Unification in Elixir
ericpfahl.com
ericpfahl.com
To this end, good coverage of decent (destructive) unification algorithms can be found in any simple resource on type inference (https://okmij.org/ftp/ML/generalization/sound_eager.ml) or the Warren Abstract Machine (https://github.com/a-yiorgos/wambook/blob/master/wambook.pdf).
Of course, there are times where you would want to reify substitutions as a data structure to compose and apply, but most of the time you just want to immediately apply them in a pervasive way.
Despite what another comment says, unification is a valid - and rather convenient - way to implement pattern matching (as in the style of ML) in an interpreter: much like how you rewrite type variables with types you are unifying them with, you can rewrite the pattern variables to refer to the parts of the (evaluated) value you are matching against (which you then use to extend the environment with when evaluating the right hand side).
The article shows that it is a valid and rather convenient way. That's why it is important to tell people that it is not necessary to use it for the "usual" pattern matching. You do (again: for the "usual" pattern matching without "variables", as in free variables, on the right side) _not_ want to use unification but "compile" the pattern matching.
EDIT: Perhaps I am missing Elixir-specific context and I apologise if this is the case. It was the unification that made me click this post.
But please don't get me wrong: I'm just saying that there exist other, specialised and (well, generally) more performant algorithms for pattern matching and people should have at least heard about them. If somebody decides to use unification for pattern matching, that still could be changed if performance or better error messages or ... matters.
https://github.com/norvig/paip-lisp
There is also an implementation in Python, but I have never tested it, just googled it now: https://github.com/dhconnelly/paip-python/blob/master/prolog...
Is unification unnecessary for pattern matching? I think that depends on what you meany by "pattern matching". In programming cirlces "pattern matching" usually means matching strings. Now, unification is Turing-complete pattern matching, meaning that it goes well beyond e.g. regular expressions that can only match the set of strings recognised by regular automata. For instance, in Prolog, you can unify a variable with an entire Prolog program and call the program at runtime, etc. So if you need "pattern matching" only to match strings then you probably don't need the full power of unification.
If I understand correctly (not an expert) Hindley-Milner type checking, the type checking that uses unification, relies exactly on that ability of unification to match entire programs. Anyway that kind of type checking is indeed pattern matching by unification. So it's not different than pattern matching on strings with regular expressions. The difference is only in what kinds of strings can be matched.
And Erlang. And Haskell. And OCaml. And F#. And...
As far as I understand, only Prolog has unification (out of more-or-less used/known languages). And in languages that require (rather awkward and verbose) workarounds for this functionality, it's hard to see the immediate value of unification.
Thanks, I didn't know that. I took the article to mean that this kind of one-sided pattern matching is unique in Elixir. My misreading.
From what I've seen it's true that most languages that have support for pattern matching have special constructs for it, whereas in Prolog it's baked in to the only data structure in the language, the term, which can be a single variable (a logic variable that can unify with ... anything).
I don't know how useful or not unification would be in a language other than Prolog to be honest. In Prolog it's an integral component of the Resolution-based theorem prover used as an interpreter of Prolog programs: resolution proceeds by unifying Horn goals to the heads of definite clauses, to produce new goals, recursively. More over, when one runs a Prolog "query" the result of the query itself is the substitution of the variables in the query, i.e. the substitution that makes the query true [1]; a substitution derived and propagated by unification.
I like how George W. Lloyd describes the use of unification in Prolog. Apologies in advance for the long-form quote:
The idea that first order logic, or at least substantial subsets of it, could be used as a programming language was revolutionary, because, until 1972, logic had only ever been used as a specification or declarative language in computer science. However, what [R. Kowalski, Predicate Logic as a Programming Language, 1974] shows is that logic has a procedural interpretation, which makes it very effective as a programming language. Briefly, a program clause A <- B1, ..., Bn is regarded as a procedure definition. If <- C1, ..., Ck is a goal, then each Cj is regarded as a procedure call. A program is run by giving it an initial goal. If the current goal is <- C1, ..., Ck, a step in the computation involves unifying some Cj with the head A of a program clause A <- B1, ... Bn and thus reducing the current goal to the goal <-(C1, ..., Cj-1, B1, ..., Bn, Cj+1, ..., Ck)θ, where θ is the unifying substitution. Unification thus becomes a uniform mechanism for parameter passing, data selection and data construction. The computation terminates when the empty goal is produced.
And that's first-order SLD-Resolution, with unification, in a nutshell! From George Lloyd: Foundations of Logic Programming, second Ed. 1993.
There's a copy of the book here:
https://archive.org/details/foundationsoflog0000lloy_o7l2
But it's a limited preview :/
________
[1] ... well, the substitution that makes the Horn goal that we call a "query" false, in reality, because Prolog proves things by refutation but we fudge it a bit.
Plus to a first approximation you can always just swap the binding from the right to the left. A deeper analysis may reveal there's some case you can't do that, there's a lot to pattern matching and I'm not going to try to sit here and construct a case where you can't, but certainly the most common cases (by far) would just require a simple swap, so it's not an onerous requirement. And if you did construct a case where you can't, you could make a very solid software engineering case that you just shouldn't do that and should do it it two parts.
To the extent this doesn't apply to prolog, it would be when prolog is functioning directly as a logic language rather than a programming language, and to be honest I'd still advise against a prolog programmer binding variables on both sides of an expression unless they're forced to for some reason just because if nothing else, you may have to understand your own code a week later. Note this specifically applies to binding variables; I'm not bothered by two already-bound variables being matched against each other, that's just prolog functioning as designed.
JavaScript (maps), Rust, Ruby, Python...
https://developer.mozilla.org/en-US/docs/Web/JavaScript/Refe...
It's not full pattern matching because as far as I know they don't support refutable patterns, but it handles destructuring and binding to variables.
> [a, 2, 2, b] = [1, 2, 2, 3]
[1, 2, 2, 3]
> a
1
> b
3
> [foo, 2, bar] = [3,3,3]
%MatchError{term: [3, 3, 3]}
> foo
%CompileError{description: "undefined function foo/0 (there is no such import)", file: "nofile", line: 1}
> The one-sided pattern matching seems to be a peculiarity of Elixir.
And Erlang. And Haskell. And OCaml. And F#. And...
The one-sided pattern matching seems to be a peculiarity of Elixir. The post above discusses an implementation of full unification without that restriction. To clarify, unification is not generally restricted to one-sided matching only.
Elixir's pattern matching is more like unification. For example:
[a, a, b, a] = [1, 1, 2, 1]
the OP is saying that this style of one sided pattern matching is indeed a peculiarity of Elixir (and Erlang)
[foo, [2, 3, bar], baz, 5] = [1,[2,3,4], 5]
If there's a match, the bindings "foo", "bar", "baz" will be bound to the values on the right hand side.
if not an exception will be thrown
(You are right, though, Erlang and Elixir have the same semantics, both being on the BEAM)
Oh, yes, that's true. "Pattern matching" as that what normally is used in "functional programming languages". This kind of pattern matching does not need unification. The IMHO best (correct, but understandable and with all needed definitions) explanation of it is in chapters 4 "STRUCTURED TYPES AND THE SEMANTICS OF PATTERN-MATCHING" and 5 "EFFICIENT COMPILATION OF PATTERN-MATCHING" of Peyton Jones et al., "The implementation of functional programming languages".
https://simon.peytonjones.org/slpj-book-1987/
> In programming cirlces "pattern matching" usually means matching strings.
Depends on the "circle" ;) https://en.wikipedia.org/wiki/Pattern_matching
I agree that what "pattern matching" means depends on a programmer's background. It's interesting that the wikipedia article that you link to does not mention "unification" at all and only has one reference in Prolog as one of a group of example languages with a "pattern matching construct"; which is not how I would describe the use of the unification algorithm in Prolog.
To be honest, I am not sure what is meant by "pattern matching". It seems to be a loosely defined term, like "Finite State Machine" (as opposed to "Finite State Automaton"). I expect that the book you linked to has a more formal treating, perhaps, but from what I can tell from a quick look the "pattern matching" here is based on types. In Prolog, unification is used to bind values to logic variables, which do not have types. So a variable, X, in Prolog can unify, i.e. "match with" absolutely anything. A "pattern" then can be formed by constructing terms (the only data structure in Prolog) with "holes" like p(X, a, b, Y,[1,2,3|Tail], f(Z)) etc. where capital letters are variables.
One use of this ability is to manipulate singly-linked lists to insert in or remove elements from their tail, where normally only the head of such a list can be accessed directly. What I mean is that in Prolog a list is denoted by the special syntax [H|T] where H is the head of a list (and can be any term, including another list) and T is the tail of the list, another list.
Normally one appends to a list in Prolog using the append/3 predicate:
append([], L, L).
append([H|T], L, [H|R]) :-
append(T, L, R).
[a,b,c,d|T]-T
append_dl(Xs-Ys, Ys-Zs, Xs-Zs).
?- Xs = [a,b,c,d|T1]-T1, Ys = [e,f,g,h|T2]-T2, append_dl(Xs,Ys,Zs).
Xs = [a,b,c,d,e,f,g,h|T2]-[e,f,g,h|T2],
T1 = [e,f,g,h|T2],
Ys = [e,f,g,h|T2]-T2,
Zs = [a,b,c,d,e,f,g,h|T2]-T2.
Of course, because this is Prolog you can also use the same predicate to split a list:
?- Xs = [a,b,c,d|T1]-T1, Zs = [a,b,c,d,e,f,g,h|T2]-T2, append_dl(Xs,Ys,Zs).
Xs = [a,b,c,d,e,f,g,h|T2]-[e,f,g,h|T2],
T1 = [e,f,g,h|T2],
Zs = [a,b,c,d,e,f,g,h|T2]-T2,
Ys = [e,f,g,h|T2]-T2.
Appending or splitting lists like this is preferred to append/3 because it appends the two lists in a single operation so it can be used to economically put a list together in a loop. The same technique is used with Prolog's Definite Clause Grammars where the "difference" in a difference list is the start and end of a string accepted by a grammar, acting as a parser.
https://en.wikipedia.org/wiki/Definite_clause_grammar
Sterling and Shapiro's "The Craft of Prolog" has a nice chapter on "Incomplete Data Structures" using difference lists and patterns with "holes" but I don't think there's a free version online, unfortunately.
I'd say that "pattern matching" is unification, where there are no free variables on the right side and the left side is the "pattern" to match.
As Prolog code, something like
?- [X|Xs] = [a, b, c]
?- [X|[b, c]] = [a|Xs]
Oh, and both unification for type checking (inference) and pattern matching are compile time actions (or done before evaluating the generated functions).
A (simple) pattern match like
match list with
| head :: tail -> do_something(head, tail)
| [] -> do_something_else()
if list != []
then do_something(head(list), tail(list))
else do_something_else()At a guess, that's something to do with the difficulty to implement unification efficiently. The first description of Unification, by J. Alan Robinson (who first described Resolution) had exponential time complexity and it took a while to find efficient unification algorithms. Perhaps the designers of functional languages simply don't need the hassle and can make do with something more efficient, less complete. That would make good, practical sense in languages that do not use Resolution as an interpreter- it's Resolution that absolutely needs unification (there's ground Resolution, without the need for unification, but it's restricted to propositional terms without first-order variables).
I am not sure if it was ever called "unification" in Erlang, it was always called "pattern matching". But the equal sign did retain its name as the "match" operator: https://www.erlang.org/doc/system/expressions.html#the-match.... And that is a legacy of Prolog. I never read X=1 as "make X equal to 1", in my head I always read it as "match X to 1".
At some point, I also wondered: why didn't they go all the way and just bring in Prolog's full unification. They obviously liked Prolog; Erlang was developed in Prolog initially, even. Well, it turns out with unification we'd have to carry the match "hole" around until we might a match. But Erlang has lightweight processes with isolated heaps which communicate via messages, so now we'd need to pass these "holes" around from process to process until say one of the processes somewhere manages to unify the value, and then have it propagate across the system to original query. That breaks isolation. So you see how the two neat features came into a conflict and they picked one over the other, and rightly so it seems.
Plus the restriction that the left hand side must be linear, that is, must not contain any variable more than once. So you usually can't write a pattern match like
match list with
| [x, y, x] -> ...
match list with
| [x, y, x'] if x = x' -> ...No. First-order syntactic unification is not Turing complete. The combination of unification and resolution is Turing complete.
> For instance, in Prolog, you can unify a variable with an entire Prolog program and call the program at runtime, etc.
Lots of languages allow you to do this. Lisp had "eval" ten years before Prolog existed. Unification is not necessary for this.
Prolog is great, unification is great. Some other things are great too.