Tuesday, May 18, 2010

SOAR workshop so far

Daniel and I have just come back from the pre-workshop dinner. Our ride (another attendee) had a delayed flight, and we had to call a taxi; we ended up being about 45 minutes late for dinner. Daniel and I ended up getting seats at a separate table, so we didn't talk to anyone right away; eventually, though, we got some good discussions.

My primary thesis has always been the importance of expressive power in AI. This has been a reaction to the overwhelming majority of AI research being done with propositional methods.  However, this method tends to fall on deaf ears for several reasons.

  1. If I don't phrase myself carefully, it sounds like I'm arguing the anti-AI position that many people abuse Gödel's theorem to argue. This is thanks, for example, to my use of words like "uncomputability" (and if I invoke Gödel's theorem, of course, that just makes it worse). If I say "system X can't represent uncomputable mathematical concepts, but humans can," it really makes people think that I mean "computers can't handle uncomputable mathematical concepts, but humans can." I don't mean that. Rather, I want to soundly answer the anti-AI position via a formal theory of which systems are capable of climbing the Tarski hierarchy on their own (and how they do it).
  2. John Laird and the other fellow in the front seat when he was giving me a ride back to the hotel (sorry, don't remember the name! EDIT: Paul Rosenbloom) pointed out that worrying about expressive completeness is far less common than trying to find less-expressive subsets of existing systems which are more efficiently manipulated. Daniel put it another way: one shouldn't just worry about what a system can do in principle but about what it can do in reasonable amounts of time. Systems which are "more expressive" might be able to do far less in practice. My response to this is just to say that I prefer systems which use heuristics to stay in the space of efficient reasoning, rather than reduced expressiveness... this is a weak-sounding point, though I do think it is the right response...
  3. If I'm already talking to someone who believes that expressive completeness is an important issue, then the remainder of my pet theory consists of ways of generalizing from a propositional system to more expressive models. This typically becomes a situation of singing to the quire, because the person will already be past propositional models and into at least first-order. All I have left to argue is the seemingly reversed position that most reasoning should actually take place at the sub-first-order level (say, 1/2 n-tuple markov models, 1/4 context-free, 1/8 context-sensitive, 1/16 first-order, 1/32 2nd-order...). This is related to the response for the 2nd point.
  4. My project is really quite an abstract one, because at present my best guess is that so long as a system is doing probabilistic relational learning (ie, at least 1st order), and so long as it approximates Solomonoff induction in its ability to discover hidden relations in data, then it satisfies the requirements to climb the Tarski hierarchy. Because of this, I can't argue that there is a pitfall of expressive incompleteness waiting to trap anyone who isn't careful. So, it seems I should only be arguing for those basic requirements rather than arguing that a general theory telling us how to climb the Tarski hierarchy must be found. If I can argue for those points on separate grounds, a general theory saying that those are sufficient is more a nice factoid for those worried about the foundations of mathematics than a guiding principle for AI. (Though, there's still the possibility that the required structure for a system is significantly more specific than Solomonoff induction, in which case the theory may have definite value.)
So, should I give up what has been my primary thesis for 4 years? The banner "Expressive completeness is the central problem!" may not be the best way of organizing and presenting my ideas at present... in any case, it's clear that I need to think about my arguments more carefully.

    Friday, May 7, 2010

    Action Logic

    Continuing a thought in this post.

    A normal axiomatic logic pins down which inferences are permissible, but makes no comment on how these inferences are to be made. For real implementations, we've come up with a plethora of inference strategies-- however, the axiomatic system is always treated as separate from the inference strategies. My idea of action logic is to merge the two, hopefully in order to do nifty things like letting the system reason about its own inference strategies in a smooth way.

    In other words, what we want is to nicely integrate logic and programming.

    Several attempts to do this already exist. One example is Prolog. Prolog does not, however, satisfy the desire to allow deductive reasoning to determine flow of inference; Prolog just performs a backtracking resolution algorithm on horn clauses, paying some heed to meta-logical control flow statements that can be inserted.

    However, I think we can take some inspiration from a predecessor of Prolog, known as Planner. Planner worked through a backtracking mechanism similar to that of Prolog, but more expressive. It used a separate notion of "goal" and "fact" to drive reasoning: if a goal is present, then the system starts reasoning about how to achieve that goal. This can involve the creation of subgoals which drive more reasoning, the execution of programs, and performance of inference steps.

    One could argue that Prolog makes this distinction, but it is much less explicit: since programs and statements are the same thing, the creation of subgoals always just corresponds to the conditions in an if-then. (In PLANNER one can specify arbitrary advice for what subgoal to create at a given step; it is not so strongly tied to the form of the logical assertions in the knowledge base.)

    PLANNER also has the feature that facts can be removed from the knowledge base when they become false, ie, the system does not need to index facts to time in order to model facts changing; this is important for efficiency, though it is less "logically pure" than the alternative.

    This is a good overview of Planner for the interested. More recent systems such as SOAR, SCREAMER, ACL2, et cetera have many of the same features; I am really picking out PLANNER mainly for its historical significance.

    In any case. I said that I was trying to merge declarative and procedural, yet I am praising PLANNER's explicit goal system over Prolog's implicit way of dealing with goals. Why? Because it is more powerful: we can provide more inference guidance in PLANNER than in Prolog.

    In any case, I am not saying that Planner is a sufficient action logic, either! In fact, I'd say, let's question the basic mechanism which it shares with Prolog: if the system has goal G, and knows a statement of the form "S implies G," then it will automatically make a subgoal S. (For the curious: Both PLANNER and Prolog will select out of several statements "X implies G", try that X, give up if that subgoal turns out infeasible, and try the next. This technique is applied recursively, creating sub-sub goals, et cetera. In Prolog this is the only mechanism at play; in PLANNER there are more conditions to be met for this to happen, there are other mechanisms at work, and the programmer can add arbitrary mechanisms.)

    What happens when we vary this mechanism? Well, there are a lot of options. We could create a reinforcement-type agent which attempted to learn what inferences to draw and what subgoals to create and when, getting rewarded whenever it meets a goal specified by the programmer. We could create a brute-force system which keeps proving theorems until it happens upon one of the form "P implies G" where P is a fully specified sequence of actions, rather than just a subgoal-- the system then does P. We could produce a system like Hutter's asymptotically optimal algorithm by creating a mechanism which searches for plans of action, replacing the current plan when the system finds a proof that switching to a different plan will result in a better outcome.

    The weakest choice is to just require that if the goal "do X" is asserted, and X is a sequence of actions, the system does them immediately. This just means that the system is programmable. (X can be an arbitrary program because it can include branching goto type statements.) Any other choice of system can be modeled in this one by telling it to behave in that way.

    Stronger choices will provide more built-in useful behavior, but will also have more built-in assumptions.

    My desire is to create an abstract framework, not something that pins things down to a specific implementation. So, it seems like what I'm looking for is an abstract "should-ness" concerning what inferences to draw when, and what subgoals to make when.

    Much of the logic discussed under the title of action logic in the Stanford Encyclopedia is somewhat relevant here, though the logics there are definitely not action logics in the sense I intend: they are just normal axiomatic systems, which must rely on outside inference procedures.

    In general, my approach to AI is very much in the line of probabilistic relational methods: I advocate a move to probabilistic reasoning where just boolean reasoning is present, and to relational methods where just propositional methods are present. In this case, that means that what I see as the logical next step is to research probabilistic generalizations of PLANNER... it seems this would force many of the simplistic methods employed in PLANNER to be "broken open," so to speak, getting out the chaff and determining what can be salvaged and improved for the probabilistic setting.