This is a reference to the "Can programming be liberated from the von Neumann style?" from 1977. It argues for functional programming, making the point that the imperative style is more common for efficiency reasons, as the programming model is close to the computer architecture. It aims to be a general thought framework inviting to step a step back on some notions that have been (hastily?) accepted in the programming world.
It makes the same analogy that Prolog (or logic programming languages in general) have been strongly influenced by the resolution algorithm. In practice that means that if you write a non-trivial program, if performance is not right you'll need to understand the execution model and adapt to it, mainly with the pruning operator (!). So while the promise is to "declare" values and not think about the implementation details, you're railroaded to think in a very specific way.
I personally found that frustrating to find good solutions essentially unworkable because of this, in comparison with either imperative or functional paradigms that are significantly more flexible. As a result, Prolog-style programming feels limited to the small problems for which it is readily a good fit, to be integrated into a general program using a general-purpose language. I may be wrong on this, but of the 50 people that learned Prolog around the same time as me, none kept up with it. Meanwhile, other niche languages like Ocaml, Haskell and Scheme had good success.
Rethinking the language foundation could remove these barriers to give the language a broader user base.
It might be a reference to that 1977 paper in name, but unlike that Backus paper using math to make its point, this reads like a shallow ad for using the Curry language. The central (and only) point is merely an example of rewriting a Prolog predicate for appending lists into Curry but without even the claim of generality. The rewritten Curry functions however trivially fix determinacy and variables to input and output roles when the entire point of logic variables in Prolog is that they're working either when bound to a value upon entering a predicate call, or can get bound to as many values indeterministically as needed until the procedure terminates succesfully (and then even more on backtracking over failed subsequent goals). The paper also glosses over the concept of unification here. I sure hope the referenced detail papers come with more substance.
The paper title doesn't even make sense? So he wants to "liberate" Logic Programming from predicates? Predicate Logic is First-Order Logic ... ie what YeGoblynQueenne says in another comment.
From a mere practical PoV, who is the author even addressing? Prolog, from the get go, was introduced with NLP, planning problems, and similar large combinatorical search spaces in mind and is used for the convenience it brings to these applications. That FP could theoretically be more broadly used is completely besides the point; Prolog's focus is its strength.
The way you write imperative programs in Prolog by exploiting the search order, using cuts, etc. seems clever when you see it in school and do a few assignments for a comparative programming languages class (the only 3 credit CS course I took) but it is painfully awkward if you have to do very much of it.
It isn't. I do most of my programming in Prolog, I write oodles of it daily, and it's not a problem. You learn to think that way easily.
The argument is basically that Prolog is not 100% declarative and that if we jump through a few hoops, and translate it all to functional notation, we can make it "more declarative". But let's instead compare the incomplete declarativeness of Prolog to a fully-imperative, zero-declarative language like Python or C#. We'll find I believe that most programmers are perfectly fine programming completely non-declaratively and don't have any trouble writing very complex programs in it, and that "OMG my language is not purely declarative" is the least of their problems. I hear some old, wizened nerds even manage to program in C where you actually can drop to the hardware level and push bits around registers entirely by hand O.o
> But let's instead compare the incomplete declarativeness of Prolog to a fully-imperative, zero-declarative language like Python or C#.
There's no such thing as "fully-imperative, zero-declarative language" -- at least not one as high level C# or Python -- because declarative/imperative are programming styles, which languages can make more natural but which are used with all (well, higher-level than assembly) languages.
Really interesting to hear!
I was quite hooked to Prolog in a previous life. Then the limiting factor was the tooling, for really practical use.
Could you tell a bit about your Prolog environment?
seconded
I use the SWI-Prolog IDE:
https://www.swi-prolog.org/PceEmacs.html
I suppose it is a bit spartan, but it has a ton of functionality that I find indispensable [1]. For example, when I place my cursor on a variable in the editor it highlights all the variables that unify with it in the same clause, and it will highlight singleton variables in a different colour so you can catch errors caused by typos easily. It is also aware of things like imported predicates, undefined or dynamic predicates, multi-file predicates etc, and will highlight them accordingly. It has some (limited) auto-complete and code-expansion etc. and a status line that gives you very good information about the kind of term you have your cursor on - where it's defined if it's a predicate, its name and arity and how many clauses etc, basically much of the information you can get from querying the program database with various program inspection built-ins. Some of my colleagues use VS Code to write Prolog instead and I keep shaking my head and grumbling about the errors they keep making that they wouldn't if they used the SWI-Prolog IDE instead. Kids, these days. In my youth, we wrote all our Prolog on Notepad! And compiled on Dos!
(nope. never)
SWI also has a graphical debugger, which however I never use:
https://www.swi-prolog.org/pldoc/doc/_SWI_/xpce/prolog/lib/g...
I know some people swear by its name but I prefer the textual debugger. Again this one looks a little spartan :)
More goodies in SWI: cross-referencer:
https://www.swi-prolog.org/gxref.md
And profiler:
1 ?- profile(between(1,10,_)).
=====================================================================
Total time: 0.000 seconds
=====================================================================
Predicate Box Entries = Calls+Redos Time
=====================================================================
between/3 1 = 1+0 0.0%
true.
________________________
[1] You can customise the IDE colours too. There's a dark theme:
https://www.swi-prolog.org/pldoc/man?section=theme
There's some screenshots here:
https://swi-prolog.discourse.group/t/questions-about-ide-the...
Do you use Prolog in Academia or you have moved to Industry?
ha, yet another emacs derivative I didn't know of
thanks for your help
I agree that not being fully declarative is okay. But lots of pure functions are written in languages like Python and C#. “Fully-imperative, zero-declarative” seems like a bit of an exaggeration?
(I’m generally of the opinion that both laziness and backtracking are bad defaults for general-purpose programming, but they’re handy when you need them.)
Serious question, how do you deal with typos in functors? And is your techniques specific to the implementation of Prolog you use? Recently I had a maddening experience chasing this down:
result(World0, move(robot(R), Dir), World) :-
dissoc(World0, at(robot(R), X0), World1),
direction_modifier(Dir, Modifier),
X #= X0+Modifier,
conj(World1, at(robot(R), X), World).
result(World0, drop_rock(robot(R), Place), World) :-
dissoc(World0, capacity(Place, Capacity0), World1),
dissoc(World1, carring_rock(robot(R)), World2),
Capacity #= Capacity0 + 1,
conj(World2, capacity(Place, Capacity), World).
result(World0, pickup_rock(robot(R), Place), World) :-
dissoc(World0, capacity(Place, Capacity0), World1),
Capacity #= Capacity0 - 1,
conj(World1, capacity(Place, Capacity), World2),
conj(World2, carrying_rock(robot(R)), World).
...
...
...
carrying_rock vs carring_rock
Right now this seems to have all the downsides of programming exclusively with "magic strings", and I haven't been able to find any cure for it or even seen this problem discussed elsewhere.
*Edit:*
I even rewrote it for SICStus and downloaded their IDE and taught myself Eclipse just to use their IDE plugin, and found that setting breakpoints didn't help the problem, because naturally due to the fact that the functor is in the predicate signature, the predicate is never stepped into in the first place!
I could linearize the arguments and include them in the body but this destroys the indexing and can put me into "defaulty representation" territory.
so so so so so so so so so much this. I have tried to make prolog a part of various systems on and off for two decades and the ergonomics and basic practical shit like this is why it never works.
the answer is always: do a lot of manual stuff that the language should do for you. I can, but I can't get a team to.
Then don't use Prolog. It's not mandatory.
For the record I never have problems like that and I'm sure I'm not special. Well, not in that way. This all comes under the heading of "learn what works". You have to do that with any language.
Edit: as a slightly less flippant answer (sorry) Prolog doesn't "do a lot of manual stuff that the language should do for you" because the Prolog community doesn't think the language should do those things for you and I agree. Take for instance inheritance and composition, like in object orientation. There's no reason Prolog should do that for you. If it did, it would shoehorn you into a certain programming paradigm that can feel like a straightjacket when you don't need it, but you absolutely need to use it because that's what the "language does for you". Prolog instead gives you the tools to do all the things you need, when you need them. It's more work, for sure, but it's also more freedom. I spent most of my career in the industry working with very rigidly OOP languages and that's one reason why I escaped into academia, where I can program in Prolog (see what I did there?) all day without having to declare a class property if I don't want to. And if I really wanted to, I could go all the way and write something like Logtalk:
Another example of something that Prolog doesn't do for you, and that you don't always need, but can always do yourself if you need it, is typing; like I point out in the sibling comment. Why should Prolog do that for you? Are we going to have the whole argument about strongly typed vs. weakly typed languages all over again? I think not. If you want types in Prolog, you can roll your own. Now try to write, say, C# code without types, when you really don't need the bother.
>> Right now this seems to have all the downsides of programming exclusively with "magic strings", and I haven't been able to find any cure for it or even seen this problem discussed elsewhere.
The cure is to not try to program with "magic strings". You don't need to, and if you really want to, then you should try to understand what exactly it is that you're doing, and do it right.
Specifically, what you call "magic strings" are what we call "functions" in First Order Logic, and that Prolog calls compound terms, like carring_rock(robot(R)) which is in fact two compound terms, nested. Prolog lets you nest compound terms infinitely and if you really want to hurt yourself, one great way to do it is to nest terms everywhere.
The alternative is to understand that when you use compound terms as arguments to predicates (or other terms) you are really _typing_ those arguments. Above, "carring_rock(_)" is the type of the second argument of result/3, and "robot(_)" is the type of the single argument of carring_rock/1. The catch is, of course, that Prolog is not a typed language and it doesn't care if you want to hurt yourself. So if you need to have types like that, then you should write some code to explicitly handle types and do some type checking. For instance, right out the top of my head:
result_type(T):-
T =.. [carring_rock,R]
,robot_type(R)
,!.
result_type(T):-
throw('Unknwon result type':T)
robot_type(T):-
T =.. [robot,R]
, % ... further typing of R
,!.
robot_type(T):-
throw('Unknown robot type':T)
A simpler thing I'd advise, but that's going into style territory, is to keep variable names as short as possible. One reason it's hard to spot the mistake in the code above is that the source code is all cluttered with long variable names and the nesting of compound terms makes it even more cluttered. An IDE with good syntax highlighting can also help. Perversly, I find that there are many people who code in Prolog either without syntax highlighting at all or in IDEs that are not aware of common results of typos, like singleton variables. The SWI-Prolog IDE is good for this and Sweep for Emacs has a good reputation also (although I haven't tried it):
https://eshelyaron.com/sweep.html
Edit: it just occurred to me that the project I'm working on currently, in my post-doc, involves quite a bit of typing using compound terms as arguments, like you do above. I've opted for a program synthesis approach where the predicates I need with typing are automatically generated from a specification where I define the arguments of predicates' types. Doing the same thing by hand is probably my number two recommendation of how not to code in Prolog. Number one is "the dynamic database is evil", but does anyone listen to me? Never.
Edit 2: Covington et al's Coding Guidelines for Prolog make the same point:
5.13 Develop your own ad hoc run-time type and mode checking system.
Many problems during development (especially if the program is large and/or there
are several developers involved) are caused by passing incorrect arguments. Even
if the documentation is there to explain, for each predicate, which arguments are
expected on entry and on successful exit, they can be, and all too often they are,
overlooked or ignored. Moreover, when a “wrong” argument is passed, erratic be-
havior can manifest itself far from where the mistake was made (and of course,
following Murphy’s laws, at the most inconvenient time).
In order to significantly mitigate such problems, do take the time to write your
own predicates for checking the legality of arguments on entry to and on exit from
your procedures. In the production version, the goals you added for these checks
can be compiled away using goal_expansion/2.
>> I could linearize the arguments and include them in the body but this destroys the indexing and can put me into "defaulty representation" territory.
Sorry, I don't know what either of those are: "linearize the arguments" and "defaulty representation".
Prolog only has first-argument indexing normally, although some implementations let you change that and use your own indexing scheme. Is that what you did?
At Uni, we had to implement the same project on different paradigms/languages.
We had to do goddamn Paint on Prolog. Yup.
There already is a pretty major effort around the prolog community to build everything as much as possible around pure, monotonic prolog, and to provide a means to support multiple search strategies depending on the best fit for the problem. CLP libraries are also pretty common and the go-to for representing algebraic expressions relationally and declaratively.
I wouldn't say that the logic or relational way of describing effects is a bad thing either. By design it allows for multiple return values (foo/1, foo/2, ...) you can build higher level predicates that return multiple resources, which is pretty common for many programs. It makes concatenative (compositional) style programming really straightforward, especially for more complex interweaving, which also ends up being quite common. Many prolog implementations also support shift/reset, so that you can easily build things like conditions and restarts, algebraic effects, and/or debugging facilities on top. Prolog is also homoiconic in a unique way compared to lisp, and it's quite nice because the pattern matching is so powerful. Prolog really is one of the best languages I ever learned, I wish it was more popular. I think prolog implementations need a better C FFI interop and a nicer library ecosystem. Trealla has a good C FFI.
I think logic programming is the future, and a lot of these problems with prolog are fixable. If it's not going to be prolog, it'll probably be something like kanren and datalog within a lisp like scheme or clojure(script).
This is a great resource for getting a good feel of prolog: https://www.youtube.com/@ThePowerOfProlog/videos
SWI-Prolog has a robust and very mature foreign interface to C++:
https://www.swi-prolog.org/pldoc/doc_for?object=section(%27p...
Man, lately, I feel like this stuff has been following me around. I'd really like to deep-dive into logic programming and related paradigms. Just recently came across Answer Set Programming[0] (via Potassco's clingo[1]), and it has made me realize just how ignorant I am of the design space that's being explored here.
More personally, I recently spent enough time with first Scheme and then APL that the paradigms clicked for me, and the effect that had on the entirety of my outlook on work was dramatically changed as a result. For whatever reason, I feel like breaking down my ingrained technical paradigms has allowed me to integrate and strengthen my soft skills.
Plus, mind-expanding experiences are just plain fun. Looking for more of that juice!
I strongly recommend checking Souffle programming language. It's a dialect of Datalog that can output bulk CSV data that can be easily imported into other databases (like Duckdb or Excel etc). It creates an extremely intuitive framework for logical programming. I.e. you can visualize logical programming as each relation being a giant table of elements, "or" operation being akin to SQL `union all`, "and" operation being akin to SQL `join`, "not" operation being akin to `outer join ... where joined isnull` etc...
The tragedy of RDF and OWL is that people don’t perceive the connection between logic and databases.
I'm literally using ASP and Clingo to do logic programming for school. And you're telling it became relevant to you in your work??
I use ASP at work! I used it as the core of a powerful code generator: I modeled the type system I wanted to implement, some base operations and derivation rules, and had it synthesize implementations for every possible operator between every possible pair of types. I run clasp and it dumps out thousands of lines of C# implementing a simple symbolic matrix linear algebra library. It's one of the most beautiful things I've made, imo.
Could have been an LSD trip description
No, it's SLD-Resolution :P
"Susie in the Lye with Diamonds"
Picture yourself as the goal of a problem
With cut-driven search and definite clause
Somebody calls you, you answer quite "no"-ly
A girl with Kowalskified eyes...Datalog does not use backtracking, and is getting ever increasingly more popular.
See: - The fastest non-incremental embedded Datalog engine https://github.com/s-arash/ascent - The state-of-the-art non-embedded and non-incremental Datalog engine https://github.com/knowsys/nemo - A python library that contains an embedded incremental Datalog engine https://github.com/brurucy/pydbsp - A Rust library that provides a embedded incremental Datalog engine over property graphs https://github.com/brurucy/materialized-view
Datalog is a nice query language but it is far more limited than prolog or general purpose logic programming.
there is a really interesting space between queries and prolog which includes mundane junk like encoding and rendering and data formatting changes that benefits in evaluation from having less power but maintains the lovely expressibility and analysis that we get from 'real' logic programming.
there's lots of exploring left to do
The Curry language (https://www.curry-language.org) does look interesting. Does anybody have practical experience with it?
while I was trying to track down the license for it[1], I found https://git.ps.informatik.uni-kiel.de/curry/curry2go (also BSD-3) which says "A compiler and run-time system to compile and run Curry programs as Go programs"
1: maybe this is it? it does include a lot of curry named submodules; anyway, BSD-3-Clause https://git.ps.informatik.uni-kiel.de/curry/pakcs/-/blob/v3....
There's an insightful critique of the paper on Reddit: https://www.reddit.com/r/ProgrammingLanguages/comments/1g1su... ...agree that it's weird the paper doesn't mention constraint logic programming, but it's perhaps pointing at it implicitly by saying "Replacing backtracking by complete search strategies"
That's a good critique.
Abstract. Logic programming has a long history. The representative of logic programming in practice, the language Prolog, has been introduced more than 50 years ago. The main features of Prolog are still present today: a Prolog program is a set of predicate definitions executed by resolution steps with a backtracking search strategy. The use of back- tracking was justified by efficiency reasons when Prolog was invented. However, its incompleteness destroys the elegant connection of logic pro- gramming and the underlying Horn clause logic and causes difficulties to teach logic programming. Moreover, the restriction to predicates hinders an adequate modeling of real world problems, which are often functions from input to output data, and leads to unnecessarily inefficient exe- cutions. In this paper we show a way to overcome these problems. By transforming predicates and goals into functions and nested expressions, one can evaluate them with a demand-driven strategy which might re- duce the number of computation steps and avoid infinite search spaces. Replacing backtracking by complete search strategies with new imple- mentation techniques closes the gap between the theory and practice of logic programming. In this way, we can keep the ideas of logic program- ming in future programming systems.
Depth-first backtracking is trivial to implement, and it's trivial to blend into the language. Breadth-first backtracking is much harder to implement, and I'm not sure that it's trivial to build into a language (though this may be a lack of imagination).
Icon had both, but depth-first backtracking was the default and trivial to use, while breadth-first backtracking required using "co-expressions" (co-routines), though at least Icon had a trivial syntax for causing procedure argument expressions to be made into co-expressions. But Icon's approach does not make breadth-first backtracking be a first-class language feature like depth-first backtracking, and this is where my imagination gets stuck. To be fair, I've not truly thought much about this problem.
I didn't read past the abstract, but it sounds like they are just transforming logic-based programs into function-based programs. But: if I wanted functional programming, I wouldn't be writing in Prolog.
What would be interesting, would be to replace depth-first search while remaining in the world of predicates and Horn clauses.
>> What would be interesting, would be to replace depth-first search while remaining in the world of predicates and Horn clauses.
For that you want tabled Prolog, or in other words Prolog executed by SLG-Resolution. The paradigmatic implementation is XSB Prolog:
SWI-Prolog also supports tabling but I think the XSB implementation is more mature.
Naïvely, writing only one logic relation of n parameters is about equivalent to writing n^2 functions (just decide for each parameter whether you give it or not as input). So there clearly is value there.
I say naïvely because on one hand you might not need all versions of the function, but on the other one you can also provide partial values, so it’s not either input or output.
functional logic programming is not equivalent to functional programming.
isn't backtracking a complete search strategy?
Depth first search is not complete if branches can be infinitely deep. Therefore if you're in the wrong infinite branch the search will never finish.
Breadth first search is complete even if the branches are infinitely deep. In the sense that, if there is a solution it will find it eventually.
Hrm. I guess the converse applies if nodes can have infinite children. That said, even if your tree is infinitely wide and deep, we're only dealing with countable children, right? Thus a complete traversal has to exist, right?
For example, each node has unique path to root, so write <n1, n2, ..., nk> where each ni is the sibling ordinal of the node at depth i in that path, i.e. it's the ni-th sibling of the n(i-1)st node. Raising each of these to the ith prime and taking a product gives each node a unique integer label. Traverse nodes in label order and voilà?
However, that all assumes we know the tree beforehand, which doesn't make sense for generic call trees. Do we just smash headfirst into Rice on this when trying to traverse in complete generality?
No breadth first search is still complete given an infinite branching factor (i.e. a node with infinite children). "Completeness" is not about finishing in finite time, it also applies to completing in infinite time.
Breadth first search would visit every node breadth first, so given infinite time, the solution would eventually be visited.
Meanwhile, say a branch had a cycle in it, even given infinite time, a naive depth first search would be trapped there, and the solution would never be found.
Suppose you have a node with two children A and B, each of which has infinitely many children. If you performed an ordinary BFS, you could get trapped in A's children forever, before ever reaching any of B's children.
Or, suppose that a node has infinitely many children, but the first child has its own child. A BFS would get stuck going through all the first-level children and never reach the second-level child.
A BFS-like approach could work for completeness, but you'd have to put lower-level children on the same footing as newly-discovered higher-level children. E.g., by breaking up each list of children into additional nodes so that it has branching factor 2 (and possibly infinite depth).
Countable infinity does not work like that: two countable infinities are not more than one countable infinity. I think it falls into the "not even wrong" category of statements.
The Wikipedia article is fairly useful: https://en.wikipedia.org/wiki/Countable_set
Yes, if you put two (or three, or countably many) countable sets together, you obtain a set that is also countable. The problem is, we want to explicitly describe a bijection between the combined set and the natural numbers, so that each element is visited at some time. Constructing such a bijection between the natural numbers and a countably-infinite tree is perfectly possible, but it's less trivial than just DFS or BFS.
If we're throwing around Wikipedia articles, I'd suggest a look at https://en.wikipedia.org/wiki/Order_type. Even if your set is countable, it's possible to iterate through its elements so that some are never reached, not after any length of time.
For instance, suppose I say, "I'm going to search through all positive odd numbers in order, then I'm going to search through all positive even numbers in order." (This has order type ω⋅2.) Then I'll never ever reach the number 2, since I'll be counting through odd numbers forever.
That's why it's important to order the elements in your search strategy so that each one is reached in a finite time. (This corresponds to having order type ω, the order type of the natural numbers.)
> "Completeness" is not about finishing in finite time, it also applies to completing in infinite time.
Can you point to a book or article where the definition of completeness allows infinite time? Every time I have encountered it, it is defined as finding a solution if there is one in finite time.
> No breadth first search is still complete given an infinite branching factor (i.e. a node with infinite children).
In my understanding, DFS is complete for finite depth tree and BFS is complete for finite branching trees, but neither is complete for infinitely branching infinitely deep trees.
You would need an algorithm that iteratively deepens while exploring more children to be complete for the infinite x infinite trees. This is possible, but it is a little tricky to explain.
For a proof that BFS is not complete if it must find any particular node in finite time: Imagine there is a tree starting with node A that has children B_n for all n and each B_n has a single child C_n. BFS searching for C_1 would have to explore all of B_n before it could find it so it would take infinite time before BFS would find C_1.
In practice, though, with BFS you'd run out of memory instead of never finding a solution.
Also, there shouldn't be many situations where you'd be able to produce infinite branches in a prolog program. Recursions must have a base case, just like in any other language.
This has to do with the ordering of search: searching a proof tree (an SLD tree, in SLD-Resolution) with DFS, as in Prolog, can get stuck when there are cycles in the tree. That's especially the case with left-recursion. The article gives an example of a left-recursive program that loops if you execute it with Prolog, but note that it doesn't loop if you change the order of the clauses.
This version of the program, taken from the article, loops (I mean it enters an infinite recursion):
last([_H|T],E) :- last(T,E).
last([E],E).
?- last_(Ls,3).
% Loops
last([E],E).
last([_H|T],E) :- last(T,E).
Ls = [3] ;
Ls = [_,3] ;
Ls = [_,_,3] ;
Ls = [_,_,_,3] ;
Ls = [_,_,_,_,3] ;
Ls = [_,_,_,_,_,3] .
% And so on forever
IIRC Markus Triska showed a trick (with a nickname i forgot) to constrain the search space by embedded a variable length into the top level goal.
I think what you mean is that he adds an argument that counts the times a goal is resolved with, thus limiting the depth of resolution? That works, but you need to give a magic number as a resolution depth limit, and if the number is too small then your program fails to find a proof that it normally should be able to find. It's not a perfect solution.
Yes, well not so much a constant value. He added an unbound variable and it was enough to alter the search. Indeed it's still more or a trick, but it got me interested if there were other more fundamental ideas beyond that.
Reminds me that Warren made a talk about prolog term domains to study resolution over infinite branches.
That phrase was badly written.
Backtracking is a complete search of the problem-space.
What is incomplete is the Horn-SAT problem space, which is a subset of SAT, that can be solved in polynomial time, and is what Prolog is based on.
A complete logic system would have to solve SAT, which is NP-complete.
At least that's what I understood they meant by that.
Yeah, it's confusing. The article is referring to the incompleteness of Prolog implemented using Depth First Search. That's what the author means by "backtracking". I know this because I know "backtracking" is used in the logic programming community to stand for DFS, but if you don't know the jargon you'd be right to be confused. You can kind of, er, glean, that meaning in the article if you see how they refer to a "fixed" search strategy, and also notice that "backtracking" is not normally a search strategy since it can't search on its own. "Backtracking" is really "DFS with backtracking".
The article is pointing out that Prolog with backtracking DFS is incomplete with respect to the completeness of SLD-Resolution. To clarify, SLD-Resolution is complete for refutation, or with subsumption. Prolog is an implementation of SLD-Resolution using DFS with backtracking. DFS is incomplete in the sense that it gets stuck in infinite loops when an SLD tree (the structure searched by DFS in Prolog) has cycles, especially left-recursive cycles. The article gives an example of a program that loops forever when executed with DFS with backtracking, in ordinary Prolog.
SLD-Resolution's completeness does not violate the Church-Turing thesis, so it's semi-decidable: SLD-trees may have infinite branches. To be honest I don't know about the equivalence with Horn-SAT, but Resolution, restricted to definite clauses, i.e. SLD-Resolution, is complete (by refutation and subsumption, as I say above, and respecting some structural constraints to do with the sharing of variables in heads and bodies of clauses). We got several different proofs of its completeness so I think we can trust it's true.
Edit: where does this knowledge about Horn-Sat come from? Do you have references? Gimme gimme gimme.
Reminded me of the “rules and schemes” concept I was thinking of about 10 years ago that would separate “pure logical” rules from the stuff it takes to turn them into a program that runs by either forward chaining (production rules) or backwards chaining (like prolog). I got so far as writing a macro compiler that would let you write a base set of rules and rewrite them for different purposes such as transforming document A into document B or going in the opposite direction.
I like how they let you write functions to control the search order because boy that is essential.
https://icfp24.sigplan.org/home/minikanren-2024#event-overvi... is related. There has been work on turning slow bidirectional code into faster functional code (in 2023, reachable from this url).
Is this about the problem that Prolog tends to ping pong infinitely between leafs? I wrote a fully declarative solitaire game solver and remember this being a big issue forcing me to memorize past states of the backtracking and basically exclude them by another predicate. This is obviously slow. I thought, why not at least have a plugin to avoid trivial cases where backtracking gets stuck switching between A and B. Or a plugin for a stochastic traversal of the solution tree.
There are. Tabling (available in most mature implementations) helps when recalculation of the same states is a problem. Meanwhile, custom search strategy is always an option to implement directly in Prolog. You'll see this in many Advent of Code solutions in Prolog when it is applied to path finding puzzles, in which depth first search is rarely a workable solution.
I'm not an expert, and I've only just skimmed the paper, but it seems to me that this is a kind of partial evaluation for Prolog, eh? (Perhaps "partial resolution" is more technically correct?)
I think some Prolog systems do something like this already as an optimization, but I could be totally off on that.
[Note I'm sick and tired; literally. I may not be firing on all cylinders in the following.]
The article is fudging things with its use of "backtracking" as a stand-in for backtracking Depth First Search (DFS)- the latter is the search strategy that is "fixed" in Prolog. And it's not really fixed. Prolog programs can also be executed by "tabling" a.k.a. SLG-Resolution, which basically replaces DFS with Breadth-First Search modulo momoization. Tabling avoids an important source of "incompleteness" in Prolog, that of non-terminating left-recursions.
To clarify, that is what makes Prolog "incomplete": that executing Prolog programs by DFS makes Prolog loop infinitely when encountering some left-recursions. The article gives the example of a last/2 predicate:
last([_H|T],E) :- last(T,E).
last([E],E).
last([E],E).
last([_H|T],E) :- last(T,E).
?- last_(Ls,3).
Ls = [3] ;
Ls = [_,3] ;
Ls = [_,_,3] ;
Ls = [_,_,_,3] ;
Ls = [_,_,_,_,3] ;
Ls = [_,_,_,_,_,3] .
The problem with that argument, which is as old as Prolog and possibly even older than that, is that it's an argument from taste, or, indeed, "style", to use the article's terminology. More specifically, for every non-terminating Prolog program that can be written, we can most likely write a Prolog program that does terminate, and that is (success-set) equivalent to the non-terminating program, by keeping in mind the structure of the search space for proofs traversed by ordinary Prolog with DFS, or tabling. And possibly with a few cuts here and there (oh, the humanity!). The article is simply arguing that it is "more declarative" to not have to do that. But, why is "more declarative" better? It's style all the way down.
As to the second axis of the argument, the horror of predicates that do not neatly map to "many" real world problems that are better represented as functions, asking whether we can liberate logic programming from predicates is like asking whether we can liberate the First Order Predicate Calculus from predicates, or, equivalently, make ice cream without the ice or the cream. Sure we can. The question is again: why? I don't see a clear justification for that. In particular, setting implementation details aside (as the article does on this point) SLD-Resolution is already sound and refutation-complete, or complete with subsumption. That is the best that can ever be achieved, and the article doesn't seem to claim anything else (or the ghosts of Alonzo Church and Alan Turing would be very, very sad). So this, too, seems to be a matter of taste: functions are more stylish than predicates. M'kay.
In fact, if I may blaspheme a little, this is what Church did with his Lambda calculus: he turned everything into typed functions to extend First Order Logic (FOL) into a higher order. And in the process completely destroyed the simple elegance of FOL. Why?
Incidentally, SQL's recursive queries completely solve the cycle problem by essentially keeping track of the results as a database, which then allows it to prune search results that have already been seen. That's what you get when using UNION in a recursive query, but don't use UNION ALL, as that turns off that pruning effect. This is elegant, but not online.
That's the problem with breadth-first search though: how to make it online?
Conversely, that's the advantage of DFS: it's online because the only state it needs is a search path, and the search path is O(log N), which is small enough to consider online.
Yes, it's called functional programming and it's been around for a while