Term order for complex Abstract Syntax Tree - prolog

I have a question about abstract syntax trees.
In particular I want to sort several trees with an particular
term order.
How can one define a term order for an AST with the following properties:
For allmost all terms, the order behaves exactly
like the standard built-in order of terms.
Deeply nested in the AST there are terms of functor
pos/6 which denote source-positions.
These functors should be ignored in the term order, i.e. all terms of functor pos should compare as equal.
Is is possible to extend the built-in term-order with
a special case for 'pos' ?
What is the most efficient solution, what is the most
readable solution ?
Maybe I should also mention that our ASTs can be quite big, I just benchmarked one AST, which has 217479 proper functors (ignoring the nullary atoms)

I would define a relation ast_without_pos/2 that relates an AST A0 to a term A that is the same as A0 except that all pos/6 subterms are replaced by the same term, say the atom t, and then use the standard term order on these resulting terms. I think this is very readable, and also reasonably efficient.

You could look at Zippers over the AST?
Here is an example over lists:
http://blog.logtalk.org/2013/04/zipper-lists-in-prolog/
If you get the sources for Termite, I think you will find a more general implementation over trees.
http://www.complang.tuwien.ac.at/adrian/termite/Manual/Contents.html
Any help to you?

Related

Prolog predicate from another perspective

I'm working with this code. I'm struggling with the prolog logic paradigm. I have these two questions.
Is there a way to change the path predicate to not only get the "best" path, but to get all the paths that lead to that node/location in order?
How can I make the path predicate more "user friendly" and just insert the two locations I want to calculate, e.g. path(london, manchester)
Is there a way to change the path predicate to not only get the "best" path, but to get all the paths that lead to that node/location in order?
The given code is rather specialized to the Dijkstra algorithm and for finding shortest paths only. I don't see a simple way to modify it to give what you ask. (Initially I thought it might suffice to just remove some cuts, but I haven't had success with that. Comments welcome!)
If you want all paths between two nodes, you don't need the complexity of the Dijkstra algorithm. Here is a mostly reasonable answer defining a predicate that enumerates all paths: https://stackoverflow.com/a/27194523/4391743 ("Mostly" reasonable because the formatting isn't pretty, and because the unvisited/2 predicate would better be expressed using a generic non_member/2 predicate such as the one given here: https://stackoverflow.com/a/10322639/4391743)
There are many other Prolog path-finding questions here, but I haven't yet found one that has the best beginner-friendly answer.
Edit: Forgot to add that the predicate given above computes all paths, but does not yield them "in order". Enumerating all paths ordered by length isn't very easy. Here the best way might be to enumerate all paths and their lengths, use findall/3 to compute a single list of all these paths, and to sort that list by length.
How can I make the path predicate more "user friendly" and just insert the two locations I want to calculate, e.g. path(london, manchester)
If your predicate has too many arguments, just define another one that hides some of them:
path(Start, End) :-
path(Start, End, _Path, _Distance).
This will simply succeed or fail but not tell you anything more:
?- path(london, manchester).
true.
In Prolog it is normal to overload predicate names with different arities like this, which is why references to predicates often include the arity. In the definition above, we define path/2 in terms of path/4.

The list [[a,b]|c] in Prolog

During my exploration of different ways to write down lists, I am intrigued by the following list [[a,b]|c] which appears in the book 'Prolog and Natural Language Analysis' by Pereira and Shieber (page 42 of the digital edition).
At first I thought that such a notation was syntactically incorrect, as it would have had to say [[a,b]|[c]], but after using write_canonical/1 Prolog returned '.'('.'(a,'.'(b,[])),c).
As far as I can see, this corresponds to the following tree structure (although it seems odd to me that structure would simply end with c, without the empty list at the end):
I cannot seem to find the corresponding notation using comma's and brackets though. I thought it would correspond to [[a,b],c] (but this obviously returns a different result with write_canonical/1).
Is there no corresponding notation for [[a,b]|c] or am I looking at it the wrong way?
As others have already indicated, the term [[a,b]|c] is not a list.
You can test this yourself, using the | syntax to write it down:
?- is_list([[a,b]|c]).
false.
You can see from write_canonical/1 that this term is identical to what you have drawn:
| ?- write_canonical([[a,b]|c]).
'.'('.'(a,'.'(b,[])),c)
In addition to what others have said, I am posting an additional answer because I want to explain how you can go about finding the reason of unexpected failures. When starting with Prolog, you will often ask yourself "Why does this query fail?"
One way to find explanations for such issues is to generalize the query, by using logical variables instead of concrete terms.
For example, in the above case, we could write:
?- is_list([[A,b]|c]).
false.
In this case, I have used the logical variable A instead of the atom a, thus significantly generalizing the query. Since the generalized query still fails, some constraint in the remaining part must be responsible for the unexpected failure. We this generalize it further to narrow down the cause. For example:
?- is_list([[A,B]|c]).
false.
Or even further:
?- is_list([[A,B|_]|c]).
false.
And even further:
?- is_list([_|c]).
false.
So here we have it: No term that has the general form '.'(_, c) is a list!
As you rightly observe, this is because such a term is not of the form [_|Ls] where Ls is a list.
NOTE: The declarative debugging approach I apply above works for the monotonic subset of Prolog. Actually, is_list/1 does not belong to that subset, because we have:
?- is_list(Ls).
false.
with the declarative reading "There is no spoon list." So, it turns out, it worked only by coincidence in the case above. However, we could define the intended declarative meaning of is_list/1 in a pure and monotonic way like this, by simply applying the inductive definition of lists:
list([]).
list([_|Ls]) :- list(Ls).
This definition only uses pure and monotonic building blocks and hence is monotonic. For example, the most general query now yields actual lists instead of failing (incorrectly):
?- list(Ls).
Ls = [] ;
Ls = [_6656] ;
Ls = [_6656, _6662] ;
Ls = [_6656, _6662, _6668] .
From pure relations, we expect that queries work in all directions!
I cannot seem to find the corresponding notation using comma's and brackets though.
There is no corresponding notation, since this is technically speaking not a real list.
Prolog has syntacical sugar for lists. A list in Prolog is, like a Lisp list, actually a linked list: every element is either an empty list [], or a node .(H,T) with H the head and T the tail. Lists are not "special" in Prolog in the sense that the intepreter handles them differently than any other term. Of course a lot of Prolog libraries do list processing, and use the convention defined above.
To make complex lists more convenient, syntactical sugar was invented. You can write a node .(H,T) like [H|T] as well. So that means that in your [[a,b]|c]. We have an outer list, which has one node .(H,c) and the ? is another list, with two nodes and an empty list H = .(a,.(b,[])).
Technically speaking I would not consider this a "real" list, since the tail of a list should have either another node ./2, or an empty list.
You can however use this with variables like: [[a,b]|C] in order to unify the tail C further. So here we have some sort of list with [a,b] as first element (so a list containing a list) and with an open tail C. If we later for instance ground C to C = [], then the list is [[a,b]].

Logic programming - Is subset with only one function symbol Turing - complete?

If I have a subset of logic programming which contains only one function symbol, am I able to do everything?
I think that I cannot but I am not sure at all.
A programming language can do anything user wants if it is a Turing-complete language. I was taught that this means it has to be able to execute if..then..else commands, recursion and that natural numbers should be defined.
Any help and opinions would be appreciated!
In classical predicate logic, there is a distinction between the formula level and the term level. Since an n-ary function can be represented as an (n+1)-ary predicate, restricting only the number of function symbols does not lessen the expressivity.
In prolog, there is no difference between the formula and the term level. You might pick an n-ary symbol p and try to encode turing machines or an equivalent notion(e.g. recursive functions) via nestings of p.
From my intution I would assume this is not possible: you can basically describe n-ary trees with variables as leaves, but then you can always unify these trees. This means that every rule head will match during recursive derivations and therefore you are unable to express any case distinction. Still, this is just an informal argument, not a proof.
P.S. you might also be interested in monadic logic, where only unary predicates are allowed. This fragment of first-order logic is decidable.

Algorithm for testing for equivalence of two algebraic expressions

I have to write a program that tests whether two algebraic expressions are equivalent. It should follow MDAS precedence and parenthesis grouping. To solve the problem about precedence, I'm thinking I should implement a Infix to Postfix Notation converter for these expressions. But by doing this, I could not conclude their equivalence.
The program should look like this:
User Input: a*(a+b) = a*a + a*b
Output : Equivalent
For this problem I'm not allowed to use Computer Algebraic Systems or any external libraries. Please don't post the actual code if you have one, I just need an idea to work this problem out.
If you are not allowed to evaluate the expressions, you will have to parse them out into expression trees.
After that, I would get rid of all parenthesis by multiplying/dividing all members so a(b - c) becomes a*b - a*c.
Then convert all expressions back to strings, making sure you have all members alphabetically sorted (a*b, not b*a) ,remove all spaces and compare strings.
That's an idea:
You need to implement building expression tree first because it's a very natural representation of expression.
Then maybe you'll need to simplify it by open brackets and etc. using associative or distributive algebraic properties.
Then you'll have to compare trees. It's not obvious because you need to take care of all branch permutations in commutative operations and etc. E.g. you can sort them (I mean branches) and then compare for equality. Also you need to keep in mind possible renaming of parameters, i.e. a + b need to be equal x + y.

Prolog - what sort of sentences can't be expressed

I was wondering what sort of sentences can't you express in Prolog? I've been researching into logic programming in general and have learned that first-order logic is more expressive compared to definite clause logic (Horn clause) that Prolog is based on. It's a tough subject for me to get my head around.
So, for instance, can the following sentence be expressed:
For all cars, there does not exist at least 1 car without an engine
If so, are there any other sentences that CAN'T be expressed? If not, why?
You can express your sentence straightforward with Prolog using negation (\+).
E.g.:
car(bmw).
car(honda).
...
car(toyota).
engine(bmw, dohv).
engine(toyota, wenkel).
no_car_without_engine:-
\+(
car(Car),
\+(engine(Car, _))
).
Procedure no_car_without_engine/0 will succeed if every car has an engine, and fail otherwise.
The most problematic definitions in Prolog, are those which are left-recursive.
Definitions like
g(X) :- g(A), r(A,X).
are most likely to fail, due to Prolog's search algorithm, which is plain depth-first-search
and will run to infinity and beyond.
The general problem with Horn Clauses however is, that they're defined to have at most one positive element. That said, one can find a clause which is limited to those conditions,
for example:
A ∨ B
As a consequence, facts like ∀ X: cat(X) ∨ dog(X) can't be expressed directly.
There are ways to work around those and there are ways to allow such statements (see below).
Reading material:
These slides (p. 3) give an
example of which sentence you can't build using Prolog.
This work (p. 10) also explains Horn Clauses and their implications and introduces a method to allow 'invalid' Horn Clauses.
Prolog is a programming language, not a natural language interface.
The sentence you show is expressed in such a convoluted way that I had hard time attempting to understand it. Effectively, I must thanks gusbro that took the pain to express it in understandable way. But he entirely glossed over the knowledge representation problems that any programming language pose when applied to natural language, or even simply negation in first order logic. These problems are so urgent that the language selected is often perceived as 'unimportant'.
Relating to programming, Prolog lacks the ability to access in O(1) (constant time) any linear data structure (i.e. arrays). Then a QuickSort, for instance, that requires access to array elements in O(1), can't be implemented in efficient way.
But it's nevertheless a Turing complete language, for what is worth. Then there are no statements that can't be expressed in Prolog.
So you are looking for sentences that can't be expressed in clausal logic that can be expressed in first order logic.
Strictly speaking, there are many, simply because clausal logic is a restriction of FOL. So that's true by definition.
What you can do though is you can rewrite any set of FOL sentences into a logic program that is not equivalent but with good properties. So for example if you want to know if p is a consequence of your theory, you can use equivalently the transformed logic program.
A few notes on the other answers:
Negation in Prolog (\+) is negation as failure and not first order logic negation
Prolog is a programming language, as correctly pointed out, we should be talking about clausal logic instead.
Left recursion is not a problem. You can easily use a different selection rule, or some other inference mechanism.

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