Open mschlund opened 11 years ago
I'm very curious about this 'context semiring' (#11): could you provide a reference (or sketch of how it works)? Is that like 3.4.4 of your dissertation?
Yeah exactly! (as you mentioned my thesis: just as a reference for any readers https://d-nb.info/109842865X/34) The definite reference is the "Handbook of Weighted Automata" (M. Droste, W. Kuich, H. Vogler, 2009) where the things are called "contexts" I think (effectively linear grammars imho). What this Newton-stuff then just shows is that you can approximate any CFG with an (infinite) sequence of linear grammars. I think the practical value is limited however :) -- you simply have a different notation for a linear CFG as this "expression-with-numbered-holes". I think it may be a kind of "weaseling-out" of the question of how to actually "solve" the linear system of equations :)
Awesome, thanks. I think that I see the connection to linear CFGs, but my worry is that linear-CFGs aren't closed under the operations needed to solve a system (addition: ok, multiplication, no, star: no). Maybe contexts are closed because they are actually weighted tree automata... which maybe are sufficiently closed to shove through a Gaussian elimination variant?
Yeah maybe I was really sloppy :) In my naive understanding "rational tree expressions"/"contexts" were essentially linear grammars but this is false of course :)
Haha, no worries, still a helpful connection!
The connection to linear CFGs is also in this paper. They appear to have a (rather expensive sounding) workaround for non-commutative semirings which lift the product operation into a Kronecker product... I haven't had time to get into the details yet (there is a talk recording)
Yeah the relationship does not run back and forth :) but rational tree expressions still suffice to represent the solution ... however I don't really know what to do with them then (and also the question of convergence in finite time remains of course). (also btw: the Kronecker-Product-Construction they use for the predicate-abstractions is also described in my thesis for matrices over commutative semirings)
(and also the question of convergence in finite time remains of course).
Oh, is it an open question?
(also btw: the Kronecker-Product-Construction they use for the predicate-abstractions is also described in my thesis for matrices over commutative semirings)
I thought it looked familiar 😉.
I am slowly working my way through your thesis. It's definitely the best resource on semiring Newton. (I am using Newton's method in a chapter of my thesis to execute our programming language Dyna https://arxiv.org/abs/2109.06966)
Requires (these are sketchy thoughts...) :