It is worth noting that lots of applications of unification do not reify explicit substitutions in their implementations. You often see introductory type inference articles (usually focused on Hindley-Milner) use algorithm W, which uses a unification procedure that returns substitutions (which must then be composed with other substitutions). All of this is less efficient and arguably more error-prone and more complicated than the destructive unification implied by algorithm J (using the union-find data structure, often implemented invasively into the type representation to imply a union-find forest).
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).
> Despite what another comment says, unification is a valid - and rather convenient - way to implement pattern matching
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.
If we are talking about the context of a compiler, I agree: you should compile pattern matching to decision trees (DAGs), by way of something like Pettersson's algorithm. In my comment, I specified "in an interpreter", as I think it's the natural way to implement an SML-like match construct in that context.
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.
Well, I'd do that with "real" interpreters (using bytecode) too. But on the other hand also _especially_ for "toy" tree walking interpreters, to learn how to build the pattern matching tree -- as unification is needed for the type checker anyways ;).
Although I would start by reading Wadler's version (that I linked in the other post), it's IMHO a bit more accessible than Peterson's https://www.classes.cs.uchicago.edu/archive/2011/spring/2262...
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.
Ah, I thought your username was familiar: you recommended Wadler's approach on a previous HN thread concerning my blog post about Pettersson/Maranget's algorithm (https://news.ycombinator.com/item?id=39241776). Yeah, I admit that Pettersson's telling of the algorithm is a bit turgid (probably why Maranget's paper - which is the same technique as its core, at least to my understanding - is cited more often).
While I'm at (free) books: a great example implementation of unification (in Lisp) is in chapter 11 "Logic Programming" of Peter Norvig's "Paradigms of Artificial Intelligence Programming: Case Studies in Common Lisp".
Type checking (type inference) is the usual application of unification in programming languages (compilers or interpreters).
Pattern matching does not need unification, because, as mentioned in the beginning of post, the "matching" is done one the left side only (no variables to "match" on the right side).
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.
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.
> I think that depends on what you meany by "pattern matching"
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".
Thanks, I don't really know much about Functional Programming. My expertise is in logic programming which btw is a different paradigm. I've heard it said that they're similar and it seems to me the reason was the support for pattern matching in functional languages.
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:
The result of appending is in Zs, but you can see the result of the intermediary unification steps in the other variables echoed in the Prolog REPL (the "listener").
Of course, because this is Prolog you can also use the same predicate to split a list:
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.
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.
> To be honest, I am not sure what is meant by "pattern matching".
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]
where X = a and Xs = [b, c] would be "pattern matching" the head and the tail of the list [a, b, c], but
?- [X|[b, c]] = [a|Xs]
where X = a and Xs = [b, c] would be "real" unification, if [X|[b, c]] would be valid syntax.
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()
is "compiled" to something like
if list != []
then do_something(head(list), tail(list))
else do_something_else()
> 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.
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] -> ...
but languages with pattern matching usually allow you to add side conditions like this:
That sounds right. I guess then "pattern matching" as in Elixir et al. is really just one-sided unification. Which makes me wonder- why don't those languages go all the way?
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).
Pattern matching as is in Erlang, or how it is in most modern languages, was not there at some point and was considered novel and exotic. Now most everyone can do [X, Y | _] = [1, [2], 3, 4] and it's not a seen as something advanced.
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.
YeGoblynQueene, would you be amenable to discussing certain questions of logic programming via e-mail? I am reading about Jean-Yves Girard's "Stellar Resolution" model of computation, and would greatly appreciate a knowledgeable counterpart in the investigation.
> The one-sided pattern matching seems to be a peculiarity of Elixir.
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.
What's not the same as what? Based on your examples it looks like you're saying that unification is not the same as pattern matching. But nobody said it was.
> The one-sided pattern matching seems to be a peculiarity of Elixir.
And Erlang. And Haskell. And OCaml. And F#. And...
All of these "languages mentioned above" have one-sided pattern matching. This was the claim. Not more, not less. Are you disagreeing with this claim? Are you trying to say something about "one-sided pattern matching"? Or are you trying to say something about something other than "one-sided pattern matching"? You are not being clear.
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.
I'm saying that, except for Erlang, the languages mentioned (Haskell, OCaml, F#, etc.) don't support one-sided pattern matching like Elixir does.
Elixir's pattern matching is more like unification. For example:
[a, a, b, a] = [1, 1, 2, 1]
This kind of pattern matching, where a variable can appear multiple times on the left-hand side and must match the corresponding parts of the list, can't be done easily in Haskell or OCaml. In those languages, each variable can only appear once on the left side of a pattern match.
the OP is saying that this style of one sided pattern matching is indeed a peculiarity of Elixir (and Erlang)
>> And Erlang. And Haskell. And OCaml. And F#. And..
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.
[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.
When you're using pattern matching to also create variables, it's just really confusing to be able to create variables with both sides of the expression. It can get confusing enough as it is. Haskell has some extensions to pattern matching that AIUI are generally considered just too "powerful" to generally use, as it makes it very difficult to read, for instance.
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.
The Curry-Howard Correspondence is more of an observation than a theorem, but insofar as it describes the general concept of relating logics and programming languages, what is the CHC for logic programming languages? If strikes me that LP directly instantiates and manipulates logical expressions, in a "self-dual" sense; or, is it that the various "proof procedures" of the logic side map to "proof search" algorithms on the computing side?
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).