David PooleDecision Theory, the Situation Calculus and Conditional Plans. 
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N:o  Question  Answer(s)  Continued discussion 

1 
30.8 Uffe Kjærulff 
14.9 David Poole 

2 
11.9 Ray Reiter 
18.9 David Poole 

3 
19.11 Hector Geffner 
Q1. Uffe Kjærulff (30.8):
The focus of the paper seems to be on representational issues, which is fine. But, being a Bayesiannetwork person, I'm would find it interesting to see how computations of expected utilities of plans are performed in the ICL framework. This issue is only touched upon very briefly in the paper, mentioning that it involves enumerating all possible cases. In my understanding that amounts to constructing the corresponding decision tree (which may be huge), and then computing the probability and utility for each leaf of the tree.
So, my question boils down to: Is there an efficient computational scheme for ICL?
Uffe Kjærulff
Dept. of Computer Science, Aalborg University, Denmark
A1. David Poole (14.9):
The focus of the paper seems to be on representational issues, which is fine. But, being a Bayesiannetwork person, I would find it interesting to see how computations of expected utilities of plans are performed in the ICL framework. This issue is only touched upon very briefly in the paper, mentioning that it involves enumerating all possible cases. In my understanding that amounts to constructing the corresponding decision tree (which may be huge), and then computing the probability and utility for each leaf of the tree.So, my question boils down to: Is there an efficient computational scheme for ICL?
Thanks for the question! First, ICL theories can be seen as representations of Bayesian networks; there is a local translation between groundings of the ICL theories and Bayesian networks. ICL theories (with only choices by nature) can be seen as firstorder rulebased representations of Bayesian networks. Thus any Bayesian network algorithms can, in principle, be used for the ICL. There are two main computational advantages of the ICL approach. The first is that they can exploit logicbased reasoning techniques for probabilistic reasoning (although it turns out that once the link is realized, it isn't restricted to ICL theories). Secondly they give us an opportunity to exploit, not only the graphical independence of a Bayesian network, but also the contextual independence that i natural in the rule based representations. I will expand briefly on each of these here.
The ICL (with only choices by nature) can be seen as a way to write firstorder Bayesian networks as rules. There is a local translation between ICL theories and Bayesian networks. Note that a Bayesian network has nothing to say about how a node depends on its parents. The ICL lets us write the conditional probabilities as a set of rules. (This is often a more compact representation than conditional probability tables when there are contextual independencies).
To translate an ICL theory into a Bayesian network: ground the theory (substituting ground terms for the free variables); the ground atoms are the nodes of the Bayesian network; the atoms in the bodes of the rules that imply a become the parents of a. There is a lot more structure that can be expressed by the rules than in the network (in particular the contextual independencies). Note that the acyclicity of the logic program ensures that the resulting network is acyclic and (even when the resulting network is infinite) there are only finitely many ancestors for any node (which means you only need a finite subset of the graph to answer any conditional probability statement).
To translate a Bayesian network to an ICL theory, you write as rules how the parents of a influence a. The atomic choices are introduced as extra conditions on the rules to let us interpret the rules logically. For more details see (Poole 1993).
The first way that the ICL can be used computationally is adapting the logicbased algorithms for probabilistic reasoning. In particular, the algorithms for consistencybased diagnosis (such as the focusing algorithms of de Kleer), based on the notions of conflicts, can be adapted to compute posterior probabilities in the ICL (and so for Bayesian networks). This works well when there are skewed probability distributions. This has been explored in (Poole 1996).
Another approach is to exploit the structure that can be expressed in a Bayesian network as well as the rule structure of the ICL. This is ongoing work with Nevin Zhang. I'll first sketch how the efficient structured Bayesian network algorithms (such as cliquetree propagation and more queryoriented algorithms such as D'Ambrosio's SPI, Zhang and Poole's VE or Dechter's bucket elimination) can be seen as ways to exploit the factorization of the joint probability distribution. The basic operation is to sum out a (nonobserved, nonquery) variable. Algorithms such as cliquetree propagation are efficient because they can compile this operation into an secondary structure (the clique tree or join tree) that works by message passing in the join tree so that the posterior distribution for all variables can be obtained in twice the time it takes to compute the posterior on one. The basic operation of the message passing from one clique to another is to sum out the variable that appears in one clique that doesn't appear in the other.
It turns out that you can carry out a similar operation on the rule base directly. We have been working on a rulebased variation of the queryoriented algorithms I call probabilistic partial evaluation (Poole 1997). Eliminating a variable turns out to be like resolution. When the rules give no structure beyond that of the Bayesian network (i.e., there is no contextual independence, and there is a rule for each assignment of variables to a parent), the algorithm reduces to the standard queryoriented Bayesian network algorithms. But when there is contextual independence (as there is in many ICL theories) the algorithm lets us exploit that. This also seems to be very promising for approximation, where different approximations can be used in different contexts (Poole 1998).
Aside: One main difference between clique tree propagation and the queryoriented algorithms is that clique tree propagation acts on marginal distributions (each clique represents the marginal distribution of the variables in the clique), whereas the queryoriented algorithms act on conditional probabilities. Probabilistic partial evaluation relies on manipulating conditional probabilities. The contextual independencies get lost quickly when you reason with marginal distributions.
Finally, there is also a promise of structured dynamic programming using the rules. The most comment methods for solving sequential decision problems is to do dynamic programming. Traditional dynamic programming explicitly reasons over state space.s and thus suffers from what Bellman calls the "curse of dimensionality". We should be able to do much better by reasoning in terms of the variables (or propositions) that define the state space and exploiting the rulestructure in dynamic programming. In the ICL, dynamic programming looks much like regression planning. We can either do this in POMDPs where we regress values through conditional plans (Boutilier and Poole 1996) or where we are building plans conditional on history (as in influence diagrams), and we can use the rule structure to determine states that all have the same utility (Poole 1995).
Thus in answer to your question, this is an active area of research by myself and others who are looking at exploiting contextual independence for probabilistic inference and sequential decision making. One of the reasons I am optimistic that this will work is that it promises nice methods of approximation, where we can approximate differently in different contexts. We will have to see if we can deliver on this promise.
The following are references to some of my papers that give more details:
Q2. Ray Reiter (11.9):
David,
I enjoyed reading your paper, and learned a lot from it. I agree with you that axiomatic action theories need an account of probability/decision theory, although I do not share your reservations about being able to do so in the sitcalc (see below). My comments and questions concern only the probabilistic (as opposed to the decision theoretic) components of your paper.
1. Examples 2.8 and 2.9.
The predicate
(a) It is not parameterized by the thing  in this case key  being picked
up. It should be
(b) In axiomatizing at on p. 20, you do not include a
2. In the conditional plan of p. 21, don't you want
3. Consider the axiom of Example 2.9 and your comment "Note that this
implies that putting down the key always succeeds".
Suppose not. Then is it the
case that the axiom should be modified by replacing the atom
_{ } ¬ (A = putdown(key)_{ ^ }putdownsucceeds(key, S)) 
4. In the axiom of Example 2.9, you include a choice
Basically, you want to describe the probability that the robot
doesn't lose the key during the interval between the occurrence of the last
action in
On p. 8, you claim "None of our representations assume that actions have
preconditions; all actions can be attempted at any time." I'm sceptical of
this claim. Consider your example domain, where you adopt the axioms
_{ } at(key, Pos, S)_{ ^ }at(robot, Pos, S) ·> P0(pickup_succeeds(S)) = .88 (*) 
Finally, you often talk about actions being
"attempted",
as in the above quotation. I found this particularly confusing
since nothing in your ontology deals with action "attempts".
Actions simply
realize their effects with certain probabilities. You do not, for example,
have an action
I must confess that I couldn't make much sense of the discussion in Section 3.3, especially your arguments that the sitcalc is inherently unsuitable for modeling multiple, possibly reactive agents. It's true that Golog lacks the right control structures for doing this, but your comments on p. 26 and p. 27 (p. 26 "In order for a situation calculus program to ... and does the appropriate concurrent actions." p. 27 "When there are multiple agents ... or complex actions.") suggest to me that you are not aware of more recent work by De Giacomo, Lesperance and Levesque on Congolog that addresses precisely these issues (Proc. IJCAI 97). Congolog provides reactive rules (interrupts) and interleaving concurrency in a semantically clean way, all formulated within the sitcalc. I believe that Congolog addresses all your criticisms of the sitcalc, except, of course, it does not (yet) incorporate utilities.
So far as I know, there have only been two proposals for augmenting the sitcalc with probabilities, yours and BHL. As I indicate below, these two approaches are considerably different, and therefore I believe it is important to elaborate on these differences more than you do (top p. 25) especially the substantial ontological differences.
For BHL, there is no
_{ } flipaCoin = flipHeadsflipTails 
_{ } pickup(key) = pickup_succeeds(key)pickup_fails(key) 
_{ } P0(pickup_succeeds(key), S) = p <> (suitable conditions on S and p) 
1. Primitive actions for BHL are entirely different than yours. BHL primitive
actions would be things like
2. Therefore, for BHL, situation terms describe the "actual" history selected by nature in response to the agent performed actions, e.g.
_{ } do(pickup_fails(key), do(flipHeads, S0)). 
_{ } do(pickup(key), do(flipaCoin, S0)). 
3. In BHL, all primitive actions (
4. For BHL, your nondeterministic agent actions like
Your paper uses a single, ongoing example to
illustrate the axiomatization style
required by your approach. Personally, I couldn't abstract, from the example,
a set of guidelines by which I could invent my own axiomatization for a
different example of my choosing. I think it would considerably enhance the
paper if you provided such guidelines. For example, for each action, tell the
reader what axioms need to be specified. What choice of fluents needs to be
made? (For example, there seem to be at least two categories of fluents 
"normal"
fluents like
A2. David Poole (18.9):
Ray,
Thanks for your comments. I will try to explain everything so that it
makes sense, and try to explain why I am doing what I am doing. I'll
answer the questions in order, although it may help to read the guide
to the perplexed first.
1. Examples 2.8 and 2.9.What it says now is that the probability that picking up something (in this case a key) actually achieves the agent carrying the object doesn't depend on what is being picked up (nor does it depend on the position, as long as the robot and key are at the same position). You are right in that this probably isn't what you want. [This would really only make a difference if theThe predicate
pickup_succeeds(S) in the first clause.(a) It is not parameterized by the thing  in this case key  being picked up. It should be
pickup_succeeds(key, S) .
Presumably, you need one of these predicates for each action type, e.g.In this caseputdown_succeeds ,goto_succeeds , etc. Therefore, why not introduce a generic predicatesucceeds(action, S) , in which case, you can writesucceeds(pickup(key), S) ,succeeds(goto(loc), S) , etc? Similarly, you can have a relationfails(action, S) .
Now,I am not sure what you mean by the union, butC_{0} containsunion(succeeds(pickup(key), S), fails(pickup(key), S)) , where the union is over all ground instances of these atoms.
Incidentally, your specifications of what sets are inYes. I mean for each situation termC_{0} , e.g. bottom p. 13, is an abuse of notation, since it looks like an object level sentence, when in fact it is a metalevel statement for which universal quantification over situations is not defined.
Here
(b) In axiomatizing at on p. 20, you do not include aYes.goto_succeeds atom in the clause body. Does the atomwould_not_fall_down_stairs(S) play this role?
2. In the conditional plan of p. 21, don't you wantNo.at(key, r101) instead ofat_key ?
This is sort of correct, but it doesn't get the probabilities right. I would expect that the probability of keeping the key to be independent of of the other two cases. There are three free parameters to be assigned probabilities, so we need three choice alternatives (when we only have binary alternatives). See the guide for the perplexed below; I do this example in more detail.3. Consider the axiom of Example 2.9 and your comment "Note that this implies that putting down the key always succeeds". Suppose not. Then is it the case that the axiom should be modified by replacing the atom
A =/ putdown(key) by something likeIf so, perhaps this should be pointed out. (See my call below for a guide for the perplexed.)
_{ } ¬ (A = putdown(key)_{ ^ }putdownsucceeds(key, S))
Yes. That is right. If the agent could drop the key at any time, we need to model (or learn) how the agent may drop the key as a function of time. Presumably this would be modelled as a Poisson process (assuming it is equally probable that the agent can drop the key at any time). As outlined on page 1, I haven't dealt here explicitly with time.4. In the axiom of Example 2.9, you include a choice
keeps_carrying(key, S) , which you introduce because "there may be a probability that the agent will drop the key". You then axiomatize this withP0(keeps_carrying(key, S)) = .95 But I can't think of any realistic way to axiomatize this probability when only the parameterS is available.Basically, you want to describe the probability that the robot doesn't lose the key during the interval between the occurrence of the last action in
S , whencarrying(key, S) was true, and the occurrence of the next actionA . In other words, you want to describe the probability that the robot is still carrying the key at situationdo(A, S) given it is carrying it in situationS . But that depends, at the very least, on WHEN this next actionA occurs. Now on one account of adding time to the sitcalc (Reiter, Proc KR'96), an action can take an additional temporal argument denoting that action's occurrence time, for example,openDoor(t) . Then one expects thatP0(keeps_carrying(key, S)) given thatopenDoor(10) is the next action should be greater than this probability given thatopenDoor(1000) is the next action. So as far as I can tell, the choicekeeps_carrying(key, S) should take another argument, namelyA , or perhaps simplytime(A) , wheretime(A) is the occurrence time of the next actionA .
The axiomatization I gave assumes that the agent can hold on to the key tightly enough so whether it drops the key isn't a function of time ;^}
On p. 8, you claim "None of our representations assume that actions have preconditions; all actions can be attempted at any time." I'm skeptical of this claim. Consider your example domain, where you adopt the axiomsThere are two separate issues here, first the meaning of preconditions, and secondly how the atomic choices act in contexts where they aren't applicable.P0(pickup_succeeds(S)) = .88 Just prior to this, you say "P0(pickup_succeeds(S)) reflects how likely it is that the agent succeeds in carrying the key given that it was at the same position as the key and attempted to pick it up". So you can't seriously writeP0(pickup_succeeds(S)) = .88 because hereS is any situation, including one in which the agent is 1000 kms from the key. I take it that what you intended to write was something likeIf this is the case, then it is exactly here where you are using action preconditions, namely, in characterizing those situations in which an action has a high probability of succeeding. It also appears that you are appealing to action preconditions in the axiom of Example 2.8 (namely,
_{ } at(key, Pos, S)_{ ^ }at(robot, Pos, S) ·> P0(pickup_succeeds(S)) = .88 (*) at(robot, Pos, S)_{ ^ }at(key, Pos, S) ). Probably, these atoms should be omitted because their effects are realized in the above axiom (*).
There are three things that could be meant by a precondition of an action:
The second issue is a bit more subtle. Let's assume that the choice
alternatives are all binary (nonbinary alternatives are useful when a
relation is functional). In this case, there are the same number of
alternatives as there are free parameters (conditional probabilities
that can be independently assigned) in the probability model. For
example, when converting a Bayesian network to an ICL theory, there
are the same number of alternatives as there are numbers to be
assigned in the Bayesian network. You typically have, for each effect
...
where the
Your question on conditions for
This should be seen in the context of an overriding theme of this work. We are not trying to see how much we can put into a formalism, but how little we can get away with. We don't need conditions on the probabilities of the atomic choices, so we don't have them.
Perhaps the term "attempted" is misleading. What I mean is that the robot sends the motor control for the action, irrespectively of whether the robot can do it (in this sense it is attempting the action). By theFinally, you often talk about actions being "attempted", as in the above quotation. I found this particularly confusing since nothing in your ontology deals with action "attempts". Actions simply realize their effects with certain probabilities. You do not, for example, have an action
try_pickup , as do other authors (e.g. James Allan) who do try to axiomatize actionattempts . In this connection, I had a hard time making sense of footnote 9, since it motivates the axiom of Example 2.9 in terms of talk about "trying" to pick up the key. In fact, even after reading this material several times, I couldn't figure out footnote 9 until, much later, I began to think about how your approach compares with Bacchus, Halpern and Levesque. See my comments below on BHL.
I must confess that I couldn't make much sense of the discussion in Section 3.3, especially your arguments that the sitcalc is inherently unsuitable for modeling multiple, possibly reactive agents. It's true that Golog lacks the right control structures for doing this, but your comments on p. 26 and p. 27 (p. 26 "In order for a situation calculus program to ... and does the appropriate concurrent actions." p. 27 "When there are multiple agents ... or complex actions.") suggest to me that you are not aware of more recent work by De Giacomo, Lesperance and Levesque on Congolog that addresses precisely these issues (Proc. IJCAI 97). Congolog provides reactive rules (interrupts) and interleaving concurrency in a semantically clean way, all formulated within the sitcalc. I believe that Congolog addresses all your criticisms of the sitcalc, except, of course, it does not (yet) incorporate utilities."It's true that Golog lacks the right control structures for doing this..." I am quite happy with the agent having simple control structures. I am concerned about modelling asynchronous external actions or events (by other agents or by nature).
It is not clear to me that Congolog addresses this, in the sense that all of the concurrency in the language is internal to the agent. It doesn't model exogenous actions (they generated externally). This would seem to mean that external events are restricted to occur at the situations defined by the primitive events. [I am looking forward to being told I am wrong.]
Moreover, with respect to the mix of the situation calculus and time, it is clear now that we can do what we want when we have both the situation calculus and time. But when you have statements saying when an event occurred, you don't need situations. Thus parsimony would suggest that we do away with situations. [I would expect that this should form a different thread.]
So far as I know, there have only been two proposals for augmenting the sitcalc with probabilities, yours and BHL. As I indicate below, these two approaches are considerably different, and therefore I believe it is important to elaborate on these differences more than you do (top p. 25) especially the substantial ontological differences.Yes, I do need to discuss this further (that is the joy of ETAI discussion of papers; we can retrospectively expand those parts that are of most interest).
First let me say that I am not claiming that mine is better than theirs, nor that it is different just for the sake of being different. We start from very different perspectives. In their introduction, they explicitly contrast their work with the work starting from Bayesian networks. In particular, they claim "...they [Bayesian networks] have difficulties in dealing with features like disjunction ...". I take a Bayesian perspective that disjunction is not a feature we want! What defines a Bayesian is that probability is a measure of belief, that any proposition can have a probability (both of which BHL would agree with) and that all uncertainty should be measured by probability. Thus you need to keep in mind that the different design choices could stem from this different perspective.
For BHL, there is noExcept that BHL don't model probabilistic actions (in their IJCAI95 paper, which is the only one I had seen) although I believe they could. All of their probabilities are within the agent, so such axioms would have to be part of the formula for updating belief. [I just found a new paper, Reasoning about Noisy Sensors and Effectors in the Situation Calculus, 1998, at Faheim Bacchus's web site that does this. I'll call this BHL98.]action_succeeds andaction_fails fluents. Instead, an action whose outcome depend on nature is represented by the nondeterministic choice of simpler, deterministic actions corresponding to each different outcome. For example, the nondeterministic actionflipaCoin is represented as the complex actionwhere
_{ } flipaCoin = flipHeadsflipTails  is GOLOG's operator for nondeterministically choosing between the two actionsflipHeads andflipTails . For your ongoing example,pickup(key) would be represented byWhen an agent executes the action
_{ } pickup(key) = pickup_succeeds(key)pickup_fails(key) pickup(key) in situationS , nature steps in and selects which of the two actionspickup_succeeds(key) andpickup_fails(key) actually occurs, and it does so with frequencies determined by an axiom of the form:In other words, your atomic choice of bottom p 13, and your axiom for
_{ } P0(pickup_succeeds(key), S) = p <> (suitable conditions on S and p) P0 top of p 14 are represented by BHL in the way I have indicated above.
One other important point to note is that the probabilistic part of the ICL is a restricted form of Bacchus's and Halpern's logics of probability as belief. The translation is that each atom that isn't an atomic choice is defined by Clark's completion of the rules defining that atom. We need statements that the elements of an alternative are exclusive and covering and that the atomic choices that aren't in the same alternative are probabilistically independent. Thus the translation doesn't require "suitable conditions on S and p".
A general framework for probability and action is intuitively straightforward (see Halpern and Tuttle (1993) for a general semantic framework). You can think about forward simulating the system. Nondeterministic/stochastic actions split the worlds and impose a probability over the resulting worlds. In the ICL we handle splitting the worlds using one simple mechanism: independent choice alternatives. I treat actions by the agent and other stochastic mechanisms (e.g., the ramifications of actions) in exactly the same way. BHL98 split the worlds by nondeterministic actions (as you outline). They would then need a different mechanism for stochastic ramifications.
These are very different ontological and representational commitments than yours, and I think they deserve a deeper analysis than your discussion provides. Here are a few important differences these commitments lead to:Yes.1. Primitive actions for BHL are entirely different than yours. BHL primitive actions would be things like
pickup_succeeds(key) ,flipHeads etc, i.e. they consist of the deterministic actions that nature can perform, together with those deterministic actions (if any) that the agent can perform. For you, primitive actions are typically nondeterministic, and correspond to those actions that the agent can perform (but with nondeterministic outcomes chosen by nature), e.g.pickup(key) ,flipaCoin .
Yes. I tried to be explicit about both of these points.2. Therefore, for BHL, situation terms describe the "actual" history selected by nature in response to the agent performed actions, e.g.
For you, situations denote the history of agent performed actions, e.g.
_{ } do(pickup_fails(key), do(flipHeads, S0)). The BHL axioms all refer to the former situations. In your ontology, only the latter are allowed.
_{ } do(pickup(key), do(flipaCoin, S0)).
3. In BHL, all primitive actions (Except I would think that my successor state axioms are perfectly straightforward. I get by with just Clark's completion, in much the same way as other logic programming representations of action. Frame and ramification axioms are treated the same (in fact I don't even think of them as different, and the paper doesn't distinguish them). I get by with Clark's completion because it is defined with respect to each world, where I just have a simple acyclic logic program. Maybe the guide to the perplexed below will help.pickup_succeeds(key) ,flipHeads ) are deterministic and successor state axioms (the causal laws + solution to the frame problem) are formulated only wrt these. For you, the causal axioms are formulated wrt the (nondeterministic) agent's actions (pickup(key) ,flipaCoin ); hence the need for your fluentspickup_succeeds(S) andpickup_fails(S) . These function as random variables that determine the effects of the nondeterministic actionpickup(key) . One consequence of this difference is that for BHL, successor state axioms are perfectly straightforward, whereas for you, because of these random variables, the causal laws become, at least conceptually, somewhat opaque (e.g. footnote 9).
Moreover, BHL use "classical" action precondition axioms, in contrast to you (see above about action preconditions).So what happens if the agent doesn't know if the precondition holds, but still wants to do the action? This is important because, with noisy sensors, agents know the actual truth values of virtually nothing outside of their internal state, and most preconditions of actions are properties of the world, not properties of the agent's beliefs.
4. For BHL, your nondeterministic agent actions likeOne major difference is that in BHL, it is the agent who does probabilistic reasoning. In my framework, the agent doesn't (have to) do probabilistic reasoning, it only has to do the right thing (to borrow the title from Russell and Wefald's book). The role of the utility is to compare agents. This is important when we consider what Good calls type 2 rationality and Russell calls bounded rationality (see the work by I.J. Good, Eric Horvitz, and Stuart Russell for example), where we must take into account the time of the computation done by the agent in order to compute utility. I would expect that optimal agents wouldn't do (exact) probabilistic reasoning at all, because it is too hard! That is why we want a language that lets us model agents and their environment, and defines the expected utility of an agent in an environment. One of the reasons I didn't want to just add "time" into the framework is that I want a way to take into account the computation time of the agent (the thinking time as well as the acting time), and that makes it much more complicated (I wanted to get the foundations debugged first).pickup(key) , are not part of the language of the sitcalc. Rather, they are nondeterministic GOLOG programs. Since you also introduce a notion of a program (Sections 2.8, 2.10), there are some natural comparisons to be made. The obvious (and perhaps only) important difference is in the nature of the primitive program actions. For BHL programs, the primitives are the deterministic actions described above (e.g.pickup_succeeds(key) ,flipHeads ) whereas for you, they are the agent actions likepickup(key) .
Your paper uses a single, ongoing example to illustrate the axiomatization style required by your approach. Personally, I couldn't abstract, from the example, a set of guidelines by which I could invent my own axiomatization for a different example of my choosing. I think it would considerably enhance the paper if you provided such guidelines. For example, for each action, tell the reader what axioms need to be specified. What choice of fluents needs to be made? (For example, there seem to be at least two categories of fluents  "normal" fluents likeThanks for the suggestion. The following is a paraphrase of how we tell people to represent knowledge in Bayesian networks, translated into the language of the ICL.carrying(X, S) , and atomic choice fluents. How does one decide on the latter, when given the primitive actions and "normal" fluents of an application domain?) Similarly, for each fluent, what axioms need to be specified? Also, what needs to be axiomatized about the initial situation? Finally, what do all these axioms look like syntactically?
First I'll do the propositional (ground) case. We totally order the
propositions. The idea is that we will define each proposition in
terms of its predecessors in the total ordering. For each proposition
If
and create an alternative
If the context is empty (
When the probability given the context is 0, we don't write any
rules.
The case for the ICL with the situation calculus is similar. Intuitively, we make the total ordering of the propositions respect the temporal ordering of situations. We write how a fluent at one situation depends on fluents (lower in the ordering) at that situation and on fluents at previous situations. Note that this rule automatically handles ramifications as well as frame axioms. The total ordering of the fluents guarantees the acyclicity of the rule base and that we don't have circular definitions. The initial situation is handled as any other, but the predecessors in the total ordering are only initial values of fluents (and perhaps atoms that don't depend on the situation).
The only thing peculiar about this is that we often have fluents that depend on values at the current as well as the previous situation. This is important when there are correlated effects of an action; we can't just define each fluent in terms of fluents at the previous situation. But this also means that we don't treat frame axioms and ramification axioms differently.
Let's look at how we would do Examples 2.8 & 2.9, with
puttingdown the key possibly failing. Lets consider when carrying would be
true. Suppose carrying doesn't depend on anything at the same time, so
we can ask what are the contexts on which
The robot would be carrying
carrying(O,do(pickup(O),S)) < at(robot,Pos,S) & at(O,Pos,S) & pickup_succeeds(O,S).
The robot would be carrying
carrying(O,do(A,S)) < carrying(O,S) & A \= putdown(O) & A \= pickup(O) & pickup_succeeds(O,S).
The robot would be carrying
carrying(O,do(putdown(O),S)) < carrying(O,S) & putdown_fails(O,S).where
Note that putdown may succeed in doing other things, it just fails in
stopping the robot from carrying
Footnote 9 is there to try to explain what the rules mean if the rule bodies aren't disjoint. We can interpret what these rules mean, but it usually isn't what you want (for the cases where both rules are applicable). When we write these rules, having nondisjoint rules is usually a bug (the implementation available from my web page warns when it finds nondisjoint rules.)
In connection with this last question, notice that on p. 20, the clause with headIt is a ramification of the robot moving. The above methodology handles these ramifications in the same way it handles frame axioms (and any other axioms).at(X, P, S) resembles neither an effect axiom (as in Example 2.8) nor a frame axiom (as in Example 2.9).
Thanks for your comments and questions. I hope this made the paper clearer. I would be happy to answer any other questions you may have (or engage in debate about the usefulness of this).
David
Q3. Hector Geffner (19.11):
David,
I've enjoyed reading your paper "Decision Theory, the Situation Calculus and Conditional Plans". I find it hard to disagree on your general point that a general model of action must combine elements from logic and probability/utility theory, so my questions and comments are addressed more to the question of how this combination is achieved in your proposal. I'll try to comment first on the more general issues.
1. In page 24, you say "The representation in this paper can be seen as a representation for POMDPs".
If so, I wonder, why not present the representation in that way from the very beginning? Wouldn't that make things much clearer?
Namely, a POMDP is described by (see [1]):
A  state space
B  actions
C  transition probabilities
D  cost function
E  observations
F  observation probabilities
Then the constructions in your language could be seen as a convenient/concise/modular way for specifying these components.
Actually, once the mapping from the language to these components is determined, no further semantics would be needed (i.e., the probabilitities of trajectories, the expected utility of policies, etc., are all defined as usual in POMDPs). This is actually the way we do things in [2].
2. I can see some reasons why not to present your representation as a "front end" to POMDPs.
3. A final comment about a nontrivial problem that results from the decision of removing all probabilities from state transitions transferring them into suitable priors in "choice space".
Consider the effect of an action "turn_off" on a single boolean fluent "on" with transition probabilities:
_{ } P(on = false  on = true ; a = turn_off) = .5  
_{ } P(on = false  on = false ; a = turn_off) = 1 
Let's say that initially
I don't see how this behavior could be captured by priors over choice space. This seems to be a strong limitation. It looks as if probabilistic actions cannot be handled after all. Is this right?
References
[1] L. Kaebling, M. Littman, and T. Cassandra. Planning and Acting in Partially Observable Stochastic Domains, AIJ 1998
[2] H. Geffner, B. Bonet. High level planning with POMDPs. Proc. 1998 AAAI Fall Symp on Cognitive Robotics (www.ldc.usb.ve/~hector)
[3] H. Levesque. What's planning in the presence of sensing? AAAI96
[4] H. Geffner and J. Wainer. A model of action, knowledge and control, Proc. ECAI 98 (www.ldc.usb.ve/~hector)
13Jun98 20:28