This talk tries to suggest how computer programming can be conceptually simplified by using abstract mathematics, in particular categorical semantics, so to achieve the 'correctness by construction' paradigm paying no price in term of efficiency. Also, it introduces an alternative point of view on what is a program and how to conceive data structures, namely as computable morphisms between models of a logical theory.

1. Does Programming Need Data Structures?
Correctness by Construction — CORCON 2014
Dr M Benini
Università degli Studi dell’Insubria
marco.benini@uninsubria.it
27th March 2014

2. Just a provocation?
The title of this talk is, of course, provocative. But, for the next 30 minutes,
I will take it seriously.
The aim is to show, through an elementary example, how one can change
the point of view on programming so to achieve a deeper understanding of
what means to compute. This deepening in knowledge has a practical
consequence, which is also the title of this workshop and of the project we
are involved in: correctness by construction.
So, the ultimate purpose of this talk is to address the impact of the project.
In the meanwhile, this talk allows me to introduce the idea behind my
contribution to the project.
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3. Concrete and abstract lists
Usually, lists are defined as the elements of the free algebra over the
signature 〈{E,L},{nil:L,cons:E ×L → L}〉.
And, in the standard practice of traditional programming, they are
represented as follows:
a cell is a record in the computer memory which contains two fields:
the head which is an element in E;
the tail of the list which is a list;
in turn, a list is pointer (memory address) to a cell;
the empty list, nil, becomes the null pointer.
Thus, cons is a procedure that allocates a cell, fills the head with its first
parameter, and the tail with the second one, finally returning the its address.
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4. Concrete manipulation of a list
As an example of program, we consider the concatenate function.
Its specification is: “Given the lists [x1,...,xn] and [y1,...,ym], concatenate
has to return the list [x1,...,xn,y1,...,ym]”.
Usually, it is implemented as
concatenate(x,y) ≡
if x = nil then return y
else q := x
while q = nil do
p := q
q := q → tail
p → tail := y
return x
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5. Correctness
The previous algorithm is correct. In fact, when x = nil, it returns y,
satisfying the specification.
When x = nil, x = [x1,...,xn]. So, at the end of the i-th iteration step,
p = [xi ,...,xn] and q = [xi+1,...,xn], as it is immediate to prove by induction.
Also, the cycle terminates after n iterations, and p = [xn].
But, in the concrete representation of x, p → tail must be nil and the
assignment p → tail := y substitutes nil with y. So x = [x1,...,xn,y1,...,ym],
as required.
The proof sketched above uses in an essential way the concrete
representation of the x list, because the algorithm uses “list surgery”.
It is evident that the algorithm, and, thus, its correctness proof, is hardwired.
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6. A functional derivation
Dropping list surgery, we can use the abstract formalisation of lists directly:
concatenate(x,y) ≡
if x = nil then return y
else return cons (hdx) (concatenate(tlx,y))
where hd and tl return the head and the tail of its argument, respectively.
Of course, this is a functional program, and it is justified by the following
reasoning, which can be immediately converted into a formal correctness
proof by induction on the structure of x:
1. we want that concatenate([x1,...,xn],[y1,...,ym]) = [x1,...,xn,y1,...,ym],
2. as before, if x = nil, the result is just y
3. when x = nil, concatenate([x1,...,xn],[y1,...,ym]) yields the same result
as consx1 (concatenate([x2,...,xn],[y1,...,ym]));
4. in the line above, the recursive application decreases the length of the
first argument, so recursion terminates after n steps.
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7. Recursion versus induction
In the functional implementation of concatenate, we may interpret the
recursive schema as the computational counterpart of an inductive schema.
It is immediate to see that such an inductive schema becomes the skeleton
of the correctness proof. So, the functional program “carries” with itself a
proof of correctness, in some sense.
Usually, the functional implementation of concatenate is regarded as
inefficient because it recursively constructs a number of intermediate lists
before yielding the final results. Often, this is said to be the inevitable effect
of dropping list surgery.
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8. Abstracting over lists
We formalised a list [x1,...,xm] as consx1 (consx2 (...(consxm nil)...)). We
can use a slightly different representation1:
λn,c. c x1 (c x2 (...(c xm n)...)) .
The key idea is to abstract over the structure of the data type, making it
part of the representation of the datum.
Alternatively, we can interpret this representation A as the abstract datum,
and the concrete one, C can be obtained by passing the instances of the
constructors A.
For example, the standard formalisation is obtained by Anilcons.
1As far as I know, the general algorithm to derive such a representation is due to
Böhm and Berarducci, and it can be traced back to Church
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9. Interpreting abstract lists
An abstract list can be thought of as representing a term in the first-order
logical language with the equality relation symbol, and the signature of the
data type of lists.
The λ-term standing for the abstract list realises the mapping from the
logical term — the list, the body of the abstraction — into some model,
which is specified when we apply to the λ-term the way to interpret the
function symbols.
If we fix this point of view, we can write a “correct by construction”
implementation of concatenate:
concatenate ≡ λx,y,n,c. x (y nc) c .
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10. Correctness by construction I
concatenate ≡ λx,y,n,c. x (y nc) c .
It is worth explaining the construction of this program:
1. it is a function, which takes two argument x and y;
2. it returns an abstract list, so a λ-term of the form λn,c. L, with L a
logical term in the language of lists;
3. the y abstract list gets interpreted in the same model as the result of
concatenate — and this is rendered by y nc;
4. the x abstract list gets interpreted in a model which has the same
interpretation for cons, but it interprets nil as the ‘concrete’ y.
We should remark that, in fact, this abstract implementation is, in essence,
the very same algorithm we have shown in the beginning, deprived from the
irrelevant details about the concrete data structure of lists.
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11. Correctness by construction II
concatenate ≡ λx,y,n,c. x (y nc) c .
The above definition is a direct coding of the explanation. In turn, the
explanation can be converted into a correctness proof by observing that
the structure depicted in point (4) is a model for the theory of lists;
there is a mapping that preserves the meaning between the standard term
model and the model above;
this mapping is just the function concatenate.
The idea behind this proof is that the function concatenate, intended as a
program, is nothing but a morphism between models of the same theory.
A non-evident aspect of the explanation of concatenate is that concatenate
correctly operates in any model for the theory of lists.
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12. One program, many meanings
For example, natural numbers, described as the structure generated by zero
and successor, are a model for lists: cons ≡ λe,l. sucl and nil ≡ 0. And
concatenate becomes just the usual addition.
For example, interpreting cons as the Cartesian product and nil as the
terminal object in a category with products, we get another model for lists.
And concatenate becomes just the Cartesian product of two products.
For example, interpreting cons as function application and nil as the identity
function, we get another model for lists. And concatenate becomes function
composition.
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13. Interpretations and computing
A hidden aspect of interpreting a data type in a model is that computational
patterns can be rendered explicitly.
For example, if we take lists of trees as our model for lists, and we define hd
as the list containing the root elements, and tl as the list of their sons, the
abstract structure of a single tree corresponds to the procedure that
sequentially scans the tree breadth-first.
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14. Generalising
Does it work only for lists?
The theory behind the abstract representation for data types has been
developed by Böhm and Berarducci, and it directly applies to all the data
structures that can be formalised as free algebras of terms over a
first-order signature. This holds for most of the elementary structures
which are used in the current practice of programming. In a similar way,
co-inductive data structures can be modelled as well.
For data structures which are not free (co-)algebras, there are still some
open problems, but, to some extent, they can be modelled in the same
spirit. That is, representing data as functions whose parameters describe
the “structure” of the data type.
Does it work in a “real” programming language?
As far as the programming language supports the dynamic creation of
functions, e.g., by providing abstraction, the technique can be
immediately used.
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15. A philosophical remark
The title of this talk was “Does programming really need data structures?”.
Now, we can say that the answer is not immediately positive:
(YES) programming needs data, and data must be structured to be
represented and manipulated by a formal entity like a program;
(NO) programming does not need concrete data structures. In fact, a
program relies only on the structural properties of a data type to perform
its computation: as far as these properties are accessible, for example, as
explicit parameters, we can do without data structures;
(YES) when we conceive a program, we assume to work on data
represented according to some structure. It is possible (and, I claim,
convenient) to make this structure abstract, but a structure is still
present, and it shapes the way the computation is performed;
(NO) the abstract structure we pass to our representation of data is
nothing but an “interpretation” of a (logical) theory into a model. In
fact, we do not need to know how the model is represented, but only how
to express the mapping from the canonical model to the intended “world”
where the computation is assumed to take place.
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16. Conclusions
In the previous slide there is a hidden assumption: that the logical theory
has a “canonical” model which can be transformed into a generic model via
a suitable mapping. This is not true in general. So the presented point of
view can be stretched only when considering logical theories having such a
classifying model — which is the case for free algebras of terms, for example.
In my recent research, I’ve shown a semantics for first-order intuitionistic
logical theories, based on a categorical setting, which has classifying models.
So, every such a theory could, in principle, be regarded as a “data structure”
in the sense of this talk.
My contribution to the CORCON research project will be to investigate
whether semantics like the above one can be effectively used to model data
structures in an programming environment.
Also, the side message of this talk is to show how even the most elementary
aspects of our project may have a non-trivial impact to the current practice
of programming. It is just a question of taking the “right” point of view,
after all. . .
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17. The end
Tramonto, Rodi — © Marco Benini (2012)
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