Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies

Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional Equality Normal form for equality proofs

Propositional Equality, Identity Typesand Reversible Rewriting Sequences as

Homotopies

Ruy de Queiroz(joint work with Anjolina de Oliveira)

Centro de InformaticaUniversidade Federal de Pernambuco (UFPE)

Recife, Brazil

Encontro Brasileiro de Logica (EBL)LNCC, Petropolis (RJ)

Apr 2014Ruy de Queiroz (joint work with Anjolina de Oliveira)Centro de Informatica Universidade Federal de Pernambuco (UFPE) Recife, Brazil

Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies


What is a proof of an equality statement?What is the formal counterpart of a proof of an equality?

In talking about proofs of an equality statement, two dichotomiesarise:

1 definitional equality versus propositional equality

2 intensional equality versus extensional equality

First step on the formalisation of proofs of equality statements: PerMartin-Lof’s Intuitionistic Type Theory (Log Coll ’73, published 1975)with the so-called Identity Type

Ruy de Queiroz (joint work with Anjolina de Oliveira)Centro de Informatica Universidade Federal de Pernambuco (UFPE) Recife, Brazil



Identity TypesIdentity Types - Topological and Categorical Structure

Workshop, Uppsala, November 13–14, 2006

“The identity type, the type of proof objects for thefundamental propositional equality, is one of the mostintriguing constructions of intensional dependent typetheory (also known as Martin-Lof type theory). Itscomplexity became apparent with the Hofmann–Streichergroupoid model of type theory. This model also hinted atsome possible connections between type theory andhomotopy theory and higher categories. Exploration of thisconnection is intended to be the main theme of theworkshop.”




Identity TypesType Theory and Homotopy Theory

Indeed, a whole new research avenue has since 2005 been exploredby people like Vladimir Voevodsky and Steve Awodey in trying tomake a bridge between type theory and homotopy theory, mainly viathe groupoid structure exposed in the Hofmann–Streicher (1994)countermodel to the principle of Uniqueness of Identity Proofs (UIP).This has open the way to, in Awodey’s words,

“a new and surprising connection between Geometry,Algebra, and Logic, which has recently come to light in theform of an interpretation of the constructive type theory ofPer Martin-Lof into homotopy theory, resulting in newexamples of certain algebraic structures which areimportant in topology”. (“Type Theory and Homotopy”,preprint, 2010, http://arxiv.org/abs/1010.1810.)




Identity TypesIdentity Types as Topological Spaces

According to B. van den Berg and R. Garner (“Topological andsimplicial models of identity types”, ACM Transactions onComputational Logic, Jan 2012),

“All of this work can be seen as an elaboration of thefollowing basic idea: that in Martin-Lof type theory, a type Ais analogous to a topological space; elements a,b ∈ A topoints of that space; and elements of an identity typep,q ∈ IdA(a,b) to paths or homotopies p,q : a→ b in A.”.




Identity TypesIdentity Types as Topological Spaces

From the Homotopy type theory collective book (2013):

“In type theory, for every type A there is a (formerlysomewhat mysterious) type IdA of identifications of twoobjects of A; in homotopy type theory, this is just the pathspace AI of all continuous maps I → A from the unitinterval. In this way, a term p : IdA(a,b) represents a pathp : a b in A.”




Identity Types: IterationFrom Propositional to Predicate Logic and Beyond

In the same aforementioned workshop, B. van den Berg in hiscontribution “Types as weak omega-categories” draws attention to thepower of the identity type in the iterating types to form a globular set:

“Fix a type X in a context Γ. Define a globular set as follows:A0 consists of the terms of type X in context Γ,modulodefinitional equality; A1 consists of terms of the typesId(X ; p; q) (in context Γ) for elements p,q in A0, modulodefinitional equality; A2 consists of terms of well-formedtypes Id(Id(X ; p; q); r ; s) (in context Γ) for elements p,q inA0, r , s in A1, modulo definitional equality; etcetera...”




Identity Types: IterationThe homotopy interpretation

Here is how we can see the connections between proofs of equalityand homotopies:

a, b : Ap, q : IdA(a,b)α, β : IdIdA(a,b)(p,q)· · · : IdIdId...(· · · )

Now, consider the following interpretation:Types SpacesTerms Maps

a : A Points a : 1→ Ap : IdA(a,b) Paths p : a⇒ b

α : IdIdA(a,b)(p,q) Homotopies α : p V q




Identity TypesUnivalent Foundations of Mathematics

From Vladimir Voevodsky (IAS, Princeton) “UnivalentFoundations: New Foundations of Mathematics”, Mar 26, 2014:

“There were two main problems with the existingfoundational systems which made them inadequate.Firstly, existing foundations of mathematics werebased on the languages of Predicate Logic andlanguages of this class are too limited.Secondly, existing foundations could not be used todirectly express statements about such objects as, forexample, the ones that my work on 2-theories wasabout.”





From Vladimir Voevodsky (IAS, Princeton) “UnivalentFoundations: New Foundations of Mathematics”, Mar 26, 2014:

“Univalent Foundations, like ZFC-based foundationsand unlike category theory, is a complete foundationalsystem, but it is very different from ZFC. To provide aformat for comparison let me suppose that anyfoundation for mathematics adequate both for humanreasoning and for computer verification should havethe following three components.”





From Vladimir Voevodsky (IAS, Princeton), “UnivalentFoundations: New Foundations of Mathematics”, Mar 26, 2014:

“The first component is a formal deduction system: alanguage and rules of manipulating sentences in thislanguage that are purely formal, such that a record ofsuch manipulations can be verified by a computerprogram.The second component is a structure that provides ameaning to the sentences of this language in terms ofmental objects intuitively comprehensible to humans.The third component is a structure that enableshumans to encode mathematical ideas in terms of theobjects directly associated with the language.”




Propositional EqualityProofs of equality as (rewriting) computational paths

Motivated by looking at equalities in type theory as arising from theexistence of computational paths between two formal objects, ourpurpose here is to offer a different perspective on the role and thepower of the notion of propositional equality as formalised in theso-called Curry–Howard functional interpretation.The main idea, i.e. proofs of equality statements as (reversible)sequences of rewrites, goes back to a paper entitled “Equality inlabelled deductive systems and the functional interpretation ofpropositional equality”, , presented in Dec 1993 at the 9th AmsterdamColloquium, and published in the proceedings in 1994.




Brouwer–Heyting–Kolmogorov InterpretationProofs rather than truth-values

a proof of the proposition: is given by:A ∧ B a proof of A and

a proof of BA ∨ B a proof of A or

a proof of BA→ B a function that turns a proof of A

into a proof of B∀xD.P(x) a function that turns an element a

into a proof of P(a)∃xD.P(x) an element a (witness)

and a proof of P(a)




Brouwer–Heyting–Kolmogorov Interpretation: FormallyCanonical proofs rather than truth-values

a proof of the proposition: has the canonical form of:A ∧ B 〈p,q〉 where p is a proof of A and

q is a proof of BA ∨ B inl(p) where p is a proof of A or

inr(q) where q is a proof of B(‘inl’ and ‘inr’ abbreviate‘into the left/right disjunct’)

A→ B λx .b(x) where b(p) is a proof of Bprovided p is a proof of A

∀xD.P(x) Λx .f (x) where f (a) is a proof of P(a)provided a is an arbitrary individual chosenfrom the domain D

∃xD.P(x) εx .(f (x),a) where a is a witnessfrom the domain D, f (a) is a proof of P(a)




Brouwer–Heyting–Kolmogorov InterpretationWhat is a proof of an equality statement?

a proof of the proposition: is given by:

t1 = t2 ?(Perhaps a sequence of rewritesstarting from t1 and ending in t2?)

What is the logical status of the symbol “=”?What would be a canonical/direct proof of t1 = t2?




Statman’s Direct ComputationsTerms, Equations, Measure

Definition (equations and systems of equations)

Let us consider equations E between individual termsa,b, c, . . ., possibly containing function variables, and finite setsof equations S.

Definition (measure)

A function M from terms to non-negative integers is called ameasure if M(a) ≤ M(b) implies M(c[a/x ]) ≤ M(c[b/x ]), and,whenever x occurs in c, M(a) ≤ M(c[a/x ]).




Statman’s Direct ComputationsKreisel–Tait’s calculus K

Definition (calculus K )

The calculus K of Kreisel and Tait consists of the axioms a = a andthe rule of substituting equals for equals:

(1)E [a/x ] a .

= bE [b/x ]

where a .= b is, ambiguously, a = b and b = a, together with the rules

(2)sa = sba = b

(3)0 = sab = c

(4)a = snab = c

H will be the system consisting only of the rule (1)




Statman’s Direct ComputationsComputations, Direct Computations

Definition (computation)

Computations T in K or H are binary trees of equationoccurrences built up from assumptions and axioms accordingto the rules.

Definition (direct computation)

If M is a measure, we say that a computation T of E from S isM-direct if for each term b occurring in T there is a term coccurring in E or S with M(b) ≤ M(c).




The Functional Interpretation of EqualitySequences of conversions

(λx .(λy .yx)(λw .zw))v .η (λx .(λy .yx)z)v .β (λy .yv)z .β zv(λx .(λy .yx)(λw .zw))v .β (λx .(λw .zw)x)v .η (λx .zx)v .β zv(λx .(λy .yx)(λw .zw))v .β (λx .(λw .zw)x)v .β (λw .zw)v .η zv

There is at least one sequence of conversions from the initial term tothe final term. (In this case we have given three!) Thus, in the formaltheory of λ-calculus, the term (λx .(λy .yx)(λw .zw))v is declared to beequal to zv .Now, some natural questions arise:

1 Are the sequences themselves normal?

2 Are there non-normal sequences?

3 If yes, how are the latter to be identified and (possibly)normalised?

4 What happens if general rules of equality are involved?Ruy de Queiroz (joint work with Anjolina de Oliveira)Centro de Informatica Universidade Federal de Pernambuco (UFPE) Recife, Brazil



The Functional Interpretation of EqualityPropositional equality

Definition (Hindley & Seldin 2008)

P is β-equal or β-convertible to Q (notation P =β Q) iff Q isobtained from P by a finite (perhaps empty) series ofβ-contractions and reversed β-contractions and changes ofbound variables. That is, P =β Q iff there exist P0, . . . ,Pn(n ≥ 0) such thatP0 ≡ P, Pn ≡ Q,(∀i ≤ n − 1)(Pi .1β Pi+1 or Pi+1 .1β Pi or Pi ≡α Pi+1).

NB: equality with an existential force.NB: equality as the reflexive, symmetric and transitive closureof 1-step contraction




The Functional Interpretation of Direct ComputationExistential Force

The same happens with λβη-equality:

Definition 7.5 (λβη-equality) (Hindley & Seldin 2008)The equality-relation determined by the theory λβη iscalled =βη; that is, we define

M =βη N ⇔ λβη ` M = N.

Note again that two terms are λβη-equal if there exists a proofof their equality in the theory of λβη-equality.




The Functional Interpretation of EqualityGentzen’s ND for propositional equality

Remark

In setting up a set of Gentzen’s ND-style rules for equality weneed to account for:

1 definitional versus propositional equality;2 there may be more than one normal proof of a certain

equality statement;3 given a (possibly non-normal) proof, the process of

bringing it to a normal form should be finite and confluent.




The Functional Interpretation of Direct ComputationEquality in Type Theory

Martin-Lof’s Intuitionistic Type Theory :Intensional (1975)Extensional (1982(?), 1984)

Remark (Definitional vs. Propositional Equality)

definitional, i.e. those equalities that are given as rewriterules, orelse originate from general functional principles(e.g. β, η, ξ, µ, ν, etc.);propositional, i.e. the equalities that are supported (orotherwise) by an evidence (a sequence of substitutionsand/or rewrites)




The Functional Interpretation of Direct ComputationDefinitional Equality

Definition (Hindley & Seldin 2008)

(α) λx .M = λy .[y/x ]M (y /∈ FV (M))(β) (λx .M)N = [N/x ]M(η) (λx .Mx) = M (x /∈ FV (M))

(ξ)M = M ′

λx .M = λx .M ′

(µ)M = M ′

NM = NM ′

(ν)M = M ′

MN = M ′N(ρ) M = M

(σ)M = NN = M

(τ)M = N N = P

M = P




The Functional Interpretation of Direct ComputationIntuitionistic Type Theory

→-introduction

[x : A]

f (x) = g(x) : Bλx .f (x) = λx .g(x) : A→ B

(ξ)

→-eliminationx = y : A g : A→ B

gx = gy : B(µ)

x : A g = h : A→ Bgx = hx : B

(ν)

→-reduction a : A[x : A]

b(x) : B(λx .b(x))a = b(a/x) : B (β)

c : A→ Bλx .cx = c : A→ B

(η)

Role of ξ: Bishop’s constructive principles.Role of η: “[In CL] All it says is that every term is equal to anabstraction” [Hindley & Seldin, 1986]




The Functional Interpretation of Direct ComputationsLessons from Curry–Howard and Type Theory

Harmonious combination of logic and λ-calculus;

Proof terms as ‘record of deduction steps’,

Function symbols as first class citizens.

Cp.

∃xP(x)

[P(t)]

CC

with

∃xP(x)

[t : D, f (t) : P(t)]

g(f , t) : C? : C

in the term ‘?’ the variable f gets abstracted from, and this enforces akind of generality to f , even if this is not brought to the ‘logical’ level.




The Functional Interpretation of Direct ComputationsEquality in Martin-Lof’s Intensional Type Theory

A type a : A b : A

Id intA (a,b) type

Id int -formation

a : A

r(a) : Id intA (a,a)

Id int -introductiona = b : A

r(a) : Id intA (a,b)

Id int -introduction

a : A b : A c : Id intA (a,b)

[x :A]d(x):C(x,x,r(x))

[x :A,y :A,z:Id intA (x,y)]

C(x,y,z) type

J(c,d) : C(a,b, c)Id int -elimination

a : A[x : A]

d(x) : C(x , x ,r(x))

[x : A, y : A, z : Id intA (x , y)]

C(x , y , z) typeJ(r(a),d(x)) = d(a/x) : C(a,a,r(a))

Id int -equality




The Functional Interpretation of Direct ComputationsEquality in Martin-Lof’s Extensional Type Theory

A type a : A b : AIdext

A (a,b) typeIdext -formation

a = b : Ar : Idext

A (a,b)Idext -introduction

c : IdextA (a,b)

a = b : AIdext -elimination

c : IdextA (a,b)

c = r : IdextA (a,b)

Idext -equality




The Functional Interpretation of Direct ComputationsThe missing entity

Considering the lessons learned from Type Theory, thejudgement of the form:

a = b : Awhich says that a and b are equal elements from domain D, letus add a function symbol:

a =s b : Awhere one is to read: a is equal to b because of ‘s’ (‘s’ beingthe rewrite reason); ‘s’ is a term denoting a sequence ofequality identifiers (β, η, ξ, etc.), i.e. a composition of rewrites.In other words, ‘s’ is the computational path from a to b.(This formal entity is missing in both of Martin-Lof’sformulations of Identity Types.)




The Functional Interpretation of Direct ComputationsPropositional Equality

Id-introductiona =s b : A

s(a,b) : IdA(a,b)

Id-elimination

m : IdA(a,b)

[a =g b : A]

h(g) : CREWR(m, g.h(g)) : C

Id-reductiona =s b : A

s(a,b) : IdA(a,b)Id-intr

[a =g b : A]

h(g) : CREWR(s(a,b), g.h(g)) : C

Id-elim

.β[a =s b : A]

h(s/g) : CRuy de Queiroz (joint work with Anjolina de Oliveira)Centro de Informatica Universidade Federal de Pernambuco (UFPE) Recife, Brazil



The Functional Interpretation of Direct ComputationsPropositional Equality: A Simple Example of a Proof

By way of example, let us prove

ΠxAΠyA(IdA(x , y)→ IdA(y , x))

[p : IdA(x , y)]

[x =t y : A]

y =σ(t) x : A(σ(t))(y , x) : IdA(y , x)

REWR(p, t(σ(t))(y , x)) : IdA(y , x)

λp.REWR(p, t(σ(t))(y , x)) : IdA(x , y)→ IdA(y , x)

λy .λp.REWR(p, t(σ(t))(y , x)) : ΠyA(IdA(x , y)→ IdA(y , x))

λx .λy .λp.REWR(p, t(σ(t))(y , x)) : ΠxAΠyA(IdA(x , y)→ IdA(y , x))




The Functional Interpretation of Direct ComputationsPropositional Equality: A Weak Version of Function Extensionality

With the formulation of propositional equality that we have justdefined, we can actually prove a weakened version of functionextensionality, namely

A→ Πf A→BΠgA→B(ΠxA.IdB(AP(f , x),AP(g, x))→ IdA→B(f ,g))

which asserts that, if the domain A is nonempty, then in case fand g agree on all points they must be considered to bepropositionally equal elements of type A→ B.




The Functional Interpretation of Direct ComputationsPropositional Equality: The Groupoid Laws

With the formulation of propositional equality that we have justdefined, we can also prove that all elements of an identity typeobey the groupoid laws, namely

1 Associativity2 Existence of an identity element3 Existence of inverses

Also, the groupoid operation, i.e. composition ofpaths/sequences, is actually, partial , meaning that not allelements will be connected via a path. (The groupoidinterpretation refutes the Uniqueness of Identity Proofs)




The Functional Interpretation of Direct ComputationsPropositional Equality: The Uniqueness of Identity Proofs

“We will call UIP (Uniqueness of Identity Proofs) the followingproperty. If a1,a2 are objects of type A then for any proofs pand q of the proposition “a1 equals a2” there is another proofestablishing equality of p and q. (...) Notice that in traditionallogical formalism a principle like UIP cannot even be sensiblyexpressed as proofs cannot be referred to by terms of theobject language and thus are not within the scope ofpropositional equality.” (Hofmann & Streicher 1996)




The Functional Interpretation of Direct ComputationsNormal form for the rewrite reasons

Strategy:

Analyse possibilities of redundancyConstruct a rewriting systemProve termination and confluence





Definition (equation)

An equation in our LNDEQ is of the form:

s =r t : A

where s and t are terms, r is the identifier for the rewrite reason, andA is the type (formula).

Definition (system of equations)

A system of equations S is a set of equations:

{s1 =r1 t1 : A1, . . . , sn =rn tn : An}

where ri is the rewrite reason identifier for the i th equation in S.Ruy de Queiroz (joint work with Anjolina de Oliveira)Centro de Informatica Universidade Federal de Pernambuco (UFPE) Recife, Brazil




Definition (rewrite reason)

Given a system of equations S and an equation s =r t : A, ifS ` s =r t : A, i.e. there is a deduction/computation of theequation starting from the equations in S, then the rewritereason r is built up from:(i) the constants for rewrite reasons: { ρ, σ, τ, β, η, ν, ξ, µ };(ii) the ri ’s;using the substitution operations:(iii) subL;(iv) subR;and the operations for building new rewrite reasons:(v) σ, τ , ξ, µ.





Definition (general rules of equality)

The general rules for equality (reflexivity, symmetry andtransitivity) are defined as follows:

x : Ax =ρ x : A

(reflexivity)

x =t y : Ay =σ(t) x : A

(symmetry)

x =t y : A y =u z : Ax =τ(t ,u) z : A

(transitivity)





Definition (subterm substitution)

The rule of “subterm substitution” is split into two rules:

x =r C[y ] : A y =s u : A′

x =subL(r ,s) C[u] : A

x =r w : A′ C[w ] =s u : AC[x ] =subR(r ,s) u : A

where C[x ] is the context in which the subterm x appears




The Functional Interpretation of Direct ComputationsReductions

Definition (reductions involving ρ and σ)

x =ρ x : Ax =σ(ρ) x : A

.sr x =ρ x : A

x =r y : Ay =σ(r) x : A

x =σ(σ(r)) y : A.ss x =r y : A

Associated rewritings:σ(ρ) .sr ρσ(σ(r)) .ss r





Definition (reductions involving τ )

x=r y :D y=σ(r)x :Dx=τ(r,σ(r))x :D

.tr x =ρ x : D

y=σ(r)x :D x=r y :Dy=τ(σ(r),r)y :D

.tsr y =ρ y : D

u=r v :D v=ρv :Du=τ(r,ρ)v :D

.rrr u =r v : D

u=ρu:D u=r v :Du=τ(ρ,r)v :D

.lrr u =r v : D

Associated equations: τ(r , σ(r)) .tr ρ, τ(σ(r), r) .tsr ρ, τ(r , ρ) .rrr r ,τ(ρ, r) .lrr r .





Definition

βrewr -×-reductionx =r y : A z : B

〈x , z〉 =ξ1(r) 〈y , z〉 : A× B× -intr

FST (〈x , z〉) =µ1(ξ1(r)) FST (〈y , z〉) : A× -elim

.mx2l1 x =r y : A

Associated rewriting:µ1(ξ1(r)) .mx2l1 r





Definition

βrewr -×-reductionx =r x ′ : A y =s z : B〈x , y〉 =ξ∧(r ,s) 〈x ′, z〉 : A× B

× -intr

FST (〈x , y〉) =µ1(ξ∧(r ,s)) FST (〈x ′, z〉) : A× -elim

.mx2l2 x =r x ′ : A

Associated rewriting:µ1(ξ∧(r , s)) .mx2l2 r





Working paper:

Propositional equality, identity types, and directcomputational pathsRuy J.G.B. de Queiroz, Anjolina G. de Oliveirahttp://arxiv.org/abs/1107.1901[v1] Sun, 10 Jul 2011 21:28:26 GMT (33kb)[v2] Thu, 1 Mar 2012 13:27:15 GMT (34kb)[v3] Mon, 5 Aug 2013 20:26:40 GMT (37kb)



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Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies