Skip to main content

Plotkin, the LGG and the MGU

By
Legend has it that a million years ago Plotkin was talking to his professor Popplestone, who said that unification (finding the most general unifier or the MGU) might have an interesting dual, and that Plotkin should find it. It turns out that the dual *is* interesting and it is known as the Least General Generalisation (LGG). Plotkin apparently described both the LGG for terms, and for clauses. I say apparently because I can't find his paper on-line.

The LGG for clauses is more complicated so we'll get back to it after we look at the LGG of terms. We can see how the MGU is related to the LGG by looking at a couple of examples and the above image. We use the prolog convention that function symbols start with lower case, and variables start with uppercase. The image above is organised as a DAG (Directed Acyclic Graph). DAGs are a very important structure in mathematics since DAGs are lattices.

Essentially what we have done is drawn an (incomplete) Hasse diagram for the lattice defined. Each of the arrows in our diagram represents a possible substitution. For instance 'p(X)' can have X substituted for either 's' or 'g(r,Y)'. Anything that can be reached from a series of downward traversals is "unifiable" and the substitution of the unification is the aggregate composition of substitutions we encountered on the path. So for instance 'p(X)' will unify with 'p(g(r,s))' under the substitution X=g(r,Y) and Y=s. Any two terms which are not connected by a path are not unifiable.

The least general generalisation is also apparent in our picture. Any two terms will have a common parent in the Hasse diagram. The least, (in the sense of distance from the top) parent of any two terms in our diagram is the LGG. So actually the connection between the two is fairly straightforward (using 20/20 hindsight).

Comments

Post a Comment

Popular posts from this blog

Decidable Equality in Agda

By
So I've been playing with typing various things in System-F which previously I had left with auxiliary well-formedness conditions. This includes substitutions and contexts, both of which are interesting to have well typed versions of. Since I've been learning Agda, it seemed sensible to carry out this work in that language, as there is nothing like a problem to help you learn a language. In the course of proving properties, I ran into the age old problem of showing that equivalence is decidable between two objects. In this particular case, I need to be able to show the decidability of equality over types in System F in order to have formation rules for variable contexts. We'd like a context Γ to have (x:A) only if (x:B) does not occur in Γ when (A ≠ B). For us to have statements about whether two types are equal or not, we're going to need to be able to decide if that's true using a terminating procedure. And so we arrive at our story. In Coq, equality is ...
2 comments
Read more

Managing state in Prolog monadically, using DCGs.

By
Prolog is a beautiful language which makes a lot of irritating rudimentary rule application and search easy. I have found it is particularly nice when trying to deal with compilers which involve rule based transformation from a source language L to a target language L'. However, the management of these rules generally requires keeping track of a context, and this context has to be explicitly threaded through the entire application, which involves a lot of irritating and error prone sequence variables. This often leads to your code looking something a bit like this: compile(seq(a,b),(ResultA,ResultB),S0,S2) :- compile(a,ResultA,S0,S1), compile(b,ResultB,S1,S2). While not the worst thing, I've found it irritating and ugly, and I've made a lot of mistakes with incorrectly sequenced variables. It's much easier to see sequence made explicitly textually in the code. While they were not designed for this task, but rather for parsing, DCGs turn out to be a conveni...
Post a Comment
Read more

Judgemental Equality, Homotopy Type Theory and Supercompilation

By
Ok, time for some incredibly preliminary thoughts on judgemental equality and what it means for Supercompilation and what supercompilation in turn means for judgemental equality. The more that I explore equality the more incredibly strange the role it plays in type theory seems to be.  There seems intuitively to be nothing more simple than imagining two things to be identical, and yet it seems to be an endless source of complexity. This story starts with my dissertation, but I'm going to start at the end and work backwards towards the beginning.  Currently I've been reading  Homotopy Type Theory , now of some niche fame.  I put off reading about it for a long time because I generally don't like getting too bogged down in foundational principles or very abstract connections with other branches of mathematics (except perhaps on Sundays).  Being more of an engineer than a mathematician, I like my types to work for me, and dislike it when I do too much work for...
Post a Comment
Read more