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| Moderated by Erik Sandewall. |
Marc Denecker, Daniele Theseider Dupré, and Kristof Van BelleghemAn Inductive Definition Approach to Ramifications |
The
article
mentioned above has been submitted to the Electronic
Transactions on Artificial Intelligence, and the present page
contains the review discussion. Click for
more
explanations and for the webpage of theauthors: Marc Denecker, Daniele Theseider Dupré, and Kristof Van Belleghem.
Overview of interactionsQ1. Murray Shanahan (7.7):
Marc, Daniele and Kristof, I haven't read your paper very closely yet. But I'm very interested to understand how your formalism (and indeed any other formalism) works with the circuit example of Figure 1 in your paper.
I can't find anywhere in the paper a precise claim about what your
formalisation of this example yields. But presumably, when switch
To relieve me of the trouble of going through your definitions and
proving it myself, can you also tell me what your formalisation yields
in the case that switch Do we get inconsistency according to your formalisation? Or do we get multiple possible outcomes? On page 7, you say that "only theories that do not entail multiple changes of the same fluent at the same time are considered as well-defined". So would this example not be well-defined? Yet this is a realisable circuit. Indeed it's a simplification of the circuit you use as an example. I'm not quite sure what would actually happen - I suppose the relays would get stuck. But something would happen. So maybe inconsistency or no-well-definedness is not what we want in such examples. Or maybe the representation just needs to be a bit more elaborate. What do you think? Murray A1. Marc Denecker, Daniele Theseider Dupré, and Kristof Van Belleghem (13.7):
Dear Murray, Thanks for your comments. You are right that we should have made more precise claims about what our formalisation yields in the circuit example. Below, we try to answer your questions. There is brief answer and an extended answer. First the brief answer.
In our circuit example, there are two stable states. In case a
suitable primitive action of opening or closing switches
In your simplified example, there is one stable state; in this state
the switch Now the extended answer with a more technical discussion. First we make a detailed analysis of your example.
In your example, there is one circuit containing a switch
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- p ----------- r1 -
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-~r2----- r --- q --
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The state constraints in this example are: r1 <-> p p <-> ~r2 r2 <-> r & q q <-> r1 If follows logically that :
{ p & r1 & q & ~r & ~r2}
This corresponds to the unique stable state:
{p, r1, q, ~r, ~r2}
If
The formalisation in our language is as follows (
r1 <-> p yields:
c(r1) <- c(p), ~h(p)
(initiating p causes r1)
c(~r1) <- c(~p), h(p)
(initiating ~p causes ~r1)
r2 <-> r & q yields:
c(r2) <- c(q), ~h(q), c(r), ~ h(r)
(initiating q&r causes r2)
c(r2) <- c(q), ~h(q), h(r), ~ c(~r)
c(r2) <- h(q), ~c(~q), c(r), ~ h(r)
c(~r2) <- c(~q), h(q)
(initiating ~(q&r) causes ~r2)
c(~r2) <- c(~r), h(r)
q <-> r1 yields:
c(q) <- c(r1), ~h(r1) ...
c(~q) <- c(~r1), h(r1)
p <-> r2 yields:
c(p) <- c(~r2), h(r2)
c(~p) <- c(r2), h(r2)
Assume that in addition, there are primitive actions c(r) <- act(close_r) c(~r) <- act(open_r)
Consider this definition in the case that a
{h(p), ~h(r2), h(r1), h(q), ~h(r)}
The set of actions is represented by the set
{act(close_r)}
The well-founded model of the above rule set, given that
First, we replace all c(r) <- c(r2) <- ~c(~q), c(r) c(~p) <- c(r2) c(~r1)<- c(~p) c(~q) <- c(~r1) c(~r2)<- c(~q)The well-founded model of this rule set is
true : c(r)
undefined : c(r2), c(~q) c(~r1) c(~p)
false : all other literals (c(~r), c(p), ...)
There are many ways to verify this. Any way to construct the
well-founded model is ok. For example, the well-founded model is known
to be a model of the 3-valued completion semantics. Verify that
the (3-)valued completion entails the following equivalences:
c(r) <-> true c(r2) <-> ~c(~q) <-> ~c(~r1) <-> ~c(~p) <-> ~c(r2) c(l) <-> false for the other fluent literals lAs a consequence, the 3-valued completion has a unique model which must be the well-founded model and which corresponds to the above model. Another way is to use our formalisation with prooftrees. The prooftree belows is the unique prooftree of c(~r2) and contains the unique prooftrees for the atoms c(~q), c(~r1), c(~p), c(r2), c(r)
c(~r2) <- c(~q) <- c(~r1) <-
c(~p) <- c(r2) <- c(r) <- true
<- ~c(~q)
(So the node
Our semantics first associates Note the close correspondence between this prooftree and the actual effect propagation. This correspondence between effect propagation and the constructiveness of inductive definitions was our main motivation for our work.
Our approach is not suitable to model this oscillation. To model
oscillation, a formalism with evolving time and effect rules with
delays seems necessary. However, in the paper we claim that our
formalisation "detects" non-stable behaviour when it produces a
3-valued model for the above state transition and also when it derives
contradictory causal literals So, the behaviour of our formalisation for this case is exactly what we hoped for and had predicted in the paper. Finally, consider the original circuit example. The circuit in the paper is slightly more complex:
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- p ----- s --- r1 -
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-~r2-----~r --- q --
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The effect rules in the circuit are related to the following state constraints:
r1 <-> p & s (if switches p and s are
closed then relay r1 is active)
r2 <-> q & r (if switches q and r are
closed then relay r2 is active)
r <-> ~s (switches r and s are
mechanically connected this way)
p <-> ~r2 (p is closed iff r2 is not active)
q <-> r1 (q is closed iff r1 is active)
It can be seen easily that this theory simplifies to
p & ~r1 & (s <-> r1 <-> q <-> ~r)Hence, the system can occur in two stable states
{p, ~r1, s, r1, q, ~r}
and
{p, ~r1, ~s, ~r1, ~q, r}.
It can be verified also that any applicable primitive action of
opening or closing r or s will yield a transition from one stable
state to the other stable state.
To formalise this, we could add primitive actions caus(r) <- act(close_r) caus(~r) <- act(open_r) caus(s) <- act(close_s) caus(~s) <- act(open_s)The other effect rules in the paper can be derived from the first set of state constraints using the influence information that fluents at the right can influence fluents at the left and r can influence s. Our formalisation yields the correct state transitions starting from the stable states and performing any of the primitive actions. We have a proof of this which we will add in a next version of the paper. Thanks again. Marc, Kristof, Daniele C1-1. Murray Shanahan (17.7):
Marc, Daniele and Kristof, Many thanks for your answer to my question about your paper. I think it's an excellent paper. I've just completed a short paper on the same topic, which you may be interested in. I've submitted it to the AIJ. Basically it shows how the event calculus from my book "Solving the Frame Problem" can be used to tackle the ramification problem, including tricky examples like Thielscher's circuit. The paper is available via the following URL. http://www.dcs.qmw.ac.uk/~mps/ramifications.ps.Z All the best, Murray Q2. Eugenia Ternovskaia (11.7):
Dear Marc, Daniele and Kristof, It was interesting to read your paper, especially because I am interested in inductive definability myself. I wrote a paper "Inductive Definability and the Situation Calculus" presented at Dynamics'97 last October. It was published in a volume of LNCS later. You can get it from http://www.cs.utoronto.ca/~eugenia/papers/indef.ps Among other things, I describe an inductive solution to the frame and ramification problems. I treat causal rules specifying direct and indirect effects of actions as rules of an inductive definition. Causal (inductive) rules specifying direct effects of actions may contain both cycles and negations. Causal rules describing indirect effects may not contain cycles. This condition can be relaxed, but in this case the correct form of successor state axioms will not be obtained. I would be grateful to you if you could read this paper and provide some comments. Here are some comments on your paper. First of all, it's a good paper. The main contribution, I think, is that you further elaborated the generalized inductive definition principle and applied it to the ramification problem. I consider Marc's result about the equivalence between this principle and the well-founded semantics significant, and it was good to see how it works in the context of the ramification problem. It was nice to see examples for the syntactic subclasses of the generalized inductive definition. I think that introducing the third truth value helps you to explain the subtle details of the iterative process which was quite difficult for me since I was working in classical first order logic with fixpoints. From my experience, it is a long way from understanding how causal (inductive) rules work and capturing it in a classical logic-based formalism such as the situation calculus. In your paper you say that your approach can be embedded in the situation calculus. Perhaps the main question to you is how the third truth value would be represented in the two-valued setting? The following will help you to make a few little improvements, I hope.
p. 28, after Th.1. "Unless nondeterminism is explicitly introduced, our theory leaves no room for ambiguity." I do not think this is an advantage of your theory. I think the truth value "u" should be obtained in two cases. First, it may be a result of a "bad axiomatization". Second, it may appear due to lack of complete information about the current state of the world. (It seems that you apply the Closed World Assumption in your definition of a state as a set of (positive) fluent literals, if I understood correctly.) I believe your definitions might be easily changed to represent the second case as well. For example, a state could be associated with a set of "known" positive and negative literals.
It also seems strange that you cannot derive anything about "holds", although I understand your motivation. Would it be difficult to introduce this feature?
p. 10 "..., where
This is a confusing notation. Usually
p. 10 Footnote 8: I did not understand the second sentence at first reading. "Initial state", "presence and absence" are confusing.
p. 23, footnote 17: A more or less meaningful "real-world" example could be something like this: an object cannot be at location
p. 28, top. "completely orthogonal to successor state calculation" is too strong, see comments for p.38-39.
p.38-39 "This approach contains several contributions: - A complete uncoupling of ramifications from state constraints..." I do not think they can be uncoupled completely. Even in your theory, ramifications impose some restriction on your function Trans, I believe. Trans maps states and actions to states. In this sense reachable states are restricted and state constraints are implied. Can you clarify? The other direction, from state constraints to ramification, as described by Thielscher, is a nice idea. As far as I know, no one claims that this is the only way to obtain causal rules specifying indirect effects. It is a useful method to obtain some of these rules, even though it does not always work. Also, in your summary you say "the standard view on causal laws as rules serving to restore the integrity of state constraints is too restrictive." From talking to different people, I never had the impression that this is the only and standard view on the role of causal rules in the community. Overall, even if you think your statement is absolutely right in its current form, it's strange to see it as a contribution #1. I think you have done more interesting things. Again, it's a good paper.
I hope my comments will be helpful. Regards, Eugenia A2. Marc Denecker, Daniele Theseider Dupré, and Kristof Van Belleghem (17.7):
Dear Eugenia, Your comments are very useful. We didn't have a close reading of your paper yet, but a first look showed that there is strong correspondence between our approaches. We have had very similar intuitions. Thanks for the careful reading and the corrections.
In your approach, you impose a syntactical restriction (no cycles) which guarantees that the effects are unique and well-defined. Our approach does not impose a syntactical restriction, but there is still the methodological constraint that the expert should design his effect theory such that it leads to well-defined transitions (at least for the above mentioned states and sets of actions). In case this methodological rule is followed, it is not necessary to introduce a third truth value in situation calculus or event calculus. This methodological rule requires from the expert that he establishes the 2-valuedness of the well-founded models of his effect theory at least in all states satisfying state constraints and actions sets satisfying action preconditions. This is a point that deserved more attention.
In our view, "u" is reservated for pointing to ill-defined causal literals. Uncertainty caused by nondeterministic actions must be modeled explicitly through nondeterministic effect rules and is modeled by having different transition states (see the section about this topic). We do not model uncertainty on the input state; we do not need to do so. We expect that in any approach which integrates our effect theories in a full temporal language (e.g. EC or SC), uncertainty on the state at a certain time point is represented by having different models with different states at that moment. In each single model, the state is well-defined and can be represented as a 2-valued set on which our transition function can be applied. In our experience, this is a better way to model uncertainty than through belief sets. We can't see in what way we would be applying the Closed World Assumption.
Observe moreover that ramifications need not imply state constraints even if they restrict Trans. Of course a restriction of Trans rules out certain transitions, but that does not need to mean that any states are ruled out: a state is ruled out only if all transitions leading to it are invalid. The "counter" example in our paper gives an example where this is not the case: whenever an output changes to true, the counter must be augmented. But a resulting state in which the counter were not updated (but all other effects applied) need not be invalid in itself. This state might well have existed at an earlier time in the history of the network. It can only not be reached from the currently given state. So ramifications need not imply any state constraints. It is absolutely true that in many cases, there is a relation between state constraints and ramifications; in that case the method of Michael Thielscher is applicable and works fine in our approach as well (we applied it in the circuit example; see our answer to Murray Shanahans question). In this case, the set of effect rules should satisfy the following correctness criterion: that any state satisfying the state constraints and any set of actions satisfying the action preconditions and occurring in this state lead to a transition state in which the state constraints are satisfied. This is not automatically satisfied but should be proven in our approach. Also we discussed with a number of people about the issue of the relationship between state constraints and effect rules; had anybody contradicted us on this point, we would not have put it the way we did. But we have no problem in weakening the point. Still, it seems true that in most approaches, state constraints are underlying or are directly involved in describing the semantics of causal rules. Maybe this is not a deliberate, conscious choice, but in the existing formalisations it is apparently a fact. Marc, Daniele, Kristof C2-1. Eugenia Ternovskaia (27.7):
Dear Marc, Daniele, Kristof, Thank you for your answers to my questions. These answers clarify an important difference between our works. Using generalized inductive definitions to detect ill-defined literals is the advantage of your work. However, after 2-valuedness is shown, all reasoning involving multiple models have to be performed on the metamathematical level. In contrast, first order logic augmented with inductive definitions allows us to work with all models at once therefore allowing for incomplete information. Perhaps your approach could be developed in this direction, but you do not address this problem in the paper. Notice I do not talk about non-determinism here. I completely agree with your point of view about non-deterministic actions. Therefore it seems that we mostly developed different parts of the problem.
To verify my method, I worked out the Gear wheels example. The approach works just fine, moreover, it produces successor state axioms in the correct form. Starting from your set of rules, I obtain the following successor state axiom:
Finally, a question about Definition 1. You say "and body B a nonempty
set of positive and negative literals of D". Do you mean "and body B, a
nonempty set of symbols from D, including Regards, Eugenia
C2-2. Marc Denecker, Daniele Theseider Dupré, and Kristof Van Belleghem (5.8):
Dear Eugenia,
Instead, we focus entirely on ramification and (instantaneous)
effect propagation. You will admit that there is merit in our
approach: it shows that the ramification problem can be investigated
independently of all these other issues - at least if one views causal
rules as we do, namely as descriptions of effect propagations
independent of say, state constraints-. In other words, our work can
be used embedded in a number of other approaches
( On the other hand, we pay a price for singling out the ramification problem; namely we do not show in our paper the interplay between causal rules and the other types of information. For instance, uncertainty on a certain state of affairs may be due to uncertainty on the initial state or on the narrative (the sequence of events that leads to the state). Obviously, in our paper, we cannot show how to model these sorts of uncertainty; however, our language and semantics is compatible with and is embeddable in many temporal logics in which such uncertainty can be described. Such a logic should satisfy only two conditions: in the logic there must be a notion of momentaneous state of the world which can be represented as a set of true fluents, and uncertainty on the state of the world at a particular moment or situation must be modeled by having different models of the theory in which the state at the time or situation is different (but always two-valued). This is precisely how uncertainty is modeled in your integrated approach. We may add that in the past we have also investigated this combination of inductive definitions and first order logic with uncertainty in a global temporal logic and in logic in general. We have deliberately kept the issue of representing uncertainty on the world (except in our study of nondeterministic effects) out of the current paper to keep the different issues separated. However, uncertainty is addressed in the technical report referred to as [38] in the paper, since there an embedding in a general theory of time is discussed. (The report, like several of the papers mentioned below, can be obtained from
As a matter of fact, the idea of integrating inductive definitions with first order logic is the main idea behind Open Logic Programming, used by Marc and Kristof in most of their previous related work (e.g. the papers by Van Belleghem, Denecker and De Schreye in ICLP'94, ICTL'94 and ICLP'97, by Denecker in LPNMR93 and LPNMR95) and is related to ideas about Abductive Logic Programming (in the paper by Console, Theseider Dupre' and Torasso in JLC'91). The idea in Open Logic Programming is that some predicates are considered "defined", and therefore a "closure" semantics, i.e. a Logic Programming (or Inductive Definition)-like semantics is applied to them, while the interpretation of other predicates is left open, possibly being constrained by partial knowledge given in the form of integrity constraints. So, OLP is to be seen as a integration of classical logic with inductive definitions and is suitable for representing uncertainty. Representing and reasoning on uncertainty in this logic is not on the metamathemathical level but on the object level. The embedding of our ramification approach in a linear time structure [38] is actually firmly based on Open Logic Programming.
We do not allow any state to contain undefined fluents just because by "state" we mean a real state, not knowledge of an agent on a state. We do not at all intend that there should be complete knowledge on states. If there is partial knowledge on a state, there will be more than one state matching this partial knowledge. It's again an issue of reasoning about different models, and for each of them our transition function gives a resulting state (it gives one as long as there are no bad definitions, and not more than one as long as there is no explicit nondeterminism).
With respect to the first interpretation, of course we have not
defined any formal proof system in the paper, just a declarative and
constructive model semantics. To derive anything about
It is evident that any derived effect rule necessarily implies some
relation between fluents in two consecutive states, e.g. for any state
Does this clarify the examples and discussions given before ?
Thanks for your comments, Kristof, Marc and Daniele
C2-3. Eugenia Ternovskaia (20.8):
Dear Marc, Daniele and Kristof, Thank you for answering my questions and clarifying the motivation for your work. A few words regarding our discussion about proof system versus model-theoretical approach. I was asking, without criticizing, about performing reasoning in your system on the object level, i.e., embedding your approach in a proof system. From your answer I understood that you consider both OLP and some logical representations as candidates for formalizing and performing reasoning. This answer is satisfactory, and it would be interesting to learn more about OLP from your papers.
Even if we choose not to extend the notion of a state constraint, it's hard for me to see in what way your observation would imply any interesting theoretical consequences. It seems obvious that a causal rule imply universal statements about situations (states) involving as many states as there are those mentioned in the causal rule. Your "counter" example only shows that there are indirect effects of actions depending on two consecutive states, not just one. It's hard to disagree. All causal rules of this king will imply universal statements about two consecutive states, not about just one, of course.
Regarding Definition 1, I do not remember whether you define what a literal
is. Maybe I just missed it. When I was reading the definition, I did not
assume that Regards, Eugenia
C2-4. Marc Denecker, Daniele Theseider Dupré, and Kristof Van Belleghem (30.8):
Dear Eugenia, Here are some thought about your comments.
The role of our counterexample was simply to show that there may be no such relation. What are interesting theoretical consequences? We do not point to some interesting new and fundamental relationship; actually we cut one: in general an effect rule may be unrelated to state constraints (but this was never your viewpoint anyway, right?). But there is a gain: the effect propagation can be described mathematically independently of state constraints and also of time topology, action preconditions, initial states, etc.. But yes, also to us
Thanks for pursuing this discussion, Eugenia. Marc, Daniele, Kristof
C2-5. Eugenia Ternovskaia (11.9):
Dear Marc, Daniele, and Kristof, Thank you for an interesting discussion regarding your paper. It clarified some subtle points of your work and helped to understand your motivation.
Best regards, Eugenia
C2-6. Michael Thielscher (16.9):
Dear Eugenia, Marc, Daniele, Christoph, I'd like to second Eugenia's view on the "Counter" counter-example, which implies that the intended behavior can be obtained after all with my approach. The natural "inter-state constraint" in this example would be
Then if you consider the causal relationship
the only consistent successor state would show the expected increment of the counter by one. (A side remark: Inter-state constraints like the above can be expressed in a novel Situation Calculus-style formulation of the Fluent Calculus; if you're interested, have a look at
Cheers. Q3. Michael Thielscher (14.7):
Dear Marc, Daniele and Kristof, I have a question concerning the notion of negative cycles. If I understand correctly, they arise whenever the absence of some causation is the cause of some effect. Now, on page 23 you consider a construction like
Also I would like to make a remark on your discussion on zero vs. non-zero delays (Section 6). This is a very interesting question, and I have recently discovered a related problem in connection with my method of using causal relationships: If effects with zero delay are not distinguished from effects with real delay, then some desirable conclusions may not follow. Since your approach has much in common with mine, I wonder if the distinction between so-called steady and stabilizing state constraints, which solves the problem in my approach, is needed here as well? (C.f. my paper entitled "Steady vs. Stabilizing State Constraints" at this year's CommonSense workshop; see http://www.ida.liu.se/ext/etai/nj/fcs-98/listing.htmlMichael A3. Marc Denecker, Daniele Theseider Dupré, and Kristof Van Belleghem (18.7):
Michael, Here is an answer to your comments. First of all, we agree with you that it is unnatural to have just the absence of an effect as a cause for anything. in fact as you can see in the paper, our high-level effect rules (which syntactically look a lot like Lin's) contain only the presence of an effect (on a complex formula) as possible trigger for further changes. But at a low level, we need to introduce negative cause literals to define initiation of complex formulas.
For example, when is the formula
One could argue that one should define that In Lin's approach (the IJCAI95 paper), absence of causation is not present because the formalisation is entirely based on the values of fluents, not on their changes. One might say that all (present or absent) causations are implicit because it is the resulting state that is considered rather than the transition, as apparent in the rule
This different view allows our approach to be more general: in cases where causal rules do not correspond to state constraints, such as example 1 in our paper (with the electronic counter), the relation is inherently dependent on the value change: it's the change of "out" from true to false that makes "count" change. This is not a case where absence of initiation is needed, since the dependency is on the single fluent "out" but how would this be modeled in Lin's causal rules?
We had another example (in our NAIC'97 paper and in Kristof's PhD thesis):
an alarm system that detects if somehow people enter a building.
While the system is active, anyone entering the building triggers
the alarm: if
Regarding your interpretation of
As indicated above, we agree that having only "not caus" literal as
the body of a causal rule is counterintuitive, precisely because such
rules are not constructive. However negative cycles can also occur
if there are positive caus literals present (just add
We wanted the constructive principle of inductive definition to model
the propagation of effects. It is hard to interpret the above two rules
as a well-defined inductive definition. As you say, it could be
interpreted as a cause for
It is true that in stable semantics the above clauses do represent the
exclusive disjunction
As regards your "Steady vs stabilizing state constraints". There,
you say that mixing the two types of ramifications is responsible for
the unintended conclusion in the example. From our point of view, the
problem in the specific example would be that the causal rules are not
correct: in our formalisation we would say that Nevertheless, the issue of delays is intriguing, and your distinction between zero and non-zero delays is an interesting contribution, even if there could be a more general view on it. First, it could be possible that even what you consider zero delays (steady constraints and ramifications) are an abstraction wrt reality. Maybe in quantum physics it is not true that an object (a sub-atomic particle, in particular) cannot be in two locations at the same time? (is any expert of quantum physics reading ETAI?) More importantly, you argue that the important distinction in qualitative reasoning is between zero and non-zero delays, but still small enough to be considered virtually instantaneous by common sense, so to be distinguished by actually "delayed" effects. That makes sense, even if one could also view a continuum between zero delays (or delays that are too small for classical physics), non-zero but "commonsense" instantaneous delays, and delayed effects. Knowledge about different delays could be still qualitative or imprecise, but still allow to conclude that a delay is smaller than another and then it is not correct to apply the (non-zero-delay) causal rules in any order. Marc, Daniele and Kristof
Q4. Anonymous Referee 1 (8.3):
The following are my answers to the standard ETAI refereeing questions:
Q5. Anonymous Referee 2 (8.3):
The results of the article, as specified in the summary, are new and interesting, and the body of the article justifies these results. The article is clearly written and well-organized. Comparison with related work in Sec. 9 would be more complete if the authors added a few words about four papers that are not included in the bibliography: [1] H. Turner, "Splitting a default theory", in Proc. AAAI-96. [2] H. Turner, "Representing actions in logic programs and default theories: a situation calculus approach", JLP, Vol. 31, 1997. [3] N. McCain and H. Turner, "Causal theories of action and change", in Proc. AAAI-97. [4] E. Giunchiglia and V. Lifschitz, "An action language based on causal explanation", Proc. AAAI-98. This direction of work is relevant, in particular, because [1] and [4] contain formalizations of the spilling water example similar to the one proposed by the authors in Sec. 5.3. |