Table Of ContentAxioms and Decidability for Type Isomorphism
in the Presence of Sums
Danko Ilik
Inria Saclay – ˆIle de France
1 rue Honore´ d’Estienne d’Orves
Campus de l’E´cole Polytechnique
91120 Palaiseau, France
Email: [email protected]
4
1
0 Abstract—We consider the problem of characterizing isomor- where 1 denotes the singleton type and xi denotes a type
2
phisms of types, or, equivalently, constructive cardinality of variable (that can be instantiated during an interpretation in a
n sets, in the simultaneous presence of disjoint unions, Cartesian concretesetting).Byabuseoflanguage,wewillsaythatatype
a products, and exponentials. Mostly relying on results about
τ belongstoE (τ ∈E)whenapolynomialwithexponentiation
J polynomials with exponentiation that have not been used in our
0 context,wederive: thattheusualfiniteaxiomatization knownas f ∈ E exists such that τ = JfK. Throughout the paper, we
2 High-SchoolIdentities(HSI)iscompleteforasignificantsubclass use the plain equality symbol “=” to stand for identity i.e.
oftypes;thatitisdecidableforthatsubclasswhentwotypesare definitional equality.
] isomorphic; that, for the whole of the set of types, a recursive ThetypesofE areinhabitedbytermsofalambdacalculus,
O extension of theaxioms of HSI exists that iscomplete; and that,
in the usual way, following the typing system shown in
for the whole of the set of types, the question as to whether
L
two types are isomorphic isdecidablewhen base typesare tobe Figure5.Theequalitybetweentwotypedtermsistherelation
.
s interpreted as finite sets. We also point out certain related open = given by the usual axioms in Figure 7.
βη
c problems. Two types τ and σ are called isomorphic (notation τ ∼=σ)
[
when there is a pair of lambda terms φτ→σ and ψσ→τ
I. INTRODUCTION
2
that are mutually inverse, that is, λx.φ(ψx) = λx.x and
βη
v The class of types built from Cartesian products (τ ×σ),
λy.ψ(φy) = λy.y. The importance of this notion in “prac-
7 disjoint unions (τ +σ) and function spaces (τ → σ) lies at βη
tice” is as follows.
6
thecoreoftypesystemsforprogramminglanguagesandcon-
5 In typed programming languages, to be able to say when
structive systems of Logic. How useful could a programming
2 two types are isomorphic amounts to being able to say when
. language be if it did not include pairing, enumeration and
1 two programs implement essentially the same type signature:
functions?Likewise,howusefulwouldaconstructivelogicbe,
0 a program of type τ can be coerced back and forth to type
4 if it did not have conjunction, disjunction, and implication? σ without loss of information.One can use this, for example,
1 It is then a strange state of affairs that we still have open
to search over a library of routines for a routine of a type
: questions regarding general properties of this basic class of
v coercible to the type needed by the programmer [1], [2].
i types. In this article, we revisit the problem of when two
X In propositional logic, knowing that two types are isomor-
types can be considered to be isomorphic and we show
phic means that the corresponding formulas (built from ∧,∨
r that this problem can be tackled using existing results from
a and →) are intuitionistically equivalent.
Mathematical Logic.
In Constructive Mathematics, type isomorphism coincides
Let us be more precise. The language of polynomials
with the notion of constructive cardinality [3], [4] that says
with exponentiation and with positive coefficients is defined
that two sets (i.e., types) are isomorphicif they have indistin-
inductively by
guishablestructure,whichisstrongerthantheclassicalnotion
E ∋f,g ::=1 | x | f +g | fg | gf, of cardinality relying on “number of elements”.
i
What is known about the isomorphism of types of E, in
where x is a variable for i ∈ N. This language determines
i general, are the following facts, both proved in [5].
the class of types that we are interested in, by the simple
translation, Theorem 1 (Soundness of HSI). If HSI⊢f =. g, then JfK∼=
JgK. In fact, we have:
J1K=1
JxiK=xi HSI⊢f =. g ⇒ JfK∼=JgK ⇒ N+ (cid:15)f ≡g.
JgfK=JfK→JgK
Theorem 2. Isomorphism is not finitely axiomatizable, that
JfgK=JfK×JgK
is, for no finite set of axioms T can we show that JfK ∼= JgK
Jf +gK=JfK+JgK, .
always implies T ⊢f =g.
.
The notation T ⊢ f = g means that there is a formal Wilkie[12]generalizedMartin’sequalitytothewholeofHSI,
.
derivation of the equation f = g in the derivation system giving the equation
shown in part (b) of Figure 3, from the axioms of the set T;
.
therefore, HSI ⊢ f = g means that the equation is derivable (Ax+Bx)y(Cy +Dy)x =(Ay +By)x(Cx+Dx)y, (1)
from the finite set of axioms shown in part (a) of Figure 3
(HSI stands for “High-School Identities”, see Subsection I-A whereA=1+x,B =1+x+x2,C =1+x3,D =1+x2+x4.
below). He showed (1) to be non-derivable in HSI, even though it is
Having only Theorem 1 and Theorem 2 is surprisingly true in N+. We thus have
little if we compare to what is known for the fragments of
.
E that do not mix gf and g +f simultaneously. For those ∀f,g ∈E(HSI⊢f =g ⇒ N+ (cid:15)f ≡g),
fragments, we have, as shown in [6], [7], [5], soundness and
completenesswithrespecttothesuitablerestrictionofHSI,we but
havemoreoverequivalencewithtruthinthestandardmodelof ∀f,g ∈E(N+ (cid:15)f ≡g 6⇒ HSI⊢f =. g),
positivenaturalnumbersN+ (seeSubsectionI-Afordefinition
of truth in N+),
which constitutes a negative solution to Tarski’s original
JfK∼=JgK ⇔ N+ (cid:15)f ≡g, question.
Gurevicˇ [13] further showed that one can not “repair” HSI
and, consequently, the decidability of τ ∼=σ for any τ and σ
by extending it with any finite list of axioms. He generalized
of those fragments.
Wilkie’s (1) to the infinite sequence of equations
In this paper, we will address both the questions of com-
pleteness and decidability for E, in simultaneous presence of (A2x +B2x)x(Cx+Dx)2x =(Ax+Bx)2x(C2x +D2x)x,
gf and g +f. In Section II, we will bring up the relevance n n n n n n
(G )
of certain subclasses of types going back to Levitz, and n
where A = x+ 1,B = 1 + x+ x2 + ··· + xn−1,C =
explain how results of Henson, Rubel, Gurevicˇ, Richardson, n n
1+xn,D =1+x2+x4+···+x2(n−1), and showed that
andMacintyre,allowtoshowthattypeisomorphismforthose n
for any finite extension T of HSI there is an odd n>3 such
subclasses are complete with respect to HSI, and decidable.
that T can not prove the equality (G ) although N+ (cid:15)G .
In Section III, using Wilkie’s positive solution of Tarski’s n n
Fiore,DiCosmoandBalat[5]showedthatGurevicˇ’sequa-
High-School Algebra Problem (see next subsection), we will
tionscanbeinterpretedastypeisomorphisms,establishingthe
establishthesamepropertiesforthewholeofE (decidabilityis
mentioned Theorem 2.
provedforbasetypesinterpretedasfinitesets.).InSectionIV,
we will mention related open problems, in particular about
having efficient means of deciding the type isomorphisms. B. Decidability of Arithmetic Equality for E
A. Tarski’s High-School Algebra Problem A separate question of more general interest is that of the
The questions that we are interested in are related to decidability of equality between polynomials with exponen-
questionsregardingpolynomialswithexponentiationfromthe tiation, that is, whether there is a procedure for deciding if
class E, posed by Skolem [8] and Tarski [9] in the 1960’s. N+ (cid:15) f ≡ g holds or not, for any f,g ∈ E. It was first
Especially relevant is the question known as Tarski’s High- addressed by Richardson [14], who proved decidability for
School Algebra Problem1: can all equations that are true in the univariate case (expressions of E in one variable). Later,
the standardmodelof positivenaturalnumbers(N+ (cid:15)f ≡g) Macintyre [15] showed the decidability for the multivariate
be derived inside the derivation system of HSI from parts (a) case i.e. for the whole of E.
.
and (b) of Figure 3 (HSI ⊢ f = g)? This is a completeness
Theorem 3. There is a recursive procedure that decides, for
question.
any f,g ∈E, whether N+ (cid:15)f ≡g holds or not.
The meaning of N+ (cid:15) f ≡ g is the standard model
theoretic one: for any replacement of the variables of f and However, we cannot use the decidability result for N+ to
g by elements of N+, one computesthe same positive natural conclude decidability of type isomorphisms for E, because,
number. The converse (soundness), although we do have that (by Theorem 1)
.
HSI⊢f =g ⇒ N+ (cid:15)f ≡g, HSI⊢f =. g ⇒ JfK∼=JgK ⇒ N+ (cid:15)f ≡g,
can easily be proved.
Martin [9] was the first to show that, if we exclude the a proof of
axioms mentioning the constant 1 from HSI, the derivation
JfK∼=JgK ⇐ N+ (cid:15)f ≡g
system is incomplete, since it can not derive the equality
(xz +xz)w(yw+yw)z =(xw+xw)z(yz+yz)w. is not known, and HSI is not complete:
1Forvariousresultsaroundthisproblem,pleaselookatthesurveyarticles .
HSI⊢f =g 6⇐ N+ (cid:15)f ≡g.
[10]and[11].
.
f =f
.
f +g =g+f
. . . . .
. f =g f =g g =h f =g f =g
(f +g)+h=f +(g+h) 1 1 2 2
. g =. f f =. h ff2 =. gg2
fg =gf 1 1
.
(fg)h=f(gh) . . . .
f =g f =g f =g f =g
. 1 1 2 2 1 1 2 2
f(g+h)=fg+fh . .
f +f =g +g f f =g g
. 1 2 1 2 1 2 1 2
1f =f
f1 =. f (a) Equality and congruence rules
1f =. 1 t =. 1
1
fg+h =. fgfh t =. x
. xi . i
(fg)h =fhgh t =t t
zu z u
. .
(fg)h =fgh t =t +t
z+u z u
.
t =t (when N+ (cid:15)z ≡u)
z u
(a) Axioms of HSI (b) Additional axioms of HSI*
Notation: f,g,h denote polynomials with exponentiation (possibly containing t terms in the congruence and equality rules),
z
while z,u denote ordinary polynomials (i.e., without non-constant exponents) with possibly negative monomial coefficients.
Fig. 3: The derivation system of (Extended) High-School Identities
II. SUBCLASSES OFE COMPLETE FORHSI where l∈Λ is defined by
Oneofthethingsthathasnotbeenexploitedintheliterature Λ∋f,g ::=s | x | f +g | fg | lf,
i 0
on type isomorphism is the line of research on subclasses of
polynomialswith exponentiationforwhichtheaxiomsofHSI and l0 ∈Λ has no variables.
are complete.
Theorem 4. For all f,g ∈L,
In [16], while studying the relation of eventual dominance
.
for polynomialswith exponentiation,Levitz isolated the class N+ (cid:15)f ≡g ⇒ HSI⊢f =g.
ofexpressionsinonevariable,builtbytheinductivedefinition
For our purposes, it suffices to take S = {1}, and we will
S ∋f,g ::=1 | x | f +g | fg | xf | nf, henceforth use L specialized to this S.
Example 1. Wilkie’s equation (1) deals with terms that do
where n is a numeral. Henson and Rubel [17] extended it to
notbelongto the classL. AlthoughA,B,C,D ∈Λ⊂L, and
the multivariate class defined by
hence Ax,Bx,Cx,Dx,Ax+Bx,Cx+Dx ∈L\Λ, we have
L(S)∋f,g ::=s | x | f +g | fg | s′f | xf | (xs′)f, (Ax+Bx)y,(Cx+Dx)y ∈/ L,becausebasesofexponentiation
i i i
arenotallowedtocontainbasesofexponentiationthatcontain
whereS isanarbitrarysetofpositiverealconstants,s,s′ ∈S,
variables.
s′ >1,andtheyprovedalltrueequalitiesbetweenexpressions
from L(S) to be derivable from HSI. They also conjectured Example 2. The term3
that the result could be extended to the class defined by
(y+z)x(y+z)(y+z)x(y+z) ∈L, (2)
R(S)∋f,g ::=s | x | f +g | fg | pf,
i but
where p is an ordinarypolynomialwith coefficientsin S, and ((y+z)x)((y+z)y+z)x y+z ∈/ L, (3)
(cid:16) (cid:17)
theyremarkedthatWilkie’scounterexamplelies“justoutside”
the class R(S). although the two terms are inter-derivable using the HSI
Finally, Gurevicˇ [18] showed that HSI are complete for the axioms. This means that, even though HSI is complete for
proper extension L of R(S) defined by2 L, there is room for extension of L to subclassesthat are still
finitely axiomatizable by HSI. In other words, L is not closed
L∋f,g ::=s | xi | f +g | fg | lf, under HSI-derivability.
2We stick to the original notations L(S) and L, although the latter also 3These terms correspond to simply typed versions of an induction axiom
depends onthesetS andwehaveL(S)(R(S)(L. fordecidable predicates, in“curried” and“uncurried” variant.
Theorems 1, 3 and 4 allow us to conclude the following. This is a statement concerning terms of E (the original
language), in the proof of which terms of E∗ (the extended
Corollary 1 (Completenessfor L). For all f,g ∈L, if JfK∼=
. language) are used. In this respect, it is reminiscent of meta-
JgK then HSI⊢f =g.
mathematicalstatements like Hilbert’s ǫ-elimination theorems
Corollary 2 (Decidability for L). There is an algorithm that [20] or Henkin’s version of Go¨del’s completeness theorem
decides, for all f,g ∈L, whether JfK∼=JgK or not. [21].
Using theorems 1 and 5, we immediately obtain the com-
III. THEEXTENDEDHIGH-SCHOOL IDENTITIES pleteness of HSI* for type isomorphism over E.
As explained in Subsection I-A, Wilkie’s negative solu- Corollary 3. Given f,g ∈ E such that JfK ∼= JgK, we have
tion of Tarski’s problem, together with the generalization of .
that HSI*⊢f =g.
Gurevicˇ, was used by Fiore, Di Cosmo, and Balat, to show
the incompleteness of HSI, and the impossibility of a finite We now move on to the decidability question. We will
axiomatization, for type isomorphism over E. show that derivations of HSI* can be interpreted as type
However, in the paper [12], a positive solution to Tarski’s isomorphisms, which suffices, since then the circuit
problem was also given. Namely, Wilkie showed that HSI is JfK∼=JgK ⇒ N+ (cid:15)f ≡g ⇒ HSI*⊢f =. g ⇒ JfK∼=JgK
almost complete: by extending it with all equations that hold
between ordinary positive polynomials4 — that is, positive
allows one to use Macintyre’s decidability results for N+
polynomials without exponents containing variables, but pos-
(Theorem 3) to conclude decidability of type isomorphism
sibly with negative monomial coefficients — one obtains an
over E.
axiomatization(HSI*fromFigure3)whichiscompleteforall
Atfirstthought,interpretingthenewt symbolsmightseem
trueequationsinN+ betweenexpressionsofE.Sinceequality z
problematic, since negative monomial coefficients in z would
of ordinary (positive) polynomials is decidable, we have a
imply the use of some kind of negative types. However, we
recursive procedure for determining if an equation belongs
also have the additional property that z is positive, which
to the set of axioms, and therefore equality between terms
means that if we instantiate its variables with positive natural
of E — although not finitely axiomatizable — is recursively
numbers (positive types), we will obtain a positive natural
axiomatizable.
number (positive type).
Tobemoreprecise,letE∗ bethelanguageextendingE with
In this paper, we will work with the restriction that base
a constant t for every ordinary positive polynomial z:
z types are finite sets. Although the method does work for base
E∗ ∋f,g ::=t | 1 | x | gf | fg | f +g. types isomorphic to ordinals in Cantor normal form5, that
z i
requires a careful constructive treatment of ordinals beyond
The system of axioms of HSI* is the extension of the sys- the scope of this paper.
tem HSI — that applies only to expressions of the original For HSI, one can keep the interpretation of base types
language E — with the axioms given in part (c) of Figure 3 implicit: soundness of HSI equations as type isomorphisms
— that apply to a strict subset of the extended language E∗. is proved uniformly, regardless of the actual interpretations
Thetwo typesofaxiomscan“interact”throughthe derivation of base types. For HSI*, we will need to be explicit about
systemofpart(b)ofFigure3,thusmakingpossibleequalities interpretation, that is, we will prove the soundness theorem
between expressions of the full language E∗. point-wise. We will thus introduce an explicit environment ρ
The left-hand side of the new axioms is always an expres- mapping variables to types and extend the interpretation J·K
sion of form tz, while the right hand side can either be an for the extra tz-terms.
expression t — whenever z and u are equal polynomials
u
(which can be decided by bringing them into canonicalform) J1Kρ =1
—oranexpressionreflectingthestructureofz whenpossible: Jx K =ρ(x )
i ρ i
whenz hasnonegativecoefficients(weuseinthatcaseletters
JgfK =JfK →JgK
p,q instead of z,u), we can fully reflect it into the language, ρ ρ ρ
. JfgK =JfK ×JgK
that is, one can prove HSI* ⊢ t = p. This representation of ρ ρ ρ
p
the axioms is inspired from the one of Asatryan [19]. Note Jf +gK =JfK +JgK
ρ ρ ρ
that,foranytz, thereare infinitelymanyaxiomshavingtz on JtzKρ =1+1+···+1=k where k =eval(tz,ρ)
the left hand side that can be used.
k-times
We can now state Wilkie’s result. | {z }
The number eval(t ,ρ) is the result of evaluating z for the
z
Theorem5(CompletenessofHSI*). Forallf,g ∈E (thatis,
variablesinterpretedinρbypositivenaturalnumbers.Wealso
all f,g of E∗ that do not contain t -symbols), we have that
. z
N+ (cid:15)f ≡g implies HSI*⊢f =g.
5Subtraction α−β between two such ordinals can be defined when we
knowthatα<β.Sincewecanalways rewritetz astp−tq,andweknow
4A polynomial is positive if it computes to a positive natural number for p < q, an ordinal in Cantor normal form can always be computed for tz
everyreplacement ofvariables bypositivenatural numbers. whenever onesforpandq aregiven.
. .
denote the type • Given interpretations of f =g and g =h by four terms
1+1+···+1
Φ :JfK →JgK Φ :JgK →JhK
1 ρ ρ 2 ρ ρ
k-times
| {z } Ψ :JgK →JfK Ψ :JhK →JgK ,
with bold-face k. 1 ρ ρ 2 ρ ρ
.
Theorem6. Letf,g ∈E∗. IfHSI*⊢f =. g thenJfKρ ∼=JgKρ wanediΨnte.rpret f = h by composing Φ1 and Φ2, and Ψ1
for any ρ that interprets variables by types of form k. 2 . .
• Given interpretations of f = g and f = g by four
1 1 2 2
Proof. The proof is by induction on the derivation. We first terms
give explicit isomorphisms for the axioms of HSI:
. Φ1 :Jf1Kρ →Jg1Kρ Φ2 :Jf2Kρ →Jg2Kρ
• f =f is interpreted with the identity lambda term λx.x
Ψ :Jg K →Jf K Ψ :Jg K →Jf K ,
in both directions; 1 1 ρ 1 ρ 2 2 ρ 2 ρ
.
• f +g =g+f is interpreted by λx.δ[x|x1.ι2x1|x2.ι2x2] we interpret ff2 =. gg2 by using the terms
1 1
in both directions;
.
• (f +g)+h=f +(g+h) is interpreted by Φ=λx.λy.Φ1(x(Ψ2y))
Ψ=λx.λy.Ψ (x(Φ y)).
δ[x|x .ι ι x|x .δ[x |x .ι ι x |x .ι x ]] 1 2
1 1 1 2 2 21 1 2 2 22 2 22
The fact that Φ and Ψ are mutually inverse w.r.t. = is
and βη
proved by using the η-axiom
δ[x|x .δ[x |x .ι x |x .ι ι x ]|x .ι ι x ];
1 1 11 1 11 12 2 1 12 2 2 2 2 λx.λy.xy = λx.x.
βη
.
• fg = gf is interpreted by λx.hπ2x,π1xi in both direc- • Given interpretations of f =. g and f =. g by four
tions; 1 1 2 2
. terms
• (fg)h=f(gh) is interpreted by
Φ :Jf K →Jg K Φ :Jf K →Jg K
λx.hhπ x,π π xi,π π xi 1 1 ρ 1 ρ 2 2 ρ 2 ρ
1 1 2 2 2
Ψ :Jg K →Jf K Ψ :Jg K →Jf K ,
1 1 ρ 1 ρ 2 2 ρ 2 ρ
and
.
λx.hπ1π1x,hπ2π1x,π2xii; we interpret f1f2 =g1g2 by using the terms
.
• f(g+h)=fg+fh is interpreted by Φ=λx.hΦ1(π1x),Φ2(π2x)i
Ψ=λy.hΨ (π y),Ψ (π y)i.
λx.δ[π x|x .ι hπ x,x i|x .ι hπ x ,ι π x i] 1 1 2 2
2 1 1 1 1 2 2 1 1 2 2 1
The fact that Φ and Ψ are mutually inverse w.r.t. = is
and βη
proved by using the η-axiom
λx.δ[x|x .hπ x ,ι π x i|x .hπ x ,ι π x i];
1 1 1 1 2 1 2 1 2 2 2 2 λy.hπ y,π yi= λy.y.
. 1 2 βη
• f1=. f is interpreted by λx.π1x and λx.hx,⋆i; • Given interpretations of f =. g and f =. g by four
• f1 =f is interpreted by λx.x⋆ and λxy.x; 1 1 2 2
. terms
• 1f =1 is interpreted by λx.⋆ and λxy.⋆;
.
• fg+h =fgfh is interpreted by Φ :Jf K →Jg K Φ :Jf K →Jg K
1 1 ρ 1 ρ 2 2 ρ 2 ρ
λx.hλy.x(ι1y),λy.x(ι2y)i Ψ1 :Jg1Kρ →Jf1Kρ Ψ2 :Jg2Kρ →Jf2Kρ,
.
and we interpret f1+f2 =g1+g2 by using the terms
λxy.δ[y|y .(π x)y |y .(π x)y ];
1 1 1 2 2 2 Φ=λx.δ[x|x .ι (Φ x )|x .ι (Φ x )]
1 1 1 1 2 2 2 2
.
• (fg)h = fhgh is interpreted by λx.hλy.π1xy,λy.π2xyi Ψ=λy.δ[y|y1.ι1(Ψ1y1)|y2.ι2(Ψ2y2)].
and λxy.h(π x)y,(π x)yi;
• (fg)h =. fg1h is int2erpreted by λxy.x(π y)(π y) and The fact that Φ and Ψ are mutually inverse w.r.t. =βη is
1 2
proved by using the η-axiom for sums twice. Once we
λxyz.xhz,yi.
use it with
The congruence and equality rules are handled using the
induction hypotheses: M :=y
.
• Given an interpretation of f = g, i.e. Φ : JfKρ → JgKρ N :=δ[ δ[z|y1.ι1(Ψ1y1)|y2.ι2(Ψ2y2)]
(cid:0) (cid:1)
and Ψ:JgK →JfK such that
ρ ρ |x .ι (Φ x )
1 1 1 1
λx.Φ(Ψx)=βη λx.x |x2.ι2(Φ2x2)],
λy.Ψ(Φy)= λy.y,
βη and the second time with M :=y, N :=z.
we just swap the order of the two equations in order to Itremainsto interpretthe restofthe axiomsofHSI*, those
.
interpret g =f. that involve t -terms.
z
Λ+ ∋M,N,P ::=x | ⋆ | λx.M | MN | ι M | ι M | δ[M|x.N|x.P] | hM,Ni | π M | π M
1 2 1 2
(a) Raw language of lambda terms
xτ ∈Γ Γ∪xτ (cid:13)Mσ Γ(cid:13)Mτ→σ Γ(cid:13)Nτ
Γ(cid:13)xτ Γ(cid:13)⋆1 Γ(cid:13)(λx.M)τ→σ Γ(cid:13)(MN)σ
Γ(cid:13)Mτ Γ(cid:13)Mσ Γ(cid:13)Mτ+σ Γ∪xτ (cid:13)Nρ Γ∪xσ (cid:13)Pρ
Γ(cid:13)(ι M)τ+σ Γ(cid:13)(ι M)τ+σ Γ(cid:13)δ[M|x.N|x.P]ρ
1 2
Γ(cid:13)Mσ Γ(cid:13)Nτ Γ(cid:13)Mσ×τ Γ(cid:13)Mσ×τ
Γ(cid:13)hM,Niσ×τ Γ(cid:13)(π M)σ Γ(cid:13)(π M)τ
1 2
(b) Typing system (well-formed lambda terms)
Fig. 5: Inhabitation of types of E with lambda terms
M = ⋆ for any term M of type 1
βη
(λx.M)N = M{N/x} where x is not a variable of N
βη
M = λx.Mx where x is not a variable of M
βη
δ[ι M|x.N|y.P]= N{M/x} where x is not a variable of M
1 βη
δ[ι M|x.N|y.P]= P{M/y} where y is not a variable of M
2 βη
N{M/z}= δ[M|x.N{ι x/z}|y.N{ι y/z}] where z is not a variable of M
βη 1 2
π hM,Ni= M
1 βη
π hM,Ni= N
2 βη
M = hπ M,π Mi
βη 1 2
The full relation = is the reflexive, symmetric, transitive, and congruent closure of the above.
βη
Fig. 7: Beta-eta equality between lambda terms of the same type
• t1 =. 1 is interpreted as the isomorphism 1 ∼= J1Kρ i.e. • When p = xi, we have eval(xi,ρ) = ρ(xi) = k, so
• 1txi∼==.1xuisiinsginttheerplraemtebddaastekrm∼=λkx,.xforinkb=othevdailr(excit,ioρn),s;by • kW∼=henJxpiK=ρ btyz,λwxe.xhainvebeovthald(tirze,cρt)io=nsk., so k∼=JtzKρ by
the lambda term λx.x in both directions; λx.x in both directions.
• tzu =. tztu is interpreted as k ∼= k1 × k2, for k = • When p=p1+p2, we have
eval(t ,ρ), k = eval(t ,ρ), and k = eval(t ,ρ), by
zu 1 z 2 u
the lambda term of Lemma 1;
• tz+u =. tz +tu is interpreted as k ∼= k1 +k2, for k = k =eval(p,ρ)=eval(p1+p2,ρ)=
eval(tzu,ρ), k1 = eval(tz,ρ), and k2 = eval(tu,ρ), by =eval(p1,ρ)+eval(p2,ρ)=k1+k2
the lambda term of Lemma 1;
• tz =. tu is interpreted as k ∼= k, for k = eval(z,ρ) = for some k1,k2 ∈ N+. By applying the induction hy-
eval(u,ρ), by the lambda term λx.x in both directions. pothesis twice, we obtain k1 ∼= Jp1Kρ and k2 ∼= Jp2Kρ,
therefore,
Lcoeemffimciaen1t.sL(peotsps∈iblEy∗cboentaaninoirndgintar-tyerpmolsy)n,okm∈iaNlw+i,thanpdosρitibvee k∼=k1+k2 ∼=Jp1Kρ+Jp2Kρ =JpKρ
z
an interpretation such that k=eval(p,ρ). Then, k∼=JpKρ. using the obvious isomorphism k∼=k1+k2, that holds
Proof. We do induction on p. when k = k + k and the lambda terms interpreting
1 2
• When p = 1, we have eval(1,ρ) = 1 = k, so k = 1 ∼= congurence for “+” from the proof of the previous theo-
1=J1K is established by usingλx.x in bothdirections. rem.
ρ
• When p=p p , we have induction on the construction of f, the difficult case being
1 2
whenf =ff2;here,theinductionhypothesisisusedtogether
1
k =eval(p,ρ)=eval(p1p2,ρ)= with the fact that each positive polynomial can be factored
=eval(p ,ρ)eval(p ,ρ)=k k as a monomial and irreducible polynomial. In fact, an enu-
1 2 1 2
merationofallpossiblepairsh irreducible , monomial iwith
for some k ,k ∈ N+. By applying the induction hy-
1 2 the right properties is used, and then, when constructing the
pothesis twice, we obtain k1 ∼= Jp1Kρ and k2 ∼= Jp2Kρ, representativeforf oneneedonlylookupintheenumeration.
therefore,
A large number of extra variables will generally be added to
k∼=k1×k2 ∼=Jp1Kρ×Jp2Kρ =JpKρ the representing “polynomial”.
The second component of Wilkie’s proof uses Differential
using the obvious isomorphism k ∼= k1+k2 that holds Algebra to show that the representation from Theorem 2.8 is
when k = k k , and the lambda terms interpreting
1 2 unique.
congurence for “×” from the proof of the previous
If we could obtain a simpler version of Wilkie’s proof of
theorem.
Theorem 2.8, that avoids an ad hoc enumeration, it would
be easier to interpret Corollary 3 as a program that given a
concrete proof of JfK ∼= JgK builds a concrete derivation of
Corollary 4. Given two types f,g ∈ E, one can decide .
whether JfKρ ∼= JgKρ or not, and this holds whenever ρ HSI*⊢f =g.
interprets variable by types of form k. 3) Efficient Decision Procedures: The decision procedure
of Macintyre [15], although primitive recursive, is of expo-
IV. CONCLUSION nential complexity. It proceeds by computing, for a given
We showed that existing results from Mathematical Logic equality one wants to test, an upper bound for the values of
allow us to conclude that type isomorphism over E is recur- the variables, and then does a brute-force search up to that
sively axiomatizable, and that a subclass L of types can be upper bound.
isolatedforwhichtypeisomorphismisevenfinitelyaxiomati- Onthe otherhand,we knowthatequalitybetweenordinary
zablebythewellknownHigh-SchoolIdentitiesanddecidable. polynomials can be decided in O(nlog(n)) [22], and in the
Our Theorem 6 allows us to conclude decidability for the contextoftypeisomorphismsdecisionalgorithmswithsimilar
whole of E when base types are finite sets. complexityhavebeengivenbyConsidine,GilandZibin[23],
Theseresultsalsoapplytoquestionsofcardinalityofsetsin [24], [25].
Constructive Mathematics, and to isomorphism of objects in Is to possible to obtain a decision algorithm for type
the corresponding category. However, further work is needed isomorphism over E with less-than-exponential complexity?
to understand fully their implications in practice. Is it at least possible to do so for some subclass of terms like
L?
A. Open Problems
B. Other Related Work
1) ExtensionsandPracticalImportanceoftheLevitzClass:
We saw that the class L of Gurevicˇ is a generalization of Bruce, Di Cosmo and Longo [26] contains a more modern
the classes R(S) and L(S) of Henson and Rubel, which are presentation of Soloviev’s result [7].
in turn generalizations of Levitz’s class S. We also saw in DosˇenandPetric´ [27]showtypeisomorphisminthecontext
Example 1 that there are two HSI-equal types, one of which of symmetric monoidal closed categories is finitely axiomati-
is in L while the other is not. zable and decidable.
Therefore, it does not seem unlikely that the class L Di Cosmo and Dufour [28] show that if one considers the
can be further extended. For example, cannot we allow the structureN+∪{0},decidabilityandnon-finite-axiomatizability
bases of exponentiation to contain variables in their bases of of arithmetical equality still hold. It is known that that there
exponentiation up to a fixed (but arbitrary) height n? Would aretruearithmeticalequalitiesconcerning0thatdonotholdas
not such a theory also be finitely axiomatizable by HSI? typeisomorphisms,anobservationattributedtoAlexSimpson
Another interesting thing to investigate would be what the in [5]. We have not treated the empty type here. This is
practical interest of these subclasses is. For example, how motivated by our own interests in using type isomorphism
many programs of a standard library for functional program- over E: in intuitionistic logic it is not necessary to have a
ming or theorem proving would fall outside (extensions of) dedicatedabsurdityproposition,sinceassoonaswehavebasic
L? arithmetic,theexfalsoquodlibetruleisderivablebyinduction.
2) SimplerCompletenessProofforHSI*: Wilkie’sproofof
completeness of HSI* relies on two components. ACKNOWLEDGMENTS
In Theorem 2.8 of [12], it is shown that each polynomial I would like to thank Olivier Danvy for being an excellent
with exponentiationf can be provedto be equal in HSI* to a host at Aarhus University, where this project finally took off.
positive polynomial with positive coefficients, but with extra I would also like to thank Alex Simpson for noticing an error
variables,someofwhichareinstantiatedwithwitnessingterms at a workshop talk that I gave on the subject, and Gurgen
τ of the form pqi. The proof of this theorem proceeds by Asatryan for sending me a copy of a hard-to-find paper.
i i
This work is supported by a fellowship from Kurt Go¨del [26] K.B.Bruce, R.D.Cosmo,andG.Longo,“Provable isomorphisms of
Society, Vienna. Last changes were made while the author types,” Mathematical Structures inComputer Science, vol.2,pp.231–
247,1992.
was working under the frame of the ProofCert ERC grant.
[27] K. Dosˇen and Z. Petric´, “Isomorphic objects in symmetric monoidal
closedcategories,”MathematicalStructuresinComputerScience,vol.7,
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