A Modal T e m p o r a l Logic for R e a s o n i n g about Change 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
s p e c i f y i n g exactly when. Secondly, our k n o w l e d g e 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.
Boolean connectives: v, -.
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 = AGp iff A b A t ( ( t ~ b & t>s) -9 <T,t>l=p) (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
some atomic condition. An event p r o p o s i t i o n 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
A. C o m p l e t i o n of History T-reg ~--> (AX T-add)
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
histories.
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
s t r o n g e r forms of future p o s s i b i l i t y can be 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
which computational methods may prove to be more 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.
Reasoning About Processes and Plans," 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 UserMisconception: Detection and Correction", Proceedings of IJCAI 83.
REFERENCES
[BMP 81] M. Ben-Ari, Z. Manna, A. Pneuli, "The Temporal Logic of Branching Time," Eighth ACM Symposium on Principles of Programming
Languages, Williamsburg, Va., January [981.
[Clark 78] K.L. Clark, "Negation as Failure," in Logic and Data Bases, H. Gallalre and J. Minker (eds.), Plenum, New York.
[GPSS 80] D. Gabbay, A. Pneull, S. Shelah, J. Stavl, "On the Temporal Analysis of Fairness, Seventh ACM Symposium on Principles of Programming Languages, 1980.
[Kamp 68] J.A.W. Kamp, Tense Logic and the Theory of Linear Order, PhD Thesis, UCLA, |968.
[Kaplan 82] S.J. Kaplan, "Cooperative Responses from a Portable Natural Language Query System," Artificial Intelligence (19, 2), October 1982.
[Mays 80] E. Mays, "Failures in Natural Language Systems: Appllcations to Data Base Query Systems," Proceedings of AAAI 80, Stanford, Ca., August [980.
[Mays 82] E. Mays, "Monitors as Responses to Questions: Determining Competence," Proceedings of AAAI 82, Pittsburgh, Pa.,
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