Link grammar does not have "rules" or "production rules". It is not a production grammar in the 1960's sense of a Chomsky context-free (or context-sensitive) grammar. In terms of linguistics, it is not a head-driven phrase-structure grammar (HPSG). It is a dependency grammar, with bi-directional dependencies. Thus, concepts like "inference" or "deduction" do not apply to link grammar or it's generalizations. There is a left-over fragment of "chaining" but that fragment is indistinguishable from parsing. The form of parsing that it does is "assembly". In terms of code, that means that the URE is incompatible with link-grammar: there are no rules.
One reason I'm excited by link-grammar is that it gets rid of ideas like deduction and inference and chaining, and replaces them by assembly. This is exciting because... well, look at the vast amounts of effort put into PLN and URE. They're problematic, they've always been problematic because of the difficulty of composing rules, and the difficulty of the notation of "input" and "output" when using lambda-calculus style notation, or Hilbert-style natural deduction, or sequent calculus, or Bayesian inference.
Worse, the "rules" are hand-curated, human-crafted carefully-engineered artifacts. I have not seen any coherent, complete proposal by which these rules could be learned by a learning system. I think those proposals are absent because trying to think of ways to learn rules and production systems is complicated.
Link-grammar nukes these concepts, replaces them with the idea of assembling parts, and nukes the lambda-calculus notation in favor of a more generic, more general connector notation. Its a more powerful, more fundamental technique. Things like lambda calc and sequent calc and production rules (and rules in general) are narrow special cases. Things like beta reduction and term re-writing and deduction and inference and reasoning are special cases.
I'll mention the nature of the solution below: link-grammar gets rid of arrows, and replaces them by bi-directional links. One no longer writes A -> B to mean "A implies B" or "A produces B" or "a function that takes A as input and returns output values of type B". Or rather, one is no longer forced to write A->B. One still can, and in certain problem domains, it's handy. It's just that now, one is liberated from a forced sequential flow. There's no forward-chaining, backward-chaining; the constraint satisfaction problem is symmetric with respect to that arrow.
Getting rid of that arrow is what nukes deduction and induction (and replaces them by the more general act of assembling)