Free structures pervade Mathematics: free monoids, free groups, the natural numbers are just the most obvious examples. These structures are characterised by a set of generators coupled with a generating process which allows to combine the generators with no constraints, roughly speaking. The generation process naturally yields an induction principle for the free structure: if a property holds for all the generators and it preserves the generating process, then the property holds for every element in the free structure. In this respect, induction can be thought of as an algebraic action of the free structure over some domain. The talk aims at illustrating and exploring induction as an algebraic action, drawing a few, very preliminary consequences mainly through examples.

1. Induction and Free Structures
Dr M Benini
Università degli Studi dell’Insubria
marco.benini@uninsubria.it
February 4th, 2020

2. Free structures
Given
a category C (of algebraic structures),
and U : C → Set the forgetful functor
then the free functor F : Set → C is the left adjoint of U, when it exists.
The free structure on X is thus F(X).
The usual examples (free monoids, free groups, path categories, free lattices,
vector spaces seen as free modules, polynomial rings, term algebras, etc.) all
arise in this way. The set X provides the generators.
Also, the U functor could be arbitrary, and Set could be replaced by any
other category.
Classically, this is a clean, neat, elegant construction. But, hidden in the
notion of adjoint lies the axiom of choice.
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3. Example: free groups
Given a set X, the free group on X is constructed as:
1. Let Y = X ∪{x−1 : x ∈ X}
2. Let Z be the collection of the finite sequences of symbols in Y and let ◦
be the concatenation of strings
3. Let G be the quotient of Z with respect to the following equations:
ε is the empty sequence
a◦a−1 = ε, a−1 ◦a = ε
if a = b then f ◦a◦g = f ◦b ◦g
Then, calling · the lifting of ◦ to the equivalence classes in the previous
quotient, the free group is 〈G;·〉.
It has to be shown that 〈G;·〉 is indeed a group, which is easy.
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4. Example: free Heyting algebras
Given a set X, the free Heyting algebra on X is constructed as:
1. Let Z be the set of formulae in the intuitionistic propositional language
over the variables X
2. Let ∼ be the relation on Z given by a ∼ b if and only if a b and b a
3. Let H be the quotient of Z with respect to ∼
Then, calling ≤ the relation on H defined by [a] ≤ [b] if and only if a b, the
free Heyting algebra is 〈H;≤〉.
It has to be shown that 〈H;≤〉 is indeed a Heyting algebra, which is the core
of the construction behind the completeness theorem for intuitionistic
propositional logic.
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5. Example: natural and integer numbers
Z is the free additive group generated by 1.
N is the free additive monoid generated by the 1.
These definitions use in an essential way the unary representation of
numbers.
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6. Example: the torus
The torus seen as an homotopy space is defined as
b : T, the torus contains a base point
l1,l2 : b =T b, there are two loops from b to b, and they are not the
identity path
h : l1 ◦l2 =b=Tb l2 ◦l1, there is a non-trivial homotopy from l1 ◦l2 to l2 ◦l1,
with ◦ path composition
This example uses homotopy type theory and, in particular, the
interpretation of witnesses in α =A α as paths from α to α.
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7. Constructions
In each of the previous examples, the free object is constructed
from a collection Z of syntactical objects1
in which Z is refined to obtain the free object23
The construction follows a somewhat general pattern. In this respect, we
can regard homotopy type theory as an attempt to regularise this
construction keeping track of the proof-relevant pieces of information.
1In the case of the torus, Z is formed by constructing the judgements about T and its
equality types
2For groups, Heyting algebras, and numbers, the refinement is a quotient
3For the torus, the refinement passes through identification of paths and homotopies in
type theory
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8. Remarks
It is worth noting that
the construction envisages a sort of completeness result, see Heyting
algebras. It suggests that the free object is a classifying model for all the
others objects of the same sort: if a property holds in the free model then
it holds in every model because every other objects can be constructed
from the free one by truth-preserving maps.
the construction envisages a sort of normalisation result. It suggests that
the elements in the free object are either values or expressions, and the
latter reduce to the former via the refinement from Z (see the previous
slide) to the free object.
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9. Completeness
The remark about completeness is obvious for Heyting algebras, leading to
the completeness theorem for intuitionistic propositional logic.
In group theory, the same remark means that every group is the quotient of
a free group. Also, the proof of Cayley’s theorem, every group is a subgroup
of a symmetric group, can be stated as a consequence of the “completeness”
of groups: the usual proof based on actions is performed on the free group
(making it concrete), and we observe that a subgroup of a subgroup of a
group is a subgroup of the group, also showing that the symmetric group of
a group is the quotient of the symmetric group of the free group via the
same quotient.
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10. Normalisation
The fundamental group of the 1-sphere is the free group generated by a
non-trivial loop on some base point. Consider its construction: every
composition of loops from the point is an “expression” and the group forces
it to reduce to an irreducible “value”. Proving that π1(S1) is the additive
group Z is as simple as noting that Z ∼ { n : n ∈ Z} and the second set is the
collection of “values” which turns out to be a group with path
concatenation as its operation.
Proving that π1(T) ∼ Z×Z, the fundamental group of the torus is Z×Z with
pointwise addition follows the same pattern.
This are, in fact, normalisation theorems.
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11. Induction
An induction principle codes the previous construction in a (set of) inference
rule(s). In fact, it does more: it codes the lifting of the previous construction
in the space of propositions. More in general, induction lifts the construction
in some space: propositions, functions, . . .
We may think that the free structure acts on the space, in analogy with the
action of a group on a set.
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12. Induction
Two important remarks are due
different representations of the same free object, although equivalent,
may be very distant
different ways of constructing the free object lead to different induction
principles, which are not necessarily equivalent
As a consequence, the action of induction over the space of propositions,
functions, etc., may be radically different, depending on the way in which
the free object construction is coded.
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13. Example: simple theory of types
Consider Church’s simple theory of types. The strong normalisation theorem
states: For every term t : T, every reduction starting from t admits only
finite extensions.
The obvious attempt is to try to prove the statement by induction on the
derivation of t : T. This attempt (and many variations one could think of)
fail. The problem lies in β-reduction: if λx : A.b : A → B and a : A are
derivable, then (λx : A.b)a β b[a/x] and we have to show that b[a/x] : B is
strongly normalisable.
It is easy to show that b[a/x] : B is derivable, and it is easy to show that
b : B whenever x : A, but b[a/x] may be constructed after b, a, and
λx : A.b, preventing to use the induction hypothesis.
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14. Example: simple theory of types
Tait and Girard proposed a proof which circumvents the problem using a
stronger form of induction.
Their principle scans the free algebra of terms in a different way:
for an atomic type T, its terms are collected;
for a product type A×B, its terms are the pairs (π1 t,π2 t) with t : A×B;
for a function space A → B, its terms are the abstractions λx : A.t x with
x new and t a : B for every a : A.
The corresponding action allows to prove the strong normalisation theorem
in a more or less straightforward way.
The fundamental observation is that the same free object can be presented
in a different but more convenient way, essentially following the inductive
structure of types (and using the η-equivalences).
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15. A modest application
Syntactically, homotopy type theory can be described as the intensional
version of Martin-Löf’s type theory with a hierarchy of universes, plus two
axioms, univalence and function extensionality, plus higher inductive types.
It is folklore that this version of Martin-Löf’s type theory enjoys the strong
normalisation property. In fact, a real proof lacks.
In her PhD thesis, R. Bonacina proved (under my supervision) a strong
normalisation theorem for a large fragment of homotopy type theory, which
comprehends Martin-Löf’s type theory, univalence, function extensionality,
and a large family of higher inductive types.
The result states that considering judgemental equality as a convertibility
relation, every reduction from a judgement cannot be indefinitely extended.
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16. A modest application
The formal system is huge and open (the collection of types is not fixed,
although they share a common pattern). Also, the presence of dependent
types and multiple universes make the usual approaches, e.g., those working
for the simple theory of types, not immediate to extend.
Therefore, our proof is long and technically complex, beyond my ability to
present it in a conference talk.
However, it is based on Tait’s technique, and it is an obvious extension of
the proof about the simple theory of types, wearing the right set of glasses.
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17. A modest application
The first part is to construct the set Z of all derivable judgements, by
combining in all the possible ways the inference rules. As before, the set Z
forms the basis on which we construct a free object.
Then, we quotient this object using judgemental equality: this is the free
object we are interested in. We want to show that in each equivalence class
there is at least one irreducible element, and every reduction from any
element in the class reaches it in a finite number of steps.
The construction of the free object comes with an induction principle: the
usual induction on the structure of formal proofs.
Our first attempts to prove the strong normalisation theorem used this
induction principle. We knew in advance that these attempts were bound to
failure. But the way in which they failed was what we were seeking.
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18. A modest application
The failures showed that, in essence, the bad behaviour of the induction
principle was the same as in the simple theory of types, except that the
whole framework was far more complex.
Then, we started to play with Girard’s variant of the notion of reducibility
candidates. Apart its formal definition, the notion defines the desired action
the induction principle should perform on judgements in order to prove the
normalisation theorem.
The idea is to scan the free object so that the structure of types governs the
order in which the terms are examined. As in the case of the simple theory
of types, the effect is that we can apply the induction hypotheses exactly
when we need them.
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19. A modest application
Precisely,
if a type converts to a variable, its terms are the terms of its possible
instances.
if a type does not convert to a variable, but it converts to a dependent
product Πx : A.B, its terms are the abstractions λx : A.t x with x new
and t a : B[a/x] for every a : A.
otherwise, the type is atomic, and its terms are collected.
The corresponding induction principle allows to prove the normalisation
theorem following Girard’s approach. The proof is long and tedious but not
too difficult.
The induction principle works because the above scan is well founded, which
is a consequence that universes are atomic types (proved via semantics).
Uniqueness of irreducible judgements follows assuming Π to be injective,
which is proved via semantics.
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20. The end
Questions?
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