cpshelley@violet.uwaterloo.ca (cameron shelley) (01/25/91)
In article <11656.9101241836@s4.sys.uea.ac.uk> jrk@information-systems.east-anglia.ac.uk (Richard Kennaway CMP RA) writes: > >-- >Richard Kennaway SYS, University of East Anglia, Norwich, U.K. >Internet: jrk@sys.uea.ac.uk uucp: ...mcsun!ukc!uea-sys!jrk > >Relevance-to-newsgroup detector now registering approx. 0.0... ^^^^ I concur! Comparisons of pragmatic, existential, positivist, etc... notions of truth are interesting but becoming both entrenched and somewhat beside whatever it was that started this. There are groups such as sci.logic and talk.philosophy (?) which are better suited and may arouse a larger volume of relevant comment. Or maybe e-mail would be sufficient. While I'm here (:-), I wouldn't mind drawing some comment on a subject I've been thinking about recently. In reviewing some short papers on some new neural net work, I was struck by the notion of 'error' being employed. One of the praises always sung of NN's is that they are "robust", ie. no matter what input you give them, they won't simply fail like symbolic programs, but will rather try to compensate and produce a meaningful output. In other words (IMHO), all errors are treated as noise and an attempt is made to ignore them. There is essentially no notion of 'ill-formed' input as opposed to 'ill-transmitted' input. I argued (in the review) that this is epistemologically inadequate, at least as a model of human cognition, since humans show the ability to do recovery from both types of error (in different fashion). The cause, I believe, is that while the representation is dynamically induced (the subject knowledge) the meta-knowledge (or domain knowledge) is fixed by the structure of the NN so that it cannot attempt more than one method of solving and therefore has no redundancy -- a common error-handling technique in both machines and people. My intial suggestion was that a system should be created in which two (or more) NN's with different structures be allowed to compete for one output. The question then is: is my analysis correct, and if so, is the suggestion e-adequate both computationally and cognitively? Cam Btw: I finished the review; you are not doing my work for me! -- Cameron Shelley | "Absurdity, n. A statement of belief cpshelley@violet.waterloo.edu| manifestly inconsistent with one's own Davis Centre Rm 2136 | opinion." Phone (519) 885-1211 x3390 | Ambrose Bierce
smoliar@isi.edu (Stephen Smoliar) (01/27/91)
In article <1991Jan25.022026.12999@watdragon.waterloo.edu> cpshelley@violet.uwaterloo.ca (cameron shelley) writes: > In reviewing some short papers on >some new neural net work, I was struck by the notion of 'error' being >employed. One of the praises always sung of NN's is that they are >"robust", ie. no matter what input you give them, they won't simply >fail like symbolic programs, but will rather try to compensate and >produce a meaningful output. In other words (IMHO), all errors are >treated as noise and an attempt is made to ignore them. There is >essentially no notion of 'ill-formed' input as opposed to 'ill-transmitted' >input. I argued (in the review) that this is epistemologically >inadequate, at least as a model of human cognition, since humans >show the ability to do recovery from both types of error (in different >fashion). I want to pick up on this and perhaps try to assimilate it with Stuart Hampshire's recent article on Wittgenstein (actually his review of the new Monk biography) in THE NEW YORK REVIEW. Hampshire offers what I feel is an excellent summary of the essence of the TRACTATUS: According to the TRACTATUS the multiplicity of elements in a sentence ought to be a picture of the multiplicity of elements in a state of affairs. The formal correspondence secures for us the reference to a particular point in reality. To some extent, the idea that it makes sense to talk about such "a particular point in reality" at all was one of the major positions Wittgenstein chose to contest in his later work; but I find it interesting that much of connectionism almost seems to have translated this idea of "a particular point in reality" into a point in some multidimensional energy landscape. What counts as robustness is nothing more than the observation that these landscapes tend to be sufficiently smooth that perturbation of a starting point will not severely disrupt its trajectory along this landscape. Cam's observation that this is but one way of viewing what "error" might be is well taken; and perhaps it obliges us to go back and think some more about appropriate metaphors for these points in "reality." Let us suppose that we still have some sort of multidimensional space as a metaphor for reality; but rather than filling it with an energy landscape, suppose we instead insert a linkage structure, sort of like a linear undirected graph, as a model of an agent's "knowledge" (whatever that may mean). The reason I wish to appear to this metaphor is as an alternative to modeling reasoning in terms of the trajectory of a point in the space which seeks out an energy sink. Think, instead, of accommodating a point in space by asking how that linkage structure might get flexed, or, perhaps expanded, in such a way that it ultimately "meets" that point; that resulting "meeting" might then be regarded as an "interpretation" of that point. Given sufficiently liberal laws of what you could do with the linkages, any given point would obviously be subject to multiple interpretations, which would mean it is not really a fixed "point in reality" (which, as I assumed above, was one of the problems Wittgenstein was trying to get away from). From Cam's point of view, the question of whether or not the point is "well-formed" ultimately boils down to whether or not the linkages can be configured to "meet" it. I realize this is all rather loose metaphor. (I also remember what John McCarthy recently had to say about metaphors.) However, since the purpose of this bulletin board is to kick around philosophical approaches, I regard this as yet another pebble to toss in the pond. Anyone who wishes to make waves is certainly welcome to do so. -- USPS: Stephen Smoliar 5000 Centinela Avenue #129 Los Angeles, California 90066 Internet: smoliar@venera.isi.edu
cpshelley@violet.uwaterloo.ca (cameron shelley) (01/28/91)
In article <16510@venera.isi.edu> smoliar@venera.isi.edu (Stephen Smoliar) writes: [...] > >I want to pick up on this and perhaps try to assimilate it with Stuart >Hampshire's recent article on Wittgenstein (actually his review of the >new Monk biography) in THE NEW YORK REVIEW. Hampshire offers what I feel >is an excellent summary of the essence of the TRACTATUS: > > According to the TRACTATUS the multiplicity of elements in > a sentence ought to be a picture of the multiplicity of > elements in a state of affairs. The formal correspondence > secures for us the reference to a particular point in reality. > >To some extent, the idea that it makes sense to talk about such "a particular >point in reality" at all was one of the major positions Wittgenstein chose to >contest in his later work; but I find it interesting that much of >connectionism almost seems to have translated this idea of "a particular >point in reality" into a point in some multidimensional energy landscape. >What counts as robustness is nothing more than the observation that these >landscapes tend to be sufficiently smooth that perturbation of a starting >point will not severely disrupt its trajectory along this landscape. Cam's >observation that this is but one way of viewing what "error" might be is well >taken; and perhaps it obliges us to go back and think some more about >appropriate metaphors for these points in "reality." > I'd like to think so. The predominant model for connectionist architectures seems to be thermodynamics, at least in networks that use relaxation methods such as "simulated annealing" in which training is thought of as lowering the 'temperature' of a system so that its 'energy' becomes minimized. Of course, in classical physics, nature does not make 'mistakes' (ie. be "ill-formed"), whereas it can at least be 'deformed' in modern physics. >Let us suppose that we still have some sort of multidimensional space as a >metaphor for reality; but rather than filling it with an energy landscape, >suppose we instead insert a linkage structure, sort of like a linear undirected >graph, as a model of an agent's "knowledge" (whatever that may mean). The >reason I wish to appear to this metaphor is as an alternative to modeling >reasoning in terms of the trajectory of a point in the space which seeks out >an energy sink. Think, instead, of accommodating a point in space by asking >how that linkage structure might get flexed, or, perhaps expanded, in such a >way that it ultimately "meets" that point; that resulting "meeting" might then >be regarded as an "interpretation" of that point. Given sufficiently liberal >laws of what you could do with the linkages, any given point would obviously >be subject to multiple interpretations, which would mean it is not really a >fixed "point in reality" (which, as I assumed above, was one of the problems >Wittgenstein was trying to get away from). From Cam's point of view, the >question of whether or not the point is "well-formed" ultimately boils down >to whether or not the linkages can be configured to "meet" it. > Hmmm. An interesting notion! Could you elabourate on "meet"? If I understand correctly, you mean the structure can be deformed by some composition of adjustments applied to its parts. This seems plausible with the proviso that the entire structure need not be completely connected (or am I missing something?). The notion "ill-formed" would be the requirement of an 'illegal' translation of the structure, while "ill-transmitted" would be uncertainty about the point being met. Would we have to know all possible translations of the structure in order to separate 'legal' from 'illegal'? Complexity could then be measured by the number of elementary translations and deformations necessary to meet the point in question. By then, the default (or acquisition of) the structure becomes an issue... Cam -- Cameron Shelley | "Absurdity, n. A statement of belief cpshelley@violet.waterloo.edu| manifestly inconsistent with one's own Davis Centre Rm 2136 | opinion." Phone (519) 885-1211 x3390 | Ambrose Bierce
smoliar@isi.edu (Stephen Smoliar) (01/29/91)
In article <1991Jan27.185935.18038@watdragon.waterloo.edu> cpshelley@violet.uwaterloo.ca (cameron shelley) writes: >In article <16510@venera.isi.edu> smoliar@venera.isi.edu (Stephen Smoliar) writes: > >>Let us suppose that we still have some sort of multidimensional space as a >>metaphor for reality; but rather than filling it with an energy landscape, >>suppose we instead insert a linkage structure, sort of like a linear >>undirected >>graph, as a model of an agent's "knowledge" (whatever that may mean). The >>reason I wish to appeal to this metaphor is as an alternative to modeling >>reasoning in terms of the trajectory of a point in the space which seeks out >>an energy sink. Think, instead, of accommodating a point in space by asking >>how that linkage structure might get flexed, or, perhaps expanded, in such a >>way that it ultimately "meets" that point; that resulting "meeting" might >>then >>be regarded as an "interpretation" of that point. Given sufficiently liberal >>laws of what you could do with the linkages, any given point would obviously >>be subject to multiple interpretations, which would mean it is not really a >>fixed "point in reality" (which, as I assumed above, was one of the problems >>Wittgenstein was trying to get away from). From Cam's point of view, the >>question of whether or not the point is "well-formed" ultimately boils down >>to whether or not the linkages can be configured to "meet" it. >> >Hmmm. An interesting notion! Could you elabourate on "meet"? If I >understand correctly, you mean the structure can be deformed by some >composition of adjustments applied to its parts. This seems plausible >with the proviso that the entire structure need not be completely >connected (or am I missing something?). The notion "ill-formed" would >be the requirement of an 'illegal' translation of the structure, while >"ill-transmitted" would be uncertainty about the point being met. >Would we have to know all possible translations of the structure in >order to separate 'legal' from 'illegal'? Complexity could then be >measured by the number of elementary translations and deformations >necessary to meet the point in question. By then, the default (or >acquisition of) the structure becomes an issue... > I think we're basically on the same channel here. I was using "meet" in the set-theoretic sense of intersection. The more I think about it, I think there are (at least) two implications to this approach: 1. There is the issue of whether or not, and how, the linkages can be manipulated in order to achieve a meeting. 2. There is the issue of WHERE on the linkages this meeting actually takes place. In a very loose sense the first issue has to do with whether or not that particular point in space is well-formed, in the notion that Cam wishes to pursue. Under the assumption that it IS well-formed, the second issue then takes on the matter of HOW it will be interpreted. This is what I was trying to get at in saying that different ways of manipulating the linkages might lead to different interpretations: Depending on the specific manipulations, the point may meet different locations on the linkage, itself. Does the structure have to be connected? I do not see that as necessary. I am more interested in the extent to which this metaphor can be useful interpreted into a set of viable rules for manipulation the structure. There is no reason to eliminate rules which would pull the structure apart unless it could be demonstrated that they led to undesirable consequences (such as, perhaps, an inability to recover information about where the meeting took place when it was finally achieved). My own sense of aesthetics seems more inclined to allowing the "limbs" of the structure to be stretched or shrunken, rather than allowing the structure to be pulled apart; but, as I said, I think it is more important to start thinking about reasonable ways to talk about the manipulation, itself. Another possibility might be that the manipulations are sufficiently powerful that the structure can always access any point in the space. In this case there would no longer be an issue of ill-formed points. This might be a way in which the manipulation of the structure reflects an adjustment in interpretation to accommodate what might have otherwise been regarded as an ill-formed point. Perhaps I can try to illustrate what I am trying to say here with a concrete example. Peter Todd recently published some of his work in trying to use connectionism for musical composition. The basic approach was to "train" a network with some examples of melodies and then, through control of some inputs, allow a trained network to synthesize new melodies. During training, Todd has some very strict rules about what constitutes well-formed input. However, when he leaves the network to its own devices, so to speak, the results it yields do not respect those rules. As an outside observer, he interprets them in a way which basically "makes sense" according to the original INTENT of his rules. To return to my metaphor, in the strictest sense Todd's network converges to points in space which are actually ill-formed. However, his human intelligence controls this metaphorical system of linkages in such a way that he can still assign interpretations to those points which are consistent with what he originally had in mind. Robustness is thus a matter of finding the best interpretation for a given situation (for some metric for "best") rather than trying to decide, in any absolute sense, whether or not that situation is "well-formed." -- USPS: Stephen Smoliar 5000 Centinela Avenue #129 Los Angeles, California 90066 Internet: smoliar@venera.isi.edu
cpshelley@violet.uwaterloo.ca (cameron shelley) (01/30/91)
In article <16537@venera.isi.edu> smoliar@venera.isi.edu (Stephen Smoliar) writes: [...] >I think we're basically on the same channel here. I was using "meet" in the >set-theoretic sense of intersection. The more I think about it, I think there >are (at least) two implications to this approach: > > 1. There is the issue of whether or not, and how, the linkages can > be manipulated in order to achieve a meeting. > > 2. There is the issue of WHERE on the linkages this meeting actually > takes place. > >In a very loose sense the first issue has to do with whether or not that >particular point in space is well-formed, in the notion that Cam wishes >to pursue. Under the assumption that it IS well-formed, the second issue >then takes on the matter of HOW it will be interpreted. This is what I was >trying to get at in saying that different ways of manipulating the linkages >might lead to different interpretations: Depending on the specific >manipulations, the point may meet different locations on the linkage, >itself. > I don't think that's too controversial. The existence of multiple interpretations is a time-honoured way of describing ambiguity: in this case, there is more than one set of translations that meet the desired point. I would then go on to distinguish "ill-transmitted", in which the desired point has been underdefined resulting in multiple points which satisfy the constraints of the transmission (and each point may be ambiguous in its own right), from "ill-formed" in which the transmitted constraints themselves cannot be interpreted (by some sense of adequacy) so that no points at all have been defined. Note that the last notion can be quite independant of errors in transmission. The concern in my original post was that connectionist architectures treat "ill-formed" as "ill-transmitted" when I don't think this does the subject justice. No doubt this is for the convenience of being able to use results from information theory in all cases. In dealing with interpretation in general, I think such (0,1,plural) boundary cases are most profitably treated as separate until evidence indicates otherwise. Another area in which (to my knowledge) the '0' case is not treated is theoretical phonology. Although the 'null' symbol turns up in phonological rules (deletion), it is never defined. This also means that the theory assumes there is some certain method of distinguishing the properties of speech sound from those of other sound before phonology is invoked. But I digress... (sorry!) >Does the structure have to be connected? I do not see that as necessary. I am >more interested in the extent to which this metaphor can be useful interpreted >into a set of viable rules for manipulation the structure. There is no reason >to eliminate rules which would pull the structure apart unless it could be >demonstrated that they led to undesirable consequences (such as, perhaps, >an inability to recover information about where the meeting took place when >it was finally achieved). My own sense of aesthetics seems more inclined to >allowing the "limbs" of the structure to be stretched or shrunken, rather than >allowing the structure to be pulled apart; but, as I said, I think it is more >important to start thinking about reasonable ways to talk about the >manipulation, itself. > Well, empirically, it is possible to lose information, and I don't see any advantage in requiring the entire 'knowledge' structure to be connected. However, this may depend on how 'knowledge' is defined. >Another possibility might be that the manipulations are sufficiently powerful >that the structure can always access any point in the space. In this case >there would no longer be an issue of ill-formed points. This might be a way >in which the manipulation of the structure reflects an adjustment in >interpretation to accommodate what might have otherwise been regarded >as an ill-formed point. > Again, I see this as ignoring a potential problem. Under-specification may be treated in this fashion, but I still see 'non-specification' as something that must be dealt with. Perhaps even a further distinction is necessary: a system shouldn't waste time interpreting random noise ('non-transmission'), but should simply baulk at nonsense ('ill-formed' transmission), and deal as mentioned above with noise in a sensible transmission. For example, the system should deal differently with: "%^%#*%^&%^%^%%^$#@!#$#" (noise), "After being run *^@r by the truck, the man said 'Ouch'!" (where "over" can be interpolated), "flying airplanes make people ill" (ambiguity), and "pick peek poke pack puck pink" (nonsense). Should I interpret the last as a description of a psychodelic hockey game, or just respond "What?" I think both should be options, but NN's (as they exist currently) do not have a choice in such a case. >Perhaps I can try to illustrate what I am trying to say here with a concrete >example. Peter Todd recently published some of his work in trying to use >connectionism for musical composition. The basic approach was to "train" >a network with some examples of melodies and then, through control of some >inputs, allow a trained network to synthesize new melodies. During training, >Todd has some very strict rules about what constitutes well-formed input. >However, when he leaves the network to its own devices, so to speak, the >results it yields do not respect those rules. As an outside observer, he >interprets them in a way which basically "makes sense" according to the >original INTENT of his rules. To return to my metaphor, in the strictest >sense Todd's network converges to points in space which are actually >ill-formed. However, his human intelligence controls this metaphorical >system of linkages in such a way that he can still assign interpretations >to those points which are consistent with what he originally had in mind. >Robustness is thus a matter of finding the best interpretation for a given >situation (for some metric for "best") rather than trying to decide, in any >absolute sense, whether or not that situation is "well-formed." This suggests that the notion of "ill-formed" is context-dependant. My response in this case would be that Peter's experiment has defined away the notion of ill-formed at the start, in the sense that the structure of the net has been controlled so that it can only produce results which have possible 'melodic' interpretations. In any case, this seems to be bringing the mountain to Mohammed; simply deforming the topology of the knowledge structure to a point doesn't appear to me to be much different than moving a point across the topology. But your point is well taken, some notion of 'systematic accomodation' seems useful. Cam -- Cameron Shelley | "Absurdity, n. A statement of belief cpshelley@violet.waterloo.edu| manifestly inconsistent with one's own Davis Centre Rm 2136 | opinion." Phone (519) 885-1211 x3390 | Ambrose Bierce
smoliar@isi.edu (Stephen Smoliar) (01/31/91)
In article <1991Jan29.165646.17764@watdragon.waterloo.edu> cpshelley@violet.uwaterloo.ca (cameron shelley) writes: > For example, the system should deal differently with: >"%^%#*%^&%^%^%%^$#@!#$#" (noise), "After being run *^@r by the truck, >the man said 'Ouch'!" (where "over" can be interpolated), "flying >airplanes make people ill" (ambiguity), and "pick peek poke pack puck pink" >(nonsense). Should I interpret the last as a description of a >psychodelic hockey game, or just respond "What?" I think both should >be options, but NN's (as they exist currently) do not have a choice >in such a case. > Cam, you interpreted my account of Peter Todd's work as suggesting "that the notion of 'ill-formed' is context-dependent." I would say that the above nonsense example supports a similar conclusion: How you choose to interpret depends upon the context in which you received it. I would even say the same of your noise example. If you saw those symbols in some avant-garde poetry magazine, you might very well try to kick in SOME attempt at interpretation, rather than just writing them off as noise (even if they had been scrupulously generated by a truly random source). Nevertheless, I think we are basically in agreement. Todd's system "works" because its architecture assumes a single context. I have not seen any convincing demonstration of a neural net which would be capable of maintaining multiple contexts, such as your nonsense example requires, let alone having a level of control for DECIDING which context is appropriate for a given interpretation task. -- USPS: Stephen Smoliar 5000 Centinela Avenue #129 Los Angeles, California 90066 Internet: smoliar@venera.isi.edu
cpshelley@violet.uwaterloo.ca (cameron shelley) (01/31/91)
In article <16558@venera.isi.edu> smoliar@venera.isi.edu (Stephen Smoliar) writes: >In article <1991Jan29.165646.17764@watdragon.waterloo.edu> >cpshelley@violet.uwaterloo.ca (cameron shelley) writes: >> For example, the system should deal differently with: >>"%^%#*%^&%^%^%%^$#@!#$#" (noise), "After being run *^@r by the truck, >>the man said 'Ouch'!" (where "over" can be interpolated), "flying >>airplanes make people ill" (ambiguity), and "pick peek poke pack puck pink" >>(nonsense). Should I interpret the last as a description of a >>psychodelic hockey game, or just respond "What?" I think both should >>be options, but NN's (as they exist currently) do not have a choice >>in such a case. >> >Cam, you interpreted my account of Peter Todd's work as suggesting "that the >notion of 'ill-formed' is context-dependent." I would say that the above >nonsense example supports a similar conclusion: How you choose to interpret >depends upon the context in which you received it. I would even say the same >of your noise example. If you saw those symbols in some avant-garde poetry >magazine, you might very well try to kick in SOME attempt at interpretation, >rather than just writing them off as noise (even if they had been scrupulously >generated by a truly random source). > Actually, I was wondering about that after sending that post. It seems to me that you're correct. If I equate 'context' with 'interpretation function' being applied, then I suppose you could have a crack at anything. I am reminded that in Montague semantics (if I recall correctly), the truth assignment of a function has a superscript identifying which of the multiple 'worlds' it is applying to. The situation seems similar here, with 'world' playing the role of 'context'. >Nevertheless, I think we are basically in agreement. Todd's system "works" >because its architecture assumes a single context. I have not seen any >convincing demonstration of a neural net which would be capable of maintaining >multiple contexts, such as your nonsense example requires, let alone having a >level of control for DECIDING which context is appropriate for a given >interpretation task. Ok, let me try this out. To summarize the information distinctions I made previously (about how well transmitted constraints can be satisfied in general), within a fixed context, let @ = noise, 0 = under- defined point, 1 = uniquely-defined point, * = multiply-defined points. [Opps! I should mention that by 'point', I mean a coordinate in a solution space...] Thus the options (under a fixed context) that a complete classification system should have are (@,0,1,*). I know of no NN which has a node dedicated to '@', ie. to explicitly recognizing over-noisy input; I am also unaware of any which explicitly recognizes '0' (which I am also calling "ill-formed"), ie. units which are individually recognizable but form no interpretable whole (regardless of noise); on the other hand, NN's are quite capable of producing 1 or * (more) interpretations by activating a single output mainly, or several equally. This implies that current NN design could well be augmented with nodes on the output indicating '@' and '0' which would be mutually inhibitory. I noticed in the Dec CACM, that Kevin Knight suggested putting noise in normal training pairs, ie. ([a,p,^,l,e],apple); what I'm suggesting is ([^,%,#,&,&],@). This also leads me to suggest pairs like ([s,t,k,a],0), bearing in mind that I'm fixing the context on english words. You may take issue with the example I've chosen, but I think the idea is at least sound. As far as varying the context goes, there are examples of I know of in which entire sub-structures of a NN are mutually exclusive but deal with the same input 'item'. My original mention of two differently structured nets independantly competing for the same 'output' seems similar. On the other hand, I have no idea how a net might be allowed to vary its context freely (or creatively!) I would be curious to know how the linkage structure you proposed might account for this. -- Cameron Shelley | "Absurdity, n. A statement of belief cpshelley@violet.waterloo.edu| manifestly inconsistent with one's own Davis Centre Rm 2136 | opinion." Phone (519) 885-1211 x3390 | Ambrose Bierce
smoliar@isi.edu (Stephen Smoliar) (02/01/91)
In article <1991Jan31.040637.15353@watdragon.waterloo.edu> cpshelley@violet.uwaterloo.ca (cameron shelley) writes: > >As far as varying the context goes, there are examples of I know of >in which entire sub-structures of a NN are mutually exclusive but >deal with the same input 'item'. My original mention of two >differently structured nets independantly competing for the same >'output' seems similar. On the other hand, I have no idea how a >net might be allowed to vary its context freely (or creatively!) >I would be curious to know how the linkage structure you proposed >might account for this. > So would I! I seem to have started with a casual metaphor, and now I have to account for it! Bearing in mind that I still do not know how seriously I want to take this metaphor (primarily because I am not sure I want to buy into viewing knowledge states in terms of points in a space of some unknown dimension and a topology which is probably not Euclidean), I would say the way to approach context is in terms of the current configuration of the linkage system. As I said before, I tend to view interpretation in terms of both how the linkages will move to meet a given point and where that meeting takes place. How the linkages will be disposed to move (and I use that word intentionally acknowledging the influence of Minsky's original K-lines paper) will depend not only on where the point is but also on how the linkages are currently configured. That configuration should, in at least a remote sense, be a representation of previous (mental) activity, which is to say a context established by what the linkages were last doing. Varying context may then be a matter of shaking the linkages around before trying to approach the point (which is beginning to sound a bit like raising the temperature before cooling down to anneal). -- USPS: Stephen Smoliar 5000 Centinela Avenue #129 Los Angeles, California 90066 Internet: smoliar@venera.isi.edu