Researching automated theorem provers

Lately I’ve been messing around with automated theorem provers. Specifically Prover9.

The moment you open it, if you use the GUI mode, you will see 2 big text boxes – assumptions and goals. Naturally, when playing one starts with simple stuff.

We leave the assumptions blank and set P \implies (Q \implies P). as our goal. If we click Start, we see that it proves it immediately.

% -------- Comments from original proof --------
% Proof 1 at 0.00 (+ 0.00) seconds.
% Length of proof is 2.
% Level of proof is 1.
% Maximum clause weight is 0.
% Given clauses 0.

1 P -> (Q -> P) # label(non_clause) # label(goal). [goal].
2 $F. [deny(1)].

============================== end of proof ==========================

That was fun, but what stood out for me is that in mathematics we often take axioms for “granted”, i.e. it’s rarely that we ever think about the logic behind it. But with Prover9, we need to express all of it using logic.

For Prover9 there is a general rule, that variables start with u through z lowercase. Everything else is a term (atom).

For start, let’s define set membership. We say \in denotes set membership, and that’s it. Because that’s all there’s to it, it’s like an atom. So let’s say that member(x, y) denotes that.

Now let’s define our sets A = {1}, B = {1, 2}, and C = {1}.

all x ((x = 1) <-> member(x, A)).
all x ((x = 1 | x = 2) <-> member(x, B)).
all x ((x = 1) <-> member(x, C)).

Now if we try to prove member(1, A), it will succeed. But it will not for member(2, A).

What do we know about the equivalence of 2 sets? They are equal if there doesn’t exist an element that is in the first and not in the second, or that is in the second and not in the first.

In other words:

set_equal(x, y) <-> - (exists z ((member(z, x) & - member(z, y)) | (- member(z, x) & member(z, y)))).

So, Prover9 can prove set_equal(A, C), but not set_equal(A, B).

Another example is that we can define the set of the naturals with just:

member(Zero, Nat).
member(x, Nat) -> member(Suc(x), Nat).

So, it can easily prove:
0: member(zero, Nat)
1: member(Suc(Zero), Nat)
2: member(Suc(Suc(Zero)), Nat), etc.

Relation between strict and non-strict identities

First, we’ll start with some definitions:

== is value check. For a and b to have the same value, we will say V(a, b). Thus a == b <=> V(a, b).

=== is value+type check. For a and b to have the same type, we will say T(a, b). Thus a === b <=> V(a, b) and T(a, b).

Now, to prove a === b => a == b, suppose that a === b. By the definitions, we have as givens V(a, b) and T(a, b). So we can conclude that V(a, b), i.e. a == b.

The contrapositive form is a != b => a !== b, which also holds.

However, note that the converse form a == b => a === b doesn’t necessarily hold. To see why, suppose a == b, that is V(a, b). Now we need to prove that V(a, b) and T(a, b). We have V(a, b) as a given, but that’s not the case for T(a, b), i.e. the types may not match.

So, whenever you see a === b you can safely assume that a == b is also true. The same holds for when you see a != b, you can safely assume that a !== b 🙂

Abstractions with Set Theory

Like in programming, building abstractions in mathematics is of equal importance.

However, the best way to understand something is to get to the bottom of it. Start by working on the foundations level upwards.
So we will start with the most basic object (the unordered collection) and work our way up to defining functions.

Set

A set is an unordered collection of objects. This might sound too abstract but that’s what it is. The objects can be anything. It is usually denoted by comma separating the list of objects and enclosing them using curly braces.

For example, one set of fruits is \{ apple, banana \}.

Since it is an unordered collection we have that \{ apple, banana \} = \{ banana, apple \}.

For membership, we denote apple \in \{ apple, banana \} to say that apple is in that collection.

Tuples

An n-tuple is an ordered collection of n objects. As with sets, the objects can be anything. It is usually denoted by comma separating the list of objects and enclosing them using parenthesis.

We can represent it using unordered collections as follows: Consider the tuple a = (a_1, a_2, ..., a_n). We can use the set A = \{ \{1, \{a_1\}\}, \{2, \{a_2\}\}, ..., \{n, \{a_n\}\} \}. Note that here I’m using natural numbers and I assume them to be unique atoms. Otherwise, if we represented naturals in terms of set theory, there would be issues with this definition. For that, we’d have to use Kuratowski’s definition, but we can skip that for simplicity’s sake.

Now to extract the k-th element of the tuple, we can pick x s.t. \{k, \{x\}\} \in A.

So now we have that (a, b) = (c, d) \leftrightarrow a = b \land c = d, that is two tuples are equal iff their first and second elements respectively are equal. This is what makes tuples ordered.

One example of a tuple is (1 pm, 2 pm, 3 pm) which represents 3 hours of a day sequentially.

Relations

An n-ary relation is just a set of n-tuples with different values. We are mostly interested in binary relations.

One example of such a relation is the “is bigger than”. So we have the following set: \{ (cat, mouse), (mouse, cheese), (cat, cheese) \}.

Functions

Now we can define functions in terms of relations.

To do that we first have to discuss subset. A is a subset of B if all elements of A are found in B (but not necessarily vice-versa). We denote it as such: A \subseteq B. So for example: \{ 1, 2 \} \subseteq \{ 1, 2, 3 \} and \{ 1, 2, 3 \} \subseteq \{ 1, 2, 3 \}. But this doesn’t hold: \{ 1, 2, 3 \} \subseteq \{ 1, 2 \}.

So with this, a function is a mapping from a set A to a set B, or in other words it is a subset of all combinations of ordered pairs whose first element is an element of A and second element is an element of B.

For example, if A = \{ a, b \} \land B = \{ 1, 2, 3 \} then the combinations are: F = \{ (a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3) \}. A function f from A to B is denoted f : A \rightarrow B and is a subset of F: f \subseteq F.

This combination of pairs is called a Cartessian product, and is defined as the set: \{ (a, b) \mid a \in A \land b \in B \}.

We have one more constraint to add to a function, namely that it cannot produce 2 or more different values for a single input. So using either of f(a) = 1 or f(a) = 2 or f(a) = 3 is okay, but not all three of them. That means that we only have to use one of \{ (a, 1), (a, 2), (a, 3) \} from our example set above. The same reasoning goes for b. So f = \{ (a, 1), (b, 2) \} with f(a) = 1 \land f(b) = 2 is one valid example.

We can also express recurrent relations like f(x) = x \cdot f(x-1) for f(0) = 0.

Conclusion

Matrices (tuples of tuples), lists (sets or tuples), graphs* (set of sets), digraphs (set of tuples), trees (graphs) can also be derived similarly.

This is what makes the set a powerful and interesting idea.

*: G=(V,E) with vertices V and edges E. E.g.: G = (\{a, b\}, \{(a, b), (b, b)\})

Soft skills

This month my professional career as a Software Engineer sums up to 10 years (although I’ve been programming longer than that). Here is a summary of what I think is really important for a good career.

Technical skills are important.
As you might have noticed, my blog is mostly technical.

But in this article I’d like to praise soft skills.

Being good with everybody, communicative, adaptable, helpful, truthful, modest, and being a good listener, observer, and learner is just some of the most important skills.

I would also like to quote our creed here at Automattic:

I will never stop learning. I won’t just work on things that are assigned to me. I know there’s no such thing as a status quo. I will build our business sustainably through passionate and loyal customers. I will never pass up an opportunity to help out a colleague, and I’ll remember the days before I knew everything. I am more motivated by impact than money, and I know that Open Source is one of the most powerful ideas of our generation. I will communicate as much as possible, because it’s the oxygen of a distributed company. I am in a marathon, not a sprint, and no matter how far away the goal is, the only way to get there is by putting one foot in front of another every day. Given time, there is no problem that’s insurmountable.

If you try to look 10 years back, you will mostly see memories of events and people and the time spent with them, not how you optimized that loop or that DB query (not that the latter is not important, thus the reason I said “mostly”).

Find inspiration and motivation in your successes, yourself, your family, nature, and other people. Maintain inner peace, and most of the soft skills will come naturally.