Some quasi-mathematical-linguistic musings... Reviewing a bunch of familiar stuff in my mind, I’m trying to take an algebraic view of Word Grammar…. This is presumably equivalent to pregroup grammar under appropriate restrictions but it’s maybe a more linguistics-ish way to look at it…
Consider a set of words interlinked by ordered dependency links (so each link has a head corresponding to the parent, and a tail corresponding to the child). For reasons including those to be described below, it is useful to consider these dependency links as typed. Word Grammar tells us how, given such a set of words and links, to construct a set of additional (ordered) “landmark links” between the words. The rules thereof are as follows… The parent is the “landmark” of the child. In some cases a word may have more than one parent. In this case, the rule is that the landmark is the one that is superordinate to all the other parents. In the rare case that two words are each others’ parents, then either may serve as the landmark. A Before landmark is one where the child is before the parent; an After landmark is one where the child is after the parent. Rules of “landmark transitivity” are: * Subordinate transitivity: If A is a Before/After landmark for B, and B is some kind of landmark for C, then A is a Before/After landmark for C * Sister transitivity: If A is a landmark for B, and A is also a landmark for C, then B is also a landmark for C * Proxy links: For certain special types T of dependency link, if A and B are joined by a link of type T, then if A is a landmark for C, B is also a landmark for C The “head” of a set of words is a root of the digraph of landmark links in that set of words Restricting attention momentarily to the case of phrases with only one head, one way to look at this is: The landmark transitivity rules tell what happens when we carry out operations such as P1 +_T P2 (putting a dependency link between the head of P1 and the head of P2, with P1 to the left and being at the child end of the link), or P1 +_T’ P2 (putting a dependency link between the head of P1 and the head of P2, with P1 to the left and being at the parent end of the link) noting that this operation is not commutative, and also that the dependency link may have a type T which may be important (e.g. due to the existence of proxy links). These operations at on the space of graphs whose nodes are words and whose linked are either typed, ordered dependency links, or ordered landmark links; and for which the landmark links are consistent according to the rules laid out above. The landmark transitivity rules tell where the landmark links go in the combined structures P1 +_T P2 and P1 +_T’ P2, in a way that will maintain the consistency of the rules regarding landmarks It is not hard to see that, according to the rules of landmark transitivity, the free algebra formed by the multiple operations +_T, +_T’ is distributive, associative, and noncommutative There is one hole in the above; we haven’t dealt with cases where a phrase has more than one head, because two words are each others’ parents. The easiest way to look at this formally seems to be to introduce operations +_Tij, where P1 +_Tij P2 builds a dependency link of type T from the i’th head of P1 to the j’th head of P2. We would also have operations of the form P1 +_Tij’ P2 We can then see that the free algebra formed by the multiple operations +_T, +_T’, +_Tij, +_T’ij is distributive, associative, and noncommutative... A next step would be to make all these links (represented here by + operators) probabilistically weighted. But I'm out of time just now so that will be saved for later ;) ... ben -- Ben Goertzel, PhD http://goertzel.org "I am God! I am nothing, I'm play, I am freedom, I am life. I am the boundary, I am the peak." -- Alexander Scriabin -- You received this message because you are subscribed to the Google Groups "opencog" group. To unsubscribe from this group and stop receiving emails from it, send an email to opencog+unsubscr...@googlegroups.com. To post to this group, send email to opencog@googlegroups.com. Visit this group at https://groups.google.com/group/opencog. To view this discussion on the web visit https://groups.google.com/d/msgid/opencog/CACYTDBftWiLCLDE3upxHwXCFeyLRw01TecZvHvSYzouQ5%3D6gdA%40mail.gmail.com. For more options, visit https://groups.google.com/d/optout.
