In NARS, "If A then B" is represented as an Implication statement "P ==> Q", whose truth value serves a similar role as P(B|A) in a Bayesian network, though the two have subtle and important differences. For detailed discussion, see http://nars.wang.googlepages.com/wang.bayesianism.pdf and http://nars.wang.googlepages.com/wang.confidence.pdf
The Implication relation is isomorphic to the Inheritance relation, but the two are not the same, and cannot exchange with each other. I don't have a short explanation on this topic, so you'd have to read the book, or at least http://nars.wang.googlepages.com/wang.abduction.pdf Pei On 10/9/07, Edward W. Porter <[EMAIL PROTECTED]> wrote: > > > > > It wasn't a question to you in particular, but to the list. > > > > You had suggested that the terms "parent" and "child" were awkward and > misleading for probabilistic implication. I was interested in seeing how > inheritance statement would represent the types of probabilistic implication > most of us are used to thinking in terms of. > > > > Bayesian probabilities provide a valuable tool for representation and > inference. If one has a probability statement such as p(A|B,C) I understand > how NARS's inheritance rules are useful in determining whether you have a B > and/or a C, and if you had an A, much of what that would entail. I also > understand how they could be used to determine when it would be appropriate > for a given perceived or conceived pattern or set of patterns to inherit > inferences from other patterns or categories. > > > > What I was asking is how categorical logic actually represents the rules of > Bayesian inference, and how it derives them from inheritance statements. I > was also interested in how the truth values for the existence of B and C, if > either or both were less than one, in the above examples, would be blended > with the conditional probability of A that p(A|B,C) would imply if the truth > values of B and C were one. > > > > I might be able to figure this out on my own, but I assume others could do > it faster than I, and if somebody has already done it, rather than spending > time trying to re-invent the wheel, it would be easier to just read it. > > > > I know Novamente has a Probabilistic Term Logic based on both inference from > inheritance rules and Bayesian analysis, and I am looking forward to > learning more about it, but until that day, perhaps somebody else, such as > Pei, has already come up with a mapping between categorical logic and > Bayesian probabilities. > > Ed Porter > > > > > -----Original Message----- > From: Mark Waser [mailto:[EMAIL PROTECTED] > Sent: Tuesday, October 09, 2007 5:32 PM > To: [email protected] > Subject: Re: [agi] Do the inference rules of categorical logic make sense? > > > I'm sorry . . . . but I don't understand the question . . . . > > ----- Original Message ----- > From: Edward W. Porter > To: [email protected] > Sent: Tuesday, October 09, 2007 4:57 PM > Subject: RE: [agi] Do the inference rules of categorical logic make sense? > > > Mark Waser, > > With regard to your statement in the below post that > > "my point was meant to be that using the terms parent and child for > probabilistic implication is very awkward and misleading," > > perhaps some one could point out how categorical logic maps into and > represents bayesian probabilities (other than the vital role it could play > in determining if you have terms corresponding to those in a given Bayesian > probability statement---the role Pei was referring to when he said > "Inference/reasoning is not about "to find/prove the absolute truth", but > "to treat one thing (e.g., a novel object/situation) as another (which is > better known in experience)"). > > Edward W. Porter > Porter & Associates > 24 String Bridge S12 > Exeter, NH 03833 > (617) 494-1722 > Fax (617) 494-1822 > [EMAIL PROTECTED] > > > > > -----Original Message----- > From: Mark Waser [mailto:[EMAIL PROTECTED] > Sent: Tuesday, October 09, 2007 4:25 PM > To: [email protected] > Subject: Re: [agi] Do the inference rules of categorical logic make sense? > > > > > > To be honest, I do not yet understand how implication is derived from the > inheritance relations in NARS. > > Implication is a form of inheritance. > > Assuming truth values of one for the child and child/parent inheritance > statement, I would guess a child implies its parent with a truth value of > one. I would assume a parent with a truth value of one implies a given > child with a lesser value that decrease the more often the parent is mapped > against other children. > > A child implies its parent with the frequency of the implication statement. > > > > Your arguments, are of the very different form : > > Given P and Q, this implies Q --> P and P --> Q > > My apologies. I wasn't even talking about inference rules yet and was > unclear. > > I assumed that you recognized the equivalence of adult and drinking age > (i.e. P <==> Q) and realized that equivalence is exactly the same as two > implication statements (P ==> Q and Q ==> P). My point was meant to be that > using the terms parent and child for equivalence statements is very awkward > and misleading. > > Similarly, I assumed that you recognized that R ==> ~S (with the frequency > of whatever "most" means) and S ==> ~R (with a low probability since a > single counter-example does very little to disprove a most). Again, > however, my point was meant to be that using the terms parent and child for > probabilistic implication is very awkward and misleading. > > > > ----- Original Message ----- > From: Edward W. Porter > To: [email protected] > Sent: Tuesday, October 09, 2007 3:44 PM > Subject: RE: [agi] Do the inference rules of categorical logic make sense? > > > > > Mark, > > > > The basic inference rules in NARS that would support an implication of the > form S is a child of P are of the form: > > > > DEDUCTION INFERENCE RULE: > Given S --> M and M--> P, this implies S --> P > > ABDUCTION INFERENCE RULE: > Given S --> M and P --> M, this implies S --> P to some degree > > INDUCTION INFERENCE RULE: > Given M --> S and M --> P, this implies S --> P to some degree > > > > where "-->" is the inheritance relations. > > > > Your arguments, are of the very different form : > > Given P and Q, this implies Q --> P and P --> Q > > > > And > > > > Given S and R, this implies S --> R and R --> S > > > > In the argument regarding drinking and being an adult, you do not > appear to use any of these NARS inference rules to show that P inherits from > Q or vice versa (unless, perhaps, one assumes multiple other NARS sentences > or terms that might help the inference along, such as an uber category such > as the "category of all categories" from which one could use the abduction > rule to imply both of the inheritances mentioned (which one would assume the > system would have learned over time was such a weak source of implication as > to be normally useless). > > > > But in that example, just from common sense reasoning, including knowledge > of the relevant subject matter, (absent any knowledge of NARS) it appears > reasonable to imply P from Q and Q from P. So if NARS did the same it would > be behaving in a common sense way. Loops in transitivity might be really > ugly, but it seems any human-level AGI has to have the same ability to deal > with them as human common sense. > > > > To be honest, I do not yet understand how implication is derived from the > inheritance relations in NARS. Assuming truth values of one for the child > and child/parent inheritance statement, I would guess a child implies its > parent with a truth value of one. I would assume a parent with a truth > value of one implies a given child with a lesser value that decrease the > more often the parent is mapped against other children. > > > > The argument claiming NARS says that R ("most ravens are black") is both the > parent and child of S ("this raven is white") (and vice versa), similarly > does not appear to be derivable from only the statements given using the > NARS inference rules. > > > > Nor does my common sense reasoning help me understand why "most ravens are > black" is both the parent and child of "this raven is white." (All though > my common sense does tell me that "this raven is black" would provide common > sense inductive evidence for "most ravens are black" and that "this raven" > that is black would be a child of the category of "most ravens" that are > black.) > > > > But I do understand that each of these two statements would tend to have > probabilistic effects on the other, as you suggested, assuming that the > fact a raven is black has implications on whether or not it is white. But > such two way probabilistic relationships are at the core of Bayesian > inference, so there is no reason why they should not be part of an AGI. > > Edward W. Porter > Porter & Associates > 24 String Bridge S12 > Exeter, NH 03833 > (617) 494-1722 > Fax (617) 494-1822 > [EMAIL PROTECTED] > > > > > -----Original Message----- > From: Mark Waser [mailto:[EMAIL PROTECTED] > Sent: Tuesday, October 09, 2007 2:28 PM > To: [email protected] > Subject: Re: [agi] Do the inference rules of categorical logic make sense? > > > > Most of the discussion I read in Pei's article related to inheritance > relations between terms, that operated as subject and predicates in > sentences that are inheritance statements, rather than between entire > statements, unless the statement was a subject or a predicate of a higher > order inheritance statement. So what you are referring to appears to be > beyond what I have read. > > Label the statement "I am allowed to drink alcohol" as P and the statement > "I am an adult" as Q. P implies Q and Q implies P (assume that age 21 > equals adult) --OR-- P is the parent of Q and Q is the parent of P. > > Label the statement that "most ravens are black" as R and the statement that > "this raven is white" as S. R affects the probability of S and, to a lesser > extent, S affects the probability of R (both in a negative direction) --OR-- > R is the parent of S and S is the parent of R (although, realistically, the > probability change is so miniscule that you really could argue that this > isn't true). > > NARS's inheritance is the "inheritance" of influence on the probability > values. > > ----- Original Message ----- > > From: Edward W. Porter > To: [email protected] > Sent: Tuesday, October 09, 2007 1:12 PM > Subject: RE: [agi] Do the inference rules of categorical logic make sense? > > > Mark, > > Thank you for your reply. I just ate a lunch with too much fat (luckily > largely olive oil) in it so, my brain is a little sleepy. If it is not too > much trouble could you please map out the inheritance relationships from > which one derives how "I am allowed to drink alcohol" is both a parent and > the child of "I am an adult." And could you please do the same with how > "most ravens are balck" is both parent and child of "this raven is white." > > Most of the discussion I read in Pei's article related to inheritance > relations between terms, that operated as subject and predicates in > sentences that are inheritance statements, rather than between entire > statemens, unless the statement was a subject or a predicate of a higher > order inheritance statement. So what you are referring to appears to be > beyond what I have read. > > Edward W. Porter > Porter & Associates > 24 String Bridge S12 > Exeter, NH 03833 > (617) 494-1722 > Fax (617) 494-1822 > [EMAIL PROTECTED] > > > > > -----Original Message----- > From: Mark Waser [mailto:[EMAIL PROTECTED] > Sent: Tuesday, October 09, 2007 12:47 PM > To: [email protected] > Subject: Re: [agi] Do the inference rules of categorical logic make sense? > > > Thus, as I understand it, one can view all inheritance statements as > indicating the evidence that one instance or category belongs to, and thus > is "a child of" another category, which includes, and thus can be viewed as > "a parent" of the other. > > Yes, that is inheritance as Pei uses it. But are you comfortable with the > fact that "I am allowed to drink alcohol" is normally both the parent and > the child of "I am an adult " (and vice versa)? How about the fact that > "most ravens are black" is both the parent and child of "this raven is > white" (and vice versa)? > > Since inheritance relations are transitive, the resulting hierarchy of > categories involves nodes that can be considered ancestors (i.e., parents, > parents of parents, etc.) of others and nodes that can be viewed as > descendents (children, children of children, etc.) of others. > > And how often do you really want to do this with concepts like the above -- > or when the evidence is substantially less than unity? > > And loops and transitivity are really ugly . . . . > > NARS really isn't your father's inheritance. > > > ----- Original Message ----- > From: Edward W. Porter > To: [email protected] > Sent: Tuesday, October 09, 2007 12:24 PM > Subject: RE: [agi] Do the inference rules of categorical logic make sense? > > > > RE: (1) THE VALUE OF "CHILD OF" AND "PARENT OF" RELATIONS & (2) DISCUSSION > OF POSSIBLE VALUE IN DISTINGUISHING BETWEEN GENERALIZATIONAL AND > COMPOSITIONAL INHERITANCE HIERARCHIES. > > Re Mark Waser's 10/9/2007 9:46 AM post: Perhaps Mark understands something I > don't. > > I think relations that can be viewed as "child of" and "parent of" in a > hierarchy of categories are extremely important (for reasons set forth in > more detail below) and it is not clear to me that Pei meant something other > than this. > > If Mark or anyone else has reason to believe that "what [Pei] means is quite > different" than such "child of" and "parent of" relations, I would > appreciate being illuminated by what that different meaning is. > > > > My understanding of NARS is that it is concerned with inheritance relations, > which as I understand it, indicate the truth value of the assumption that > one category falls within another category, where category is broadly > defined to included not only what we normally think of as categories, but > also relationships, slots in relationships, and categories defined by a sets > of one or more properties, attributes, elements, relationships, or slot in > relationships. Thus, as I understand it, one can view all inheritance > statements as indicating the evidence that one instance or category belongs > to, and thus is "a child of" another category, which includes, and thus can > be viewed as "a parent" of the other. Since inheritance relations are > transitive, the resulting hierarchy of categories involves nodes that can be > considered ancestors (i.e., parents, parents of parents, etc.) of others and > nodes that can be viewed as descendents (children, children of children, > etc.) of others. > > I tend to think of similarity as a sibling relationship under a shared > hidden parent category -- based on similar aspects of the sibling's > extensions and/or intensions. > > In much of my own thinking I have thought of such categorization relations > as is generalization, in which the parent is the genus, and the child is the > species. Generalization is important for many reasons. First, perception > is trying to figure which in category or generalization of things, actions, > or situations various parts of a current set of sensory information might > fit. Secondly, Generalization is important because it is necessary for > implication. All those Bayesian probabilities we are used to thinking about > such as P(A|B,C), are totally useless unless we have some way of knowing the > probability the situation being considered contains a B or C. To do that > you have to have categories that help you determine the extent to which a B > or a C is present. To understand the implication of P(A|B,C) you have to > have some meaning for the category A. Generalization is important for > behavior because one uses generalization learned from past experiences to > develop plans for how to achieve goals, and because most action schema are > usually generalization that have to be instantiated in a context specific > way. > > One of the key problems in AI has been non-literal matching. That is why > representation schemes that have a flexibility something like that of NARS > are necessary for any intelligence capable of operating well in anything > other than limited domains. That is why so-called "invariant" or > "hierarchical memory" representations are so valuable. This is indicated in > writings of Jeff Hawkins, Thomas Serre ("Learning a Dictionary of > Shape-Components in Visual Cortex: Comparison with Neurons, Humans and > Machines", by Thomas Serre, the google-able article I have cited so many > times), and many others. Such hierarchical representations achieve their > flexibility though a composition/generalization hierarchy which presumably > maps easily into NARS. > > Another key problem in AI is context sensitivity. A hierarchical > representation scheme that is capable of computing measures of similarity, > fit, and implications throughout multiple levels in such a hierarchical > representation scheme of multiple aspects of a situation in real time can be > capable of sophisticated real time context sensitivity. In fact, the > ability to perform relative extensive real time matching and implication > across multiple levels of compositional and generalization hierarchies has > been a key feature of the types of systems I have been thinking of for > years. > > That is one of the major reasons why I have argued for "BREAKING THE SMALL > HARDWARE MINDSET." > > I understand NARS's inheritance (or categorizations) as being equivalent two > both of what I have considered two of the major dimensions in an AGI's self > organizing memory, (1) generalization/similarity and (2) composition. I > was, however, aware, that down in the compositional (comp) hierarchy can be > viewed as up in the generalization (gen) hierarchy, since the set of things > having one or more properties or elements of a composition can be viewed as > a generalization of that composition (i.e., the generalization covering the > category of things having that one or more properties or elements). > > Although I understand there is an importance equivalence between down in the > comp hierarchical and up in the gen hierarchy, and that the two could be > viewed as one hierarchy, I have preferred to think of them as different > hierarchies, because the type of gens one gets by going up in the gen > hierarchy tend to be different than the type of gens one gets by going down > in the comp hierarchy. > > Each possible set in the powerset (the set of all subsets) of elements > (eles), relationships (rels), attributes (atts) and contextual patterns > (contextual pats) could be considered as possible generalizations. I have > assumed, as does Goertzel's Novamente, that there is a competitive ecosystem > for representational resources, in which only the fittest pats and gens -- > as determined by some measure of usefulness to the system -- survive. There > are several major uses of gens, such as aiding in perception, providing > inheritance of significant implication, providing appropriate level of > representation for learning, and providing invariant representation in > higher level comps. Although temporary gens will be generated at a > relatively high frequency, somewhat like the inductive implications in NARS, > the number of gens that survive and get incorporated into a lot of comps and > episodic reps, will be an infinitesimal fraction of the powerset of eles, > rels, atts, and contextual features stored in the system. Pats in the up > direction in the Gen hierarchy will tend to be ones that have been selected > for the usefulness as generalizations. They will often have reasonable > number of features that correspond to that of their species node, but with > some of them more broadly defined. The gens found by going down in the comp > hierarchy are ones that have been selected for their representational value > in a comp, and many of them would not normally be that valuable as what we > normally think of as generalizations. > > In the type of system I have been thinking of I have assumed there will be > substantially less multiple inheritance in the up direction in the gen > hierarchy than in the down direction in the comp hierarchy (in which there > would be potential inheritance from every ele, rel, att, and contextual > feature of in a comp's descendant nodes at multiple levels in the comp > hierarchy below it. Thus, for spreading activation control purposes, I > think it is valuable to distinguish between generalization and compositional > hierarchies, although I understand they have an important equivalence that > should not be ignored. > > I wonder if NARS makes such a distinction. > > These are only initial thoughts. I hope to become part of a team that gets > an early world-knowledge computing AGI up and running. Perhaps when I do > feedback from reality will change my mind. > > I would welcome comments, not only from Mark, but also from other readers. > > > Edward W. Porter > Porter & Associates > 24 String Bridge S12 > Exeter, NH 03833 > (617) 494-1722 > Fax (617) 494-1822 > [EMAIL PROTECTED] > > > > -----Original Message----- > From: Mark Waser [mailto:[EMAIL PROTECTED] > Sent: Tuesday, October 09, 2007 9:46 AM > To: [email protected] > Subject: Re: [agi] Do the inference rules of categorical logic make sense? > > > > I don't believe that this is the case at all. NARS correctly > > handles > > cases where entities co-occur or where one entity implies another only due > > to other entities/factors. "Is an ancestor of" and "is a descendant of" > > has nothing to do with this. > > Ack! Let me rephrase. Despite the fact that Pei always uses the words of > inheritance (and is technically correct), what he means is quite different > from what most people assume that he means. You are stuck on the "common" > meanings of the terms "is an ancestor of" and "is a descendant of" and it's > impeding your understanding. > > > ----- > This list is sponsored by AGIRI: http://www.agiri.org/email > To unsubscribe or change your options, please go to: > http://v2.listbox.com/member/?&; ________________________________ > This list is sponsored by AGIRI: http://www.agiri.org/email > To unsubscribe or change your options, please go to: > http://v2.listbox.com/member/?&; ________________________________ > This list is sponsored by AGIRI: http://www.agiri.org/email > To unsubscribe or change your options, please go to: > http://v2.listbox.com/member/?&; ________________________________ > This list is sponsored by AGIRI: http://www.agiri.org/email > To unsubscribe or change your options, please go to: > http://v2.listbox.com/member/?&; ________________________________ > This list is sponsored by AGIRI: http://www.agiri.org/email > To unsubscribe or change your options, please go to: > http://v2.listbox.com/member/?&; ________________________________ > This list is sponsored by AGIRI: http://www.agiri.org/email > To unsubscribe or change your options, please go to: > http://v2.listbox.com/member/?&; ________________________________ > This list is sponsored by AGIRI: http://www.agiri.org/email > To unsubscribe or change your options, please go to: > http://v2.listbox.com/member/?&; ________________________________ > This list is sponsored by AGIRI: http://www.agiri.org/email > To unsubscribe or change your options, please go to: > http://v2.listbox.com/member/?&; ________________________________ > This list is sponsored by AGIRI: http://www.agiri.org/email > To unsubscribe or change your options, please go to: > http://v2.listbox.com/member/?&; > ________________________________ > This list is sponsored by AGIRI: http://www.agiri.org/email > To unsubscribe or change your options, please go to: > http://v2.listbox.com/member/?&; ----- This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?member_id=8660244&id_secret=51704647-d23e66
