

In a first order theory, T, the c o m p l e t i o n

a x i o m with respect to the predicate Q where

(Q cl) (Q cn) are the only occurences of Q in T

is:

Ax((Q x) ~-~ x=cl v v x=cn) From right to left

on the bicondltional this just says what the

orginal theory T did, that Q is true of cl cn

The c o m p l e t i o n occurs from left to right,

a s s e r t i n g that cl cn are the only constants for

which Q holds Thus for some c' which is not equal

to any of cl cn, it is provable in the completed

theory that ~(Q c'), which was not provable in the

original theory T This axiom captures our

intuitive notions about Q 2 The completion axiom

for temporal logic is developed by introducing

time propositions The idea is that a conjunct of

a time proposition, T, and some other proposition,

Q, denotes that Q is true at time T If time

propositions are linearly ordered, and Q occurs

only in the form

P(Q & TI) & & P(Q & Tn) in some theory M, then

the h~story c o m p l e t i o n axiom for M with respect to

Q is

H(Q 4 > T1 v v Tn) Analogous to the first-

order completion axiom, the d i r e c t i o n from left to

right is the completion of Q An equivalent first-

order theory to M in which each temporal

proposition Ti is a first-order constant tl and Q

is a monadic predicate,

(Q tl) & & (Q tn), has the flrst-order

c o m p l e t i o n axiom (with Q restricted to time

constants of the past, where tO is now):

Ax<t0 ((Q x) ~-+ x=tl v v x=tn)

B Example

The propositional variables T-reg, T-add, T-

drop, T-enroll, and T-break are time points

intended to denote periods in the academic semster

on which certain activities regarding enrollment

for courses is dependent The event p r o p o s i t i o n

are Qe-reg, Qe-pass, Qe-fail, and Qe-drop; for

registering for a course, passing a course,

f a i l i n g a c o u r s e , a n d d r o p p i n g a c o u i r s e ,

respectively The only state i s Qs-reg, which

means that a student is registered for a course

2[Clark 781 contains a general discussion of

predicate completion [Reiter 82] discusses the

completion axiom with respect to circumscription

T-add ~ > (AX T-drop) - drop follows add

T-drop ~-~ (AX T-enroll) - enroll follows drop

T-enroll (-~ ( A X T-break) - b r e a k follows enroll

((T-reg v T-add) & ~Qs-reg & -(P Qe-pass)) ~-~ (EX Qe-reg) - if the period is reg or add and

a student is not registered and has not passed the course then the student may next register for the course

((T-add v T-drop) & Qs-reg) ~-) (EX Qe-drop) - if the period is add or drop and a student is registered for a course then the student may next drop the course

(T-enroll & Qs-reg) ~-+ (EX Qe-pass)) - if the period is enroll and a student is registered for a course then the student may next pass

t h e c o u r s e

(T-enroll & Qs-reg) ~-~ (EX Qe-fail)) - if the period is enroll and a student is registered for a course then the student may next fail the course

Qe-reg -+ (Os-reg AU (Qe-pass v Qe-fail v Qe-drop)) - if a student registers for a course then e v e n t u a l l y the student will pass

or fall or drop the course and until then the student will be registered for the course

((L -Qs-reg) & -Qe-reg) > -Qs-reg) - not registering maintains not being registered

AX(Qe-reg & Qe-pass & Qe-fail & Qe-drop & Qe-null)

- one of these events must next happen

-(Qe-i & Qe-j), for -l=j (e.g -(Qe-reg & Qe- pass)) - but only one

IV C O U N T E R H I S T O R I C A L S

A counterhistorlcal may be thought of as a special case of a counterfactual, where rather than asking the counterfactual, "If kangaroos did not have tails would they topple over?", one asks instead "Could I have taken CSEII0 last

semester?" That is, c o u n t e r f a c = u a l s suppose that the present state of affairs is slightly different and then q u e s t i o n the consequences

Counterhlstorlcals, on the other hand, question how a course of events might have proceeded otherwise If we picture the u n d e r l y i n g temporal structure, we See that althouKh there are no branches into the past, there are branches from the past into the future These are a l t e r n a t i v e histories to the one we are actually in

C o u n t e r h l s t o r i c a l s explore these alternate

I n t u i t i v e l y , a c o u n t e r h i s t o r l c a l may be

e v a l u a t e d by " r o l l i n g back" to some previous state

and then reasoning forward, d l s r e g a r d i n g any

events that actually took place after that state,

to d e t e r m i n e w h e t h e r the s p e c l f i e d c o n d i t i o n might

arise For the question, "Could I have r e g i s t e r e d

for C S E I I 0 last semester?", we access the state

s p e c i f i e d by last semester, and from that state

d e s c r i p t i o n , reason forward regarding the

p o s s i b i l i t y of r e g i s t e r i n g for CSEII0

However, a c o u n t e r h i s t o r l c a l is really only

i n t e r e s t i n g if there is some way in w h i c h the

c o u r s e of events is c o n s t r a i n e d These c o n s t r a i n t s

may be legal, physical, moral, b u r e a u c r a t i c , or a

w h o l e host of others The set of axioms in the

previous s e c t i o n is one example The formalism

does not provide any facility to d l s t i n q u i s h

b e t w e e n various sorts of c o n s t r a i n t s Thus the

mortal i n e v i t a b i l i t y that everyone e v e n t u a l l y dies

is g i v e n the same i m p o r t a n c e as a u n i v e r s i t y rule

that you can't take the same course twice

In the logic, the general c o u n t e r h i s t o r i c a l

has the form: P(EFp) That is, is there some time

in the past at w h i c h there is a future time when p

m i g h t p o s s i b l y be true C o n s t r a i n t s may be placed

on the prior time:

P(q & EFp), e.g "When I was a sophomore, could I

have taken Phil 6?" One might wish to require

that some other c o n d i t i o n still be a c c e s s i b l e :

P(EF(p & EFq)), e.g "Could I have taken C S E 2 2 0

and then CSEII0?"; or that the c o u n t e r h i s t o r i c a l

be immediate from the most recent state:

L(EXp) (The latter is interesting in what it has

to say about possible a l t e r n a t i v e s to or the

i n e v i t a b i l i t y of what is the case now [WM 831

shows its use in r e c o g n i z i n g and c o r r e c t i n g e v e n t -

related m i s c o n c e p t i o n s ) For example, in the

r e g i s t r a t i o n d o m a i n if we know that someone has

passed a course then we can derive from the axioms

above the c o u n t e r h i s t o r i c a l that they could have

not passed:

((P Qe-pass) -+ P ( E F - Q e - p a s s )

V F U T U R I T I E S

A query r e g a r d i n g future p o s s i b i l i t y has the

general logical form: EFp That is, is there some

future time in which p is true The basic

v a r i a t i o n s are: AFp, must p e v e n t u a l l y be true;

EGp, can p remain true; AGp, must p remain true

These can be nested to produce infinite variation

However, a n s w e r i n g direct questions about future

p o s s i b i l i t y is not the only use to be made of

futurities In addition, futurities permit the

query system to c o m p e t e n t l y offer m o n i t o r s as

responses to questions (A m o n i t o r watches for

some specified c o n d i t i o n to arise and then

performs some action, usually n o t i f i c a t i o n that

the condition has occurred.) A m o n i t o r can only be

offered c o m p e t e n t l y if it can be shown that the

c o n d i t i o n might p o s s i b l y arise, given the present

state of the data base Note that if any of the

derived it w o u l d be d e s i r a b l e to provide

i n f o r m a t i o n to that effect

For example, if a student is not r e g i s t e r e d for a course and has not passed the course and the time w a s p r i o r to e n r o l l m e n t , a m o n i t o r for the student r e g i s t e r i n g w o u l d be c o m p e t e n t l y made

g i v e n some q u e s t i o n about r e g i s t r a t i o n , since ((~Qs-reg & -(P Q e - p a s s ) & ~ X ( A F Te)) -+

(EF Qe-reg)) However, if the student had

p r e v i o u s l y passed the course, the m o n i t o r o f f e r

w o u l d not be c o m p e t e n t , since ( ( - Q s - r e g & (P Q e - p a s s ) & A X ( A F Te)) -+

-(EF Qe-reg))

Note that if a m o n i t o r was e x p l i c i t y requested, "Let me know when p h a p p e n s , " a futurity may be used to d e t e r m i n e w h e t h e r p might ever happen In a d d i t i o n to the p r o c e s s i n g

e f f i c i e n c y gained by d i s c a r d i n g m o n i t o r s that can never be satisfied, one is also in a p o s i t i o n to correct a user's m i s t a k e n belief that p might ever happen, since in order to make such a request s/he must believe p could happen C o r r e c t i o n s of this sort arise from I n t e n s i o n a l failures of

p r e s u m p t i o n s in the sense of [Mays gOl and [WM 8~I If at some future time from the m o n i t o r request, due to some i n t e r v e n i n g events p can no longer happen, but was o r i g i n a l l y possible, an

e x t e n s i o n a l failure of the p r e s u m p t i o n (in the sense of [Kaplan 82]) might be said to have occurred

The a p p l i c a t i o n of the c o n s t r a i n t s when

a t t e m p t i n g to d e t e r m i n e the v a l i d i t y of an u p d a t e

to the data base is important to the d e t e r m i n a t i o n

of m o n i t o r competence The a p p r o a c h we have

a d o p t e d is to require that when some formula p is

c o n s i d e r e d as a potential a d d i t i o n to the data base that it be provable that EXp A l t e r n a t i v e l y one could just require that the update not be inconsistent, that is not provable chat ~X~p The former a p p r o a c h is preferred since it does not make any requirement on d e c i d a b i l i t y Thus, in order to say that a monitor for some c o n d i t i o n p [s competent, it must be provable that EFp

VI D I S C U S S I O N

This work has been influenced most s t r o n g l y

by w o r k within theory of c o m p u t a t i o n on p r o v i n g

p r o g r a m c o r r e c t n e s s (IBMP 811 and [GPSS 801) and within p h i l o s o p h y on temporal logic [RU 711 The work within AI that is most relevant is that of [McDermott 821 Two of M c D e r m o t t ' s m a j o r points are regarding the openess of the future and the

c o n t i n u i t y of time With the first of these we are

in agreement, but on the second we differ This

d i f f e r e n c e is largely due to the intended

a p p l i c a t i o n of the logic Ours is applied to changes in data base states (which are d i s c r e t e ) ,

w h e r e a s M c D e r m o t t ' s is physical systems (which are

c o n t i n u o u s ) But even w i t h i n the d o m a i n of physical systems it may be w o r t h w h i l e to c o n s i d e r discrete s t r u c t u r e s as a tool for a b s t r a c t i o n , for

tractable At least by considering modal temporal

logics we may be able to gain some insight into

the reasoning process whether over discrete or

continuous structures

We have not made at serlous effort towards

implementation thus far A tableau based theorem

prover has been implemented for the future

fragment based on the procedure given in [BMP 81]

It is able to do problems about one-half the size

of the example given here Based on this limited

experience we have a few Ideas which might improve

its abilities Another procedure based on the

tableau method which is based on ideas from [BMP

81] and [RU 71] has been developed but we are not

sufficiently confident In its correctness to

present ft at this point

ACKNOWLEDGEMENTS

I have substantially benefited from comments,

suggestions, and discussions wlth Aravlnd Joshi,

Sltaram Lanka, Kathy McCoy, Gopalan Nadathur,

David Silverman, Bonnie Webber, and Scott

Weinstein

Cognitive Science (6), I982

[Moore 82] R.C Moore, "The Role of Logic in Knowledge Representation and Commensense Reasoning," Proceedings of AAAI 82, Pittsburgh, Pa., August 1982

[RU 7 1 1 N Rescher and A Urquhart, Temporal Logic, Sprlnger-Verlag, New York, 1971

[Relter 82] R Relter, "Circumscription Implies Predicate Completion (Sometimes),"

Proceedings of AAAI 82, Pittsburgh, Pa., August [982

[WJMM 83] B Webber, A Joshi, E Mays,

K McKeown, "Extended Natural Language Data Base Interactions," International Journal of Computers and Mathematics, Spring 83

[W'M 83] B Webber and E Mays, "Varieties of User Misconception: Detection and Correction", Proceedings of IJCAI 83

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Temporal Logic of Branching Time," Eighth ACM

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[Clark 78] K.L Clark, "Negation as Failure," in

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[Kamp 68] J.A.W Kamp, Tense Logic and the Theory

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