Tuple Notation

Consider the set \(S = \{a, b\}\). Now lets take the cross product of itself. \(S \times S = \{(a,a), (a,b), (b,a), (b,b)\} = S^2\). By the same logic \(S^3 = \{(a, a,a), (a, a,b), (a, b,a), (a,b,b), (b,a,a), (b,a,b), (b,b,a), (b,b,b)\}\). This is somewhat strange since \(S^3 = S^2 \times S = S \times S^2\) meaning that these two should also be the same as \(S^3\):

\(\{(a, (a,a)), (a, (a,b)), (a, (b,a)), (a,(b,b)), (b,(a,a)), (b,(a,b)), (b,(b,a)), (b,(b,b))\} = \\ \{((a, a),a), ((a, a),b), ((a, b),a), ((a,b),b), ((b,a),a), ((b,a),b), ((b,b),a), ((b,b),b)\}\)

Let's step back a bit and look at the identity, meaning \(S^1\). We know that the power one is the identity but following the same logic as before we have the following: \(S = \{a, b\} = S^1 = \{(a), (b)\}\). The parantheses do not matter and we are supposed to think of them as just as sequence of values. Let's take a look at the empty sequence: \(S^0 = \{()\}\). Note that this is not the empty set but it is the neutral element for the cartesian product: \(\{()\} \times \{a, b\} = \{((), a), ((), b)\} = \{a, b\}\).

What is going on here? The cartesian product is, as is the normal product, an extension of some kind of addition. In our case addition is concatenation of sequences and every value is a sequence of one element (singleton sequence). This is different to the singleton set which is distinct from the value it contains. \(\{a, b\} \times \{c, d\} = \begin{Bmatrix}a \circ c \ & a \circ d\\b \circ c & b \circ d\end{Bmatrix}\) looks very much like the cross product for vectors and of course, the empty sequence is the neutral element (or identity) of that operation.

So actually we are supposed to take nested tuples as sequences of values that - in our mind - should be flattened. This also means, there is no one-tuple (it is the same as the value) and there ist just one zero-tuple.

By the way: this notation fits nicely as it just maps the cardinality of the sets. So if you just take \(S=2\) you see that all the formulas just magically work out.

Why do we even use this tuple notation? This has to do with something we call coding.

Coding

I will define code the following way: A set is a code if no two different sequences of elements of that look the same when written down consecutively (this is a layman's definition that shall suffice here). Take the set \(B = \{11, 100, 00, 001\}\). Both these sequences look the same when written consecutively: \((001,100)=(00,11,00)=001100\). Whether a set is a code is a special instance of the post correspondence problem. I leave it to the reader to define the mapping.

If the carrier set is a code, we do not need to use the tuple syntax. For the following reasons a set might trivially be a code: All symbols have the same length. There are no overlaps (no non-empty prefix of one symbol is a suffix of another) between the symbols.

So for the first set \(S\) we do not need to use the tuple notation as it is always quite clear what is meant: \(S^2 = \{aa,ab,ba,bb\}\). Using this notation we now need a new symbol for the empty sequence and we usually use \(\varepsilon\) (epsilon). Note that concatenation follows these axioms: \(a\circ\varepsilon = \varepsilon\circ a = a\) (neutral element) and \((ab)c=a(bc)\) (associativity). The concatenation operation forms a monoid under the carrier set (a semigroup with identity).

Tuple Types in Programming

Where does this confusion stem from? I think large part of it is that most programming languages do not properly support tuples in their type system in the way we want to use them in computer science theory. Programming langues do have tuple types but they are more rigid than what we want to use.

Let us look at python first:

# Python does have a native tuple type
>>> (3, 6)
(3, 6)
# We have an empty unit sequence
>>> ()
()
# Using the correct syntax we can nest this arbitrarily deep
>>> (((((),),),),)
(((((),),),),)
# We can construct one-tuples which are distinct from their value
>>> (3,) == 3
False
# The standard library has a sequence-product operation
>>> from itertools import product
# the cartesian product with itself works as we expect
>>> list(product([1,2,3], repeat=2))
[(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)]
# Having a zero-product returns the sequence that contains unit, which we expect
>>> list(product([1,2,3], repeat=0))
[()]
# Somewhat consistently, one-product returns the sequence of one-tuples but as seen above, this is not eqal to the original sequence
>>> list(product([1,2,3], repeat=1))
[(1,), (2,), (3,)]
# Also we do not have associativity
>>> list(product([1,2],product([3,4],[5,6])))
[(1, (3, 5)), (1, (3, 6)), (1, (4, 5)), (1, (4, 6)), (2, (3, 5)), (2, (3, 6)), (2, (4, 5)), (2, (4, 6))]
>>> list(product(product([1,2],[3,4]),[5,6]))
[((1, 3), 5), ((1, 3), 6), ((1, 4), 5), ((1, 4), 6), ((2, 3), 5), ((2, 3), 6), ((2, 4), 5), ((2, 4), 6)]

I also want to take a look at Haskell which is also interesting.

-- Haskell has native tuple types
> :t (3,5)
(3,5) :: (Num a, Num b) => (a, b)
-- But there is no singleton tuple type
> :t (5)
(5) :: Num p => p
> (5) == 5
True
-- We have the unit type that is inhabited with the single instance unit which we also can not nest
> :t ()
() :: ()
> :t ((((()))))
((((())))) :: ()
-- Checking for associativity is a type error as both nestings are distinct types and can not be compared via equals
Prelude> (1,(2,3)) == ((1,2),3)
<interactive>:6:2: error:
Prelude> :t (1,(2,3))
(1,(2,3)) :: (Num a1, Num a2, Num b) => (a1, (a2, b))
Prelude> :t ((1,2),3)
((1,2),3) :: (Num a, Num b1, Num b2) => ((a, b1), b2)

The type system does not allow us to even construct a type-safe product function as it is in python as the tuple size of the result sequence is dependent on the value of the argument (something like a -> Int:Val -> (a,)*Val). This would need dependent types. There is a generic function for the cartesian product with the type [a] -> [b] -> [(a, b)] which, as we have seen from the interactive session, would not obey the associativity law either.

Conclusion and Context

We have seen that the tuple notation is just that: notation. It is not used for structure, only for disambiguity. If you want structure, you need to use uninterpreted functions \(t(1,t(2,3)) \neq t(t(1,2),3)\) that are not flattened. But programming languages and the type systems do see a structural difference in the tuple types. The 3-product in Haskell could have, depending on the implementation, three different types: [a] -> [b] -> [c] -> [(a,(b,c))] or [a] -> [b] -> [c] -> [((a,b),c)] or [a] -> [b] -> [c] -> [(a,b,c)] where we would expect all types to be the same.

Why did this article come to pass? Two reasons. During the database lectures we wanted to write up some database state (a set of rows for each table, a row being a sequence of column values) and while one table had three columns (\(\{(1,2,3),(4,5,6)\}\)) one table only had one column and for symmetry \(\{(1),(2),(3)\}\) was brought to paper which I remarked to be at least confusing for students as it might convey something not meant (there being a difference between the singleton tuple and the value).

The second example comes from computer science didactics, a module for the aspiring teachers. In a homework where some lesson had to be designed, one student had the topic of languages and grammars. In an example of a regular language they took this as an example alphabet: \(A=\{la,li,lu\}\). This is fine as the symbol of the alphabet are just syntax and are only of interest (as I've written above), in terms of whether the alphabet is a code or not and therefore which notation is used for concatenation operations. Now on a worksheet the following question was asked: \(lul \in A^*\). And this, I would claim, is a type error. The kleene closure of the alphabet only contains sequences of symbols from the alphabet (including the empty sequence) but "lul" is no sequence of symbols from the alphabet so it can not be of the same type (sequence from the alphabet) as the set (set of sequences from the alphabet). As the original alphabet was a code, all sequence can be written consecutively but it is unclear how "lul" can be deconstructed into elements of the alphabet. A possibiliy is, that the original alphabet was \(\{l,a,i,u\}\) and \(A\) was just a language over that alphabet but that was not written. For teachers, I believe, formalisms in that area are quite important.