Date: Sun 24 Jul 1988 02:00-EDT From: AIList Moderator Nick Papadakis Reply-To: AIList@mc.lcs.mit.edu Us-Mail: MIT Mail Stop 38-390, Cambridge MA 02139 Phone: (617) 253-2737 Subject: AIList Digest V8 #23 To: AIList@mc.lcs.mit.edu Status: R AIList Digest Sunday, 24 Jul 1988 Volume 8 : Issue 23 Today's Topics: Philosophy: lim(facts about the world) -> ding an sich ? Critique of Systems Theory Are all Reasoning Systems Inconsistent? Generality in Artificial Intelligence Metaepistemology and unknowability ---------------------------------------------------------------------- Date: Sun, 17 Jul 88 11:23:37 EDT From: George McKee Subject: lim(facts about the world) -> ding an sich ? In AIList Digest v7 #41 John McCarthy wrote > There is no reason to expect a mathematical theorem about > cellular automata in general or the Life cellular automaton > in particular that says that a physicist program will be able > to discover the fundamental physics of its world is the > Life cellular automaton. This may be true if one thinks in terms of cellular automata, but if one thinks in other terms that are convertible to statements about automata, say the lambda calculus, the possible existence of such a theorem is not such a far-fetched idea. I'm enough of a newcomer to this kind of thinking myself that I won't pretend to understand this in exhaustive detail, but the concepts seem to fit together well enough at low resolution... One of the most fascinating results to come out of work in denotational semantics is that one can look at the sequence of statements that represent each step in the evaluation of lambda expressions and see that not only do the steps follow in a way that makes it proper to say that the expressions are related by a partial order, but that this partial order has enough additional structure that it can be called a continuous lattice, where the definition of "continuous" is closely related to the definition that topologists use to describe more familiar sorts of mathematical things, like surfaces in space. How close "closely related" has to be in order to be convincing is unclear to me at this time. It's this property of continuity that makes people much more comfortable with calling the lambda calculus a "calculus" than they used to be, and forms the basis for the rest of this argument. (Duke Briscoe's remarks in v8 #2 suggest that he may be thinking along these lines as well.) It means that a reasoning system based on the lambda calculus is halfway to being able to model real systems. Without going into quantum mechanics, which would lead to a discussion of a different aspect of computability, real systems in addition to being continuous, are also dense. That is, given an appropriate definition of "nearby", it's always possible to find or generate a new element between any two nearby elements. In this sense, real systems contain an infinite amount of detail. The real numbers, of course, contain infinite numbers of values like pi or the square root of 2, that fail to be computable functions in the sense that they can only be fully actualized by nonterminating computations. But a system like the lambda calculus that is able to evaluate data as programs doesn't have to compute forever in order to be able to claim to know about irrational numbers. Such a system can represent dense structures even though the representational system itself may not be dense. The issue of density is not so important in a cellular automaton universe as it is in our own, where at human scales of measurement the world is in fact dense, and a physics of partial differential equations based on real numbers has been marvelously successful. Things become really interesting when one considers a device made of dense physical material, functioning as a digital, non-dense computer system, attempting to discover and represent its own structure. The device, at the physical level, is itself, a ding an sich. At the representational level, a finite representation can exist that is not the ding an sich, but can approximate its behavior and explain its structure to whatever degree of accuracy and detail you want, given enough time. Can a device (or evolutionary series of devices) that starts out without a representation of the world devise a series of progressively more accurate representations? As long as the structure of the world (the ding an sich, not the (tentative) representation) is consistent from time to time and place to place, I can't see why not. After all, this is just an abstract, semantical way of looking at learning. But what about the last step, recognizing the convergence of the series of world-models and representing the limit of that series, i.e. representing *the* ding an sich rather than a set of approximations? The properties of continuity and density in the lambda calculus suggest that enough analogies with the differential calculus might exist to make this plausible, and that farther on, a sufficiently self-referential analog computer (the human brain may be one of this type) might be able to "compile" the representation back into a form suitable for direct action. My knowledge of either kind of calculus is not deep enough to allow me to do much more than guess about how to express this rigorously. In other words, even though it may not be possible to duplicate the universe in calculo (why bother, when the world is there to be examined?), it seems to me that it's likely to be possible to _understand_ its principles of organization, no matter how discrete your logic is. Acting congruently with that understanding is a different question. - George McKee NU Computer Science ------------------------------ Date: 19 Jul 88 05:40:06 GMT From: bingvaxu!vu0112@cs.buffalo.edu Reply-to: vu0112@bingvaxu.cc.binghamton.edu (Cliff Joslyn) Subject: Re: Philosophy: Critique of Systems Theory In a previous article, larry@VLSI.JPL.NASA.GOV writes: >Using Gilbert Cockton's references to works critical of systems theory, over >the last month I've spent a few afternoons in the CalTech and UCLA libraries >tracing down those and other criticisms. I'm very sorry to have missed the original discussion. Gilbert: could you re-mail to me? >General Systems Theory was founded by biologist Ludwig von Bertallanfy in >the late '40s. It drew heavily on biology, borrowed from many areas, and >promised a grand unified theory of all the sciences. The newer term is "Systems Science." For example, I study in a Systems Science department (one of a very few in the country), and the International Society for General Systems Research is changing its name to the Int. Soc. for the Systems Sciences. >The ideas gained >momentum till in the early '70s in the "humanics" or "soft sciences" it had >reached fad proportions. Sad but true. >What seems to have happened is that the more optimistic promises of GST >failed and lost it the support of most of its followers. Its more success- >ful ideas were pre-empted by several fields. These include control theory >in engineering, taxonomy of organizations in management, and the origins of >psychosis in social psychology. It should not be lost sight of that "Systems Science" and "Cybernetics" are different views of the same field. They showed the same course of development, especially in Norbert Weiner's career. With the rise of Chaos theory, fractals, connectionism, family therapy, global politics, and so many other things, GST/Cybernetics is implicitly achieving the kinds of results they always claimed. The body of GST work stands as a testament to the vision of those who could see the future of science, even though they couldn't claim a corner for themselves. >For me the main benefit of GST has been a personally satisfactory resolution >of the reduction paradox, which follows. > [ excellent description omitted ] It is a very difficult task to defend the discipline, which I do, because it is not clear that it is a discipline in the traditional sense. While it has a body of knowledge and a variety of specific claims about the world, and especially about dialectical philosopy, it is inherently interdisciplinary. George Klir, one of my teachers, describes it as a "second dimension" of science, studying the similarities of systems across systems types. This in itself is addressing the problem of reduction by talking about systems at different scales. >This is what many physi- >cists have done with the conflict between quantum and wave views of energy. I refere you to an article I am currently reading, by another of my professors, Howard Pattee, "The Complementarity Principle in Biological and Social Structures," in _JOurnal of Social and Bio. Structures_, vol. 1, 1978. >New kinds of systems exhibit synergy: >attributes and abilities not observed in any of their constituent elements. >But where do these attributes/abilities come from? Some Systems Scientists claim emergent phenomena in the traditional sense. Others say that that concept is not necessary, but rather "emergent" phenomena is just a problem of observing at multiple scales. The physical unity of a rock is a physical property of the electrical "synergy" of its constituent atoms. Same for a hurricane, an organism, an economy, or a society, only with different constituents. In dynamical systems it is common for there to be a complex interplay between global and local effects and phenomena. -- O----------------------------------------------------------------------> | Cliff Joslyn, Cybernetician at Large | Systems Science, SUNY Binghamton, vu0112@bingvaxu.cc.binghamton.edu V All the world is biscuit shaped. . . ------------------------------ Date: Tue, 19 Jul 88 10:16:06 EDT From: jon@XN.LL.MIT.EDU (Jonathan Leivent) Subject: Are all Reasoning Systems Inconsistent? Within any (finite) reasoning system, it is possible to construct a sentence S from any preposition A such that S = (S -> A) using Godel-like methods to establish the recursion. However, such a sentence leads to the inconsistent conclusion that A is true - any A! 1. S = (S -> A) ; the definition of S, true by construction 2. S -> (S -> A) ; a weaker form of 1. [U = V, so U -> V] 3. S -> S ; an obvious tautology 4. S -> (S ^ (S -> A)) ; from 2. and 3. by conjunction of the consequents [U -> V and U -> W, so U -> (V ^ W)] 5. (S ^ (S -> A)) -> A ; modus ponens 6. S -> A ; from 4. and 5. by transitivity of -> [U -> V and V -> W, so U -> W] 7. S ; from 1. and 6. [U = V and V, so U] 8. A ; from 6. and 7. by modus ponens Am I doing something wrong, or did logic just fall down the rabbit hole? -- Jon Leivent ------------------------------ Date: Thu, 21 Jul 88 16:05:59 EDT From: jon@XN.LL.MIT.EDU (Jonathan Leivent) Subject: more 'Are all Reasoning Systems Inconsistent?' I have found a mistake in my original proof. Here is a revised version that should more aptly be titled "Are all Reasoning Systems Impractical?": Godel's method of creating self-referential sentences allows us to create the sentence S such that S = (P(n) -> A), where A is any proposition and P(x) is true if the sentence represented by the Godel number x is provable in this reasoning system. The self-reference comes from the fact that S can be so constructed that n is the Godel number of S, hence P(n) asserts the provability of S. Originally, I managed to induce inconsistency in a reasoning system by using the sentence S = (S -> A), the inconsistency being that A is proven true regardless of what proposition it is (even a false one would do). The subtle mistake with that proof is that such a sentence is not constructable by Godel numbering techniques. The statement S = (P(n) -> A), where n is the Godel number of S itself, is constructable, and yields rather grave consequences: 1. S = (P(n) -> A) ; definition of S, true by construction [n is the Godel number of S itself] 2. P(n) -> S ; if something is provable, then it is true [the definition of P] 3. P(n) -> (P(n) -> A) ; from 1 and 2 by substitution for S 4. P(n) -> P(n) ; tautology 5. P(n) -> (P(n) ^ (P(n) -> A)) ; conjunction of the consequents in 3 and 4 6. (P(n) ^ (P(n) -> A)) -> A ; modus ponens 7. P(n) -> A ; transitivity of -> from 5 and 6 8. S ; from 1 and 7 9. P(n) ; the fact that steps 1 thru 8 prove S is sufficient to prove P(n) 10. A ; from 7 and 9 by modus ponens So, it seems on the surface that the same inconsistency is constructable: that regardless of what A is, it can be proven to be true. However, the conclusion in step 9 that P(n) is true based on the derivation of S in steps 1 thru 8, combined with the axiom P(n) -> S used in step 2, may be the source of the inconsistency. Perhaps, in order for P(n) to imply S, it must not lead to inconsistency (a proof is not a proof if it leads to a contradiction). This insistance seems to be quite self-serving, but it does the trick - the derivation of S in steps 1 thru 8 is not a proof of S because it "eventually" leads to inconsistency in step 10, hence step 9 is not valid (not that if A is instantiated to be a true propostion, then no contradiction is reached - only if A is false or uninstantiated is there a contradiction). We seem to have saved the day, except for one thing: we are requiring that a true proof of any statement involve the exhaustive search for inconsistency (contradictions). The penalty is that this forces reasoning systems to take infinite time to generate "true" theorems (otherwise, there may be a contradiction lurking under the next stone). There is no simple heuristic to use to determine that the search for inconsistency can end (some would suggest that only theorems that make statements about a reasoning system's own proof procedure are in doubt, but the above theorem can be transformed isomorphically into a theorem entirely about number theory with no reference to a proof procedure (using Godel numbering techniques again) - the new theorem would still have the same problems). So, any reasoning system that can do things in finite time should be doubted (the theorems may be true, but then again ...). Leivent's Theorem: Doubt all theorems (including this one). There's something rather sinister about this: how can this theorem be disproven? If one succeeds in proving the contrary in finite time, perhaps the theorem is still true, and the proof to the contrary would eventually lead to a contradiction. Perfect proof is, in practice, impossible. -- Jon Leivent ------------------------------ Date: 20 Jul 88 08:32 PDT From: hayes.pa@Xerox.COM Subject: Re: Generality in Artificial Intelligence Steve Easterbrook gives us an old idea: that the way to get all the knowledge our systems need is to let them experience it for themselves. It doesnt work like that, however, for several reasons. First, most of our common sense isnt in episodic memory. That candles burn more often than tables isnt something from episodic memory, for example. Or is the suggestion is that we only store episodes and do the inductive generalisations whenever we need to by remarkably efficient internal access machinery? Apart from the engineering difficulties ( I can imagine 1PC being reinvented as a handy device to save memory ), this has the problem that lots of what we know CANT be induced from experiences. Ive never been to Africa or ancient Rome, for example, but I know a fair bit about them. But the more serious problem is that the very idea of storing experiences assumes that there is some way to encode the experiences, some episodic memory which can represent episodes. I dont mean which knows how to index them efficiently, just put them into memory in the first place. You know that, in your `..experience, candles are burnt much more often than tables.' How are all those experiences of candle-combustion represented in your head? Put your wee robot in front of a candle, and what happens in its head? Now think of all the other stuff your episodic memory has to be able to represent. How is this representing done? Maybe after a while following this thought you will begin to see McCarthys footsteps on the trail in front of you. Pat Hayes ------------------------------ Date: Thu, 21 Jul 88 21:19 O From: Subject: metaepistemology and unknowability Distribution-File: AILIST@AI.AI.MIT.EDU In AIList Digest V8 #9, David Chess writes: >Can anyone complete the sentence "The actual world is unknowable to >us, because we have only descriptions/representations of it, and not..."? I may have misused the word "unknowable". I'm applying a mechanistic model of human thinking: it is an electrochemical process, neuron activation patterns representing objects which one thinks of. The heart of the matter is if you can say a person or a robot *knows* something if all it has is a representation, which may be right or wrong, and there is no way for it to get absolute knowledge. Well, the philosophy of science has a lot to say about describing the reality with a theory or a model. Note that there are two kinds of models here. The human brain utilizes electrochemical, intracranial models without us being aware of it; the philosophy of science involves written theories and models which are easy to examine, manipulate and communicate. I would say that the actual world is unknowable to us because we have only descriptions of it, and not any kind of absolutely correct, totally reliable information involving it. >(I would tend to claim that "knowing" is just (roughly) "having > the right kind of descriptions/representations of", and that > there's no genuine "unknowability" here; but that's another > debate...) The unknowability here is uncertainty about the actual state of the world very much in the same sense as scientific theories are theories, not pure, absolute truths. Andy Ylikoski ------------------------------ End of AIList Digest ********************