Welcome!
Subrecursive degrees of honest functions
and provably recursive functions.
Lars Kristiansen, Department of Mathematics, University of Oslo.
Overview
The structure of my talk
PART I:
Introduction to honest elementary degree theory.
PART II:
Generalisation of the degree theory introduced in Part I.
PART III:
The relationship to provability in PA.
The honest elementary degrees
PART I
The honest elementary degrees.
The honest elementary degrees
We will work with the the subrecursive reducibility relation
≤E (being Kalmar elementary in).
The honest elementary degrees
We will work with the the subrecursive reducibility relation
≤E (being Kalmar elementary in).
Definition. f ≤E g iff
f can be generated from the initial functions
g , 2x , max (maximum), S (successor), Iin (projections)
by
composition
and bounded primitive recursion
The honest elementary degrees
We will work with degrees of honest functions.
The honest elementary degrees
We will work with degrees of honest functions.
Definition. f is honest iff
f (x) ≥ 2x
f (x) ≤ f (x + 1)
f has elementary graph, that is, the function φ where
0 if f (x) = y
φ(x, y ) =
1 otherwise
is elementary.
The honest elementary degrees
Remarks
The honest elementary degrees
Remarks
The growth of an honest function reflects the function’s
computational complexity.
The honest elementary degrees
Remarks
The growth of an honest function reflects the function’s
computational complexity.
The notion of honesty is related to the notions of
time-constructibility and space-constructibility in complexity
theory.
The honest elementary degrees
Remarks
The growth of an honest function reflects the function’s
computational complexity.
The notion of honesty is related to the notions of
time-constructibility and space-constructibility in complexity
theory.
Any set not being elementary, is an example of a dishonest
function (sets are 0-1 valued functions).
The honest elementary degrees
Remarks
The growth of an honest function reflects the function’s
computational complexity.
The notion of honesty is related to the notions of
time-constructibility and space-constructibility in complexity
theory.
Any set not being elementary, is an example of a dishonest
function (sets are 0-1 valued functions).
The Ackermann function is honest.
The honest elementary degrees
Remarks
The growth of an honest function reflects the function’s
computational complexity.
The notion of honesty is related to the notions of
time-constructibility and space-constructibility in complexity
theory.
Any set not being elementary, is an example of a dishonest
function (sets are 0-1 valued functions).
The Ackermann function is honest.
The Hardy function Hǫ0 is honest.
The honest elementary degrees
Remarks
The growth of an honest function reflects the function’s
computational complexity.
The notion of honesty is related to the notions of
time-constructibility and space-constructibility in complexity
theory.
Any set not being elementary, is an example of a dishonest
function (sets are 0-1 valued functions).
The Ackermann function is honest.
The Hardy function Hǫ0 is honest.
All the backbone functions of a natural subrecursive hierarchy
will be honest (at least the ones that dominate 2x ).
The honest elementary degrees
Let
f ≡E g :⇔ f ≤E g ∧ g ≤E f .
The honest elementary degrees
Let
f ≡E g :⇔ f ≤E g ∧ g ≤E f .
Now, ≡E is an equivalence relation.
The honest elementary degrees
Let
f ≡E g :⇔ f ≤E g ∧ g ≤E f .
Now, ≡E is an equivalence relation.
The elementary honest degrees are the equivalence classes induced
by ≡E on the honest functions. (We use a, b, c, . . . to denote such
degrees.)
The honest elementary degrees
Let
f ≡E g :⇔ f ≤E g ∧ g ≤E f .
Now, ≡E is an equivalence relation.
The elementary honest degrees are the equivalence classes induced
by ≡E on the honest functions. (We use a, b, c, . . . to denote such
degrees.)
We use deg(f ) to denote the degree of f , i.e.
deg(f ) = { g | g ≡E f } .
The honest elementary degrees
The combination of honest functions and ≤E -reducibility yields a
neat and very useful theorem.
The honest elementary degrees
The combination of honest functions and ≤E -reducibility yields a
neat and very useful theorem.
The Growth Theorem (Kristiansen 1996). Let f and g be
honest functions. Then,
f ≤E g ⇔ f (x) ≤ g k (x) for some fixed k .
The honest elementary degrees
The combination of honest functions and ≤E -reducibility yields a
neat and very useful theorem.
The Growth Theorem (Kristiansen 1996). Let f and g be
honest functions. Then,
f ≤E g ⇔ f (x) ≤ g k (x) for some fixed k .
It is easy to prove the theorem . . .
The honest elementary degrees
Assume f (x) ≤ g k (x).
The honest elementary degrees
Assume f (x) ≤ g k (x).
We have
f (x) = (µy ≤ g k (x)) [ f (x) = y ] .
The honest elementary degrees
Assume f (x) ≤ g k (x).
We have
f (x) = (µy ≤ g k (x)) [ f (x) = y ] .
Hence f ≤E g as the elementary functions are closed under
composition and bounded minimalisation. (Use that f has
elementary graph.)
The honest elementary degrees
Assume f (x) ≤ g k (x).
We have
f (x) = (µy ≤ g k (x)) [ f (x) = y ] .
Hence f ≤E g as the elementary functions are closed under
composition and bounded minimalisation. (Use that f has
elementary graph.)
This proves the implication
f ≤E g ⇐ f (x) ≤ g k (x) for some fixed k .
The honest elementary degrees
To prove the converse implication, we need the following claim.
The honest elementary degrees
To prove the converse implication, we need the following claim.
(Claim) For any ψ ≤E g , there exists k ∈ N such that
ψ(~x ) ≤ g k (max(~x )) .
The honest elementary degrees
To prove the converse implication, we need the following claim.
(Claim) For any ψ ≤E g , there exists k ∈ N such that
ψ(~x ) ≤ g k (max(~x )) .
The claim is proved by induction over the build-up of ψ. (Use that
g is monotone and dominates 2x .)
The honest elementary degrees
To prove the converse implication, we need the following claim.
(Claim) For any ψ ≤E g , there exists k ∈ N such that
ψ(~x ) ≤ g k (max(~x )) .
The claim is proved by induction over the build-up of ψ. (Use that
g is monotone and dominates 2x .)
It follows straightforwardly from this claim that
f ≤E g ⇒ f (x) ≤ g k (x) for some fixed k .
The honest elementary degrees
The Growth Theorem makes it possible to study the structure of
honest elementary degrees without resorting to classical
computability-theoretic constructions (involving enumerations,
diagonalisations, etc.).
The honest elementary degrees
The Growth Theorem makes it possible to study the structure of
honest elementary degrees without resorting to classical
computability-theoretic constructions (involving enumerations,
diagonalisations, etc.).
To prove f ≤E g : provide k such that f (x) ≤ g k (x).
The honest elementary degrees
The Growth Theorem makes it possible to study the structure of
honest elementary degrees without resorting to classical
computability-theoretic constructions (involving enumerations,
diagonalisations, etc.).
To prove f ≤E g : provide k such that f (x) ≤ g k (x).
To prove f 6≤E g : prove that such a k does not exists, that is,
prove that for any k we have f (x) > g k (x) for some x.
The honest elementary degrees
The Growth Theorem makes it possible to study the structure of
honest elementary degrees without resorting to classical
computability-theoretic constructions (involving enumerations,
diagonalisations, etc.).
To prove f ≤E g : provide k such that f (x) ≤ g k (x).
To prove f 6≤E g : prove that such a k does not exists, that is,
prove that for any k we have f (x) > g k (x) for some x.
Assume f <E g . How can we prove that there exists two
incomparable functions strictly between f and g ? . . . USE THE
BLACKBOARD . . .
The honest elementary degrees
Some operators on honest functions:
max[f , g ] . . . defined by max[f , g ](x) = max(f (x), g (x))
The honest elementary degrees
Some operators on honest functions:
max[f , g ] . . . defined by max[f , g ](x) = max(f (x), g (x))
min[f , g ] . . . defined by min[f , g ](x) = min(f (x), g (x))
The honest elementary degrees
Some operators on honest functions:
max[f , g ] . . . defined by max[f , g ](x) = max(f (x), g (x))
min[f , g ] . . . defined by min[f , g ](x) = min(f (x), g (x))
f ′ . . . defined by f ′ (x) = f x+1 (x).
The honest elementary degrees
Some operators on honest functions:
max[f , g ] . . . defined by max[f , g ](x) = max(f (x), g (x))
min[f , g ] . . . defined by min[f , g ](x) = min(f (x), g (x))
f ′ . . . defined by f ′ (x) = f x+1 (x).
It turns out that these operators on functions induce operators on
the honest elementary degrees.
The honest elementary degrees
Some operators on honest functions:
max[f , g ] . . . defined by max[f , g ](x) = max(f (x), g (x))
min[f , g ] . . . defined by min[f , g ](x) = min(f (x), g (x))
f ′ . . . defined by f ′ (x) = f x+1 (x).
It turns out that these operators on functions induce operators on
the honest elementary degrees.
Furthermore, when f and g are honest, then
deg(max[f , g ]) is the l.u.b. of deg(f ) and deg(g )
The honest elementary degrees
Some operators on honest functions:
max[f , g ] . . . defined by max[f , g ](x) = max(f (x), g (x))
min[f , g ] . . . defined by min[f , g ](x) = min(f (x), g (x))
f ′ . . . defined by f ′ (x) = f x+1 (x).
It turns out that these operators on functions induce operators on
the honest elementary degrees.
Furthermore, when f and g are honest, then
deg(max[f , g ]) is the l.u.b. of deg(f ) and deg(g )
deg(min[f , g ]) is the g.l.b. of deg(f ) and deg(g )
The honest elementary degrees
Some operators on honest functions:
max[f , g ] . . . defined by max[f , g ](x) = max(f (x), g (x))
min[f , g ] . . . defined by min[f , g ](x) = min(f (x), g (x))
f ′ . . . defined by f ′ (x) = f x+1 (x).
It turns out that these operators on functions induce operators on
the honest elementary degrees.
Furthermore, when f and g are honest, then
deg(max[f , g ]) is the l.u.b. of deg(f ) and deg(g )
deg(min[f , g ]) is the g.l.b. of deg(f ) and deg(g )
deg(f ′ ) is an honest degree strictly above deg(f ).
The honest elementary degrees
Some operators on honest functions:
max[f , g ] . . . defined by max[f , g ](x) = max(f (x), g (x))
min[f , g ] . . . defined by min[f , g ](x) = min(f (x), g (x))
f ′ . . . defined by f ′ (x) = f x+1 (x).
It turns out that these operators on functions induce operators on
the honest elementary degrees.
Furthermore, when f and g are honest, then
deg(max[f , g ]) is the l.u.b. of deg(f ) and deg(g )
deg(min[f , g ]) is the g.l.b. of deg(f ) and deg(g )
deg(f ′ ) is an honest degree strictly above deg(f ).
So, the structure is a lattice, and we have a join, a meet and a
jump operator.
The honest elementary degrees
Let a and b be honest degrees, and let f and g be any honest
functions such that deg(f ) = a and deg(g ) = b.
The honest elementary degrees
Let a and b be honest degrees, and let f and g be any honest
functions such that deg(f ) = a and deg(g ) = b.
We define the meet of a and b, written a ∩ b, by
a ∩ b = deg(min[f , g ]) .
The honest elementary degrees
Let a and b be honest degrees, and let f and g be any honest
functions such that deg(f ) = a and deg(g ) = b.
We define the meet of a and b, written a ∩ b, by
a ∩ b = deg(min[f , g ]) .
We define the join of a and b, written a ∪ b, by
a ∪ b = deg(max[f , g ]) .
The honest elementary degrees
Let a and b be honest degrees, and let f and g be any honest
functions such that deg(f ) = a and deg(g ) = b.
We define the meet of a and b, written a ∩ b, by
a ∩ b = deg(min[f , g ]) .
We define the join of a and b, written a ∪ b, by
a ∪ b = deg(max[f , g ]) .
We define the jump of a, written a′ , by
a′ = deg(f ′ ) .
The honest elementary degrees
Let a and b be honest degrees, and let f and g be any honest
functions such that deg(f ) = a and deg(g ) = b.
We define the meet of a and b, written a ∩ b, by
a ∩ b = deg(min[f , g ]) .
We define the join of a and b, written a ∪ b, by
a ∪ b = deg(max[f , g ]) .
We define the jump of a, written a′ , by
a′ = deg(f ′ ) .
Furthermore, we define the zero degree, written 0, by 0 = deg(2x ).
The honest elementary degrees
Remarks
We can prove that our jump operator is equivalent to a classic
computability-theoretic jump operator based on enumeration and
diagonalisation.
The honest elementary degrees
Obviously, we have the canonical degrees
0 < 0′ < 0′′ < 0′′′ < . . .
The honest elementary degrees
Obviously, we have the canonical degrees
0 < 0′ < 0′′ < 0′′′ < . . .
What more do we know about the structure of honest elementary
degrees?
The honest elementary degrees
We know a lot about the structure of elementary honest degrees.
The honest elementary degrees
We know a lot about the structure of elementary honest degrees.
There exists Lown and Highn degrees (for any n ∈ N).
The honest elementary degrees
We know a lot about the structure of elementary honest degrees.
There exists Lown and Highn degrees (for any n ∈ N).
There exists intermediate degrees.
The honest elementary degrees
We know a lot about the structure of elementary honest degrees.
There exists Lown and Highn degrees (for any n ∈ N).
There exists intermediate degrees.
We have jump inversion theorems.
The honest elementary degrees
We know a lot about the structure of elementary honest degrees.
There exists Lown and Highn degrees (for any n ∈ N).
There exists intermediate degrees.
We have jump inversion theorems.
Density results: For any degrees a and b such that a < b, the
exists degrees c1 and c2 strictly between a and b such that
a = c1 ∪ c2 and b = c1 ∩ c2 .
The honest elementary degrees
We know a lot about the structure of elementary honest degrees.
There exists Lown and Highn degrees (for any n ∈ N).
There exists intermediate degrees.
We have jump inversion theorems.
Density results: For any degrees a and b such that a < b, the
exists degrees c1 and c2 strictly between a and b such that
a = c1 ∪ c2 and b = c1 ∩ c2 .
Cupability and capability results.
The honest elementary degrees
We know a lot about the structure of elementary honest degrees.
There exists Lown and Highn degrees (for any n ∈ N).
There exists intermediate degrees.
We have jump inversion theorems.
Density results: For any degrees a and b such that a < b, the
exists degrees c1 and c2 strictly between a and b such that
a = c1 ∪ c2 and b = c1 ∩ c2 .
Cupability and capability results.
. . . and a lot more. All our proofs are based on the Growth
Theorem.
The honest elementary degrees
Open Problems
Is the first order theory over the language {≤} decidable?
The honest elementary degrees
Open Problems
Is the first order theory over the language {≤} decidable?
Is the first order theory over the language {≤, ·′ } decidable?
The honest elementary degrees
Open Problems
Is the first order theory over the language {≤} decidable?
Is the first order theory over the language {≤, ·′ } decidable?
These two open problems are easy to state. There are more (very
natural) open problems . . .
The honest α-elementary degrees
PART II
Generalisations to ordinal recursion.
The honest α-elementary degrees.
The honest α-elementary degrees
Notation and basic definitions:
We will work with ordinals less than ǫ0 . The letters α, β, γ, . . .
always denote ordinals less than ǫ0 .
The honest α-elementary degrees
Notation and basic definitions:
We will work with ordinals less than ǫ0 . The letters α, β, γ, . . .
always denote ordinals less than ǫ0 .
We define the norm of α, written Nα or N(α), by induction over
the structure of α’s Cantor normal form:
N(0) = 0
N(β + γ) = Nβ + Nγ
N(ω β ) = 1 + Nβ.
The honest α-elementary degrees
Notation and basic definitions:
We will work with ordinals less than ǫ0 . The letters α, β, γ, . . .
always denote ordinals less than ǫ0 .
We define the norm of α, written Nα or N(α), by induction over
the structure of α’s Cantor normal form:
N(0) = 0
N(β + γ) = Nβ + Nγ
N(ω β ) = 1 + Nβ.
Thus, Nα is the number of omegas needed to write α in Cantor
normal form.
The honest α-elementary degrees
Notation and basic definitions:
Now, note that the set
{β | β < α ∧ Nβ ≤ Nα + x}
is finite when α and x ∈ N are given.
The honest α-elementary degrees
Notation and basic definitions:
Now, note that the set
{β | β < α ∧ Nβ ≤ Nα + x}
is finite when α and x ∈ N are given.
We define the α-iterate of the unary function f , written fα , by
f0 (x) = f (x) and
fα (x) = max{fβ fβ (x) | β < α ∧ Nβ ≤ Nα + x}
for any α such that 0 < α < ǫ0 .
The honest α-elementary degrees
Notation and basic definitions:
Let the Cantor normal for of α and β be given by
α = ω α1 + . . . + ω αn and β = ω β1 + . . . + ω βm .
We define the natural sum of α and β, written α#β, by
α#β = ω γ1 + . . . + ω γn+m + 0
where γ1 , . . . , γn+m is a permutation of α1 , . . . , αn , β1 , . . . , βm
such that γ1 ≥ γ2 ≥ . . . ≥ γn+m .
The honest α-elementary degrees
Notation and basic definitions:
Let the Cantor normal for of α and β be given by
α = ω α1 + . . . + ω αn and β = ω β1 + . . . + ω βm .
We define the natural sum of α and β, written α#β, by
α#β = ω γ1 + . . . + ω γn+m + 0
where γ1 , . . . , γn+m is a permutation of α1 , . . . , αn , β1 , . . . , βm
such that γ1 ≥ γ2 ≥ . . . ≥ γn+m .
Let SLim denote the class of all infinite additive principal numbers,
that is,
SLim = {α | α > 1 ∧ ∀β, γ < α[ β#γ < α ]} = {ω β | β > 0} .
The honest α-elementary degrees
Now recall the definition of
≤E (being elementary in).
Definition. f ≤E g iff
f can be generated from the initial functions
g , 2x , max (maximum), S (successor), Iin (projections)
by
composition
bounded primitive recursion
The honest α-elementary degrees
Here is the definition of
≤αE (being α-elementary in).
Definition. f ≤αE g iff
f can be generated from the initial functions
g , 2x , max (maximum), S (successor), Iin (projections)
by
composition
bounded primitive recursion
β-iteration where β < α.
The honest α-elementary degrees
Now recall the Growth Theorem:
The Growth Theorem. Let f and g be honest functions. Then,
f ≤E g ⇔ f (x) ≤ g k (x) for some fixed k .
The honest α-elementary degrees
Now recall the Growth Theorem:
The Growth Theorem. Let f and g be honest functions. Then,
f ≤E g ⇔ f (x) ≤ g k (x) for some fixed k .
Here is a generalised version:
The Generalised Growth Theorem. (Kristiansen, Weiermann,
Schlage-Puchta 2011). Let f and g be honest functions, and let
α ∈ SLim. Then,
f ≤αE g ⇔ f (x) ≤ gβ (x) for some fixed β < α .
The honest α-elementary degrees
We can define a degree structure for each α ∈ SLim:
The honest α-elementary degrees
We can define a degree structure for each α ∈ SLim:
We define the relation ≡αE by
f ≡αE g ⇔ f ≤αE g ∧ g ≤αE f .
The honest α-elementary degrees
We can define a degree structure for each α ∈ SLim:
We define the relation ≡αE by
f ≡αE g ⇔ f ≤αE g ∧ g ≤αE f .
≡αE is an equivalence relation on the honest functions.
The honest α-elementary degrees
We can define a degree structure for each α ∈ SLim:
We define the relation ≡αE by
f ≡αE g ⇔ f ≤αE g ∧ g ≤αE f .
≡αE is an equivalence relation on the honest functions.
The ≡αE -equivalence classes of honest functions are the
honest α-elementary degrees.
The honest α-elementary degrees
We can define a degree structure for each α ∈ SLim:
We define the relation ≡αE by
f ≡αE g ⇔ f ≤αE g ∧ g ≤αE f .
≡αE is an equivalence relation on the honest functions.
The ≡αE -equivalence classes of honest functions are the
honest α-elementary degrees.
. . . and we can investigate these structures by applying the
Generalised Growth Theorem.
The honest α-elementary degrees
We can prove that the structure of honest α-elementary degrees is
a lattice.
The honest α-elementary degrees
We can prove that the structure of honest α-elementary degrees is
a lattice.
The proof is a generalisation of the one for elementary degrees.
The honest α-elementary degrees
We can prove that the structure of honest α-elementary degrees is
a lattice.
The proof is a generalisation of the one for elementary degrees.
The operators min[·, ·] and max[·, ·] induce meet and join operators
on the honest α-elementary degrees.
The honest α-elementary degrees
What about jump operators?
The honest α-elementary degrees
What about jump operators?
For any honest functions f , g and α ∈ SLim, we can prove that
fα is honest
f ≤αE g ⇒ fα ≤αE gα (and thus f ≡αE g ⇒ fα ≡αE gα )
f < α fα
The honest α-elementary degrees
What about jump operators?
For any honest functions f , g and α ∈ SLim, we can prove that
fα is honest
f ≤αE g ⇒ fα ≤αE gα (and thus f ≡αE g ⇒ fα ≡αE gα )
f < α fα
Hence, α-iteration induce a jump operator on the honest
α-elementary degrees.
The honest α-elementary degrees
What about jump operators?
For any honest functions f , g and α ∈ SLim, we can prove that
fα is honest
f ≤αE g ⇒ fα ≤αE gα (and thus f ≡αE g ⇒ fα ≡αE gα )
f < α fα
Hence, α-iteration induce a jump operator on the honest
α-elementary degrees.
Indeed, if α ≤ β ∈ SLim, then β-iteration induce a jump operator
on the honest α-elementary degrees.
The honest α-elementary degrees
What about jump operators?
For any honest functions f , g and α ∈ SLim, we can prove that
fα is honest
f ≤αE g ⇒ fα ≤αE gα (and thus f ≡αE g ⇒ fα ≡αE gα )
f < α fα
Hence, α-iteration induce a jump operator on the honest
α-elementary degrees.
Indeed, if α ≤ β ∈ SLim, then β-iteration induce a jump operator
on the honest α-elementary degrees.
Thus, we can define many jump operators on the α-elementary
degrees. But the α-jump seems to be the most natural.
The honest α-elementary degrees
Thus, the honest α-elementary is a lattice (for α ≤ ǫ0 and
α ∈ SLim), with canonical degrees
0 < 0′ < 0′′ < 0′′′ < . . .
The honest α-elementary degrees
Thus, the honest α-elementary is a lattice (for α ≤ ǫ0 and
α ∈ SLim), with canonical degrees
0 < 0′ < 0′′ < 0′′′ < . . .
The Generalised Growth Theorem will be a very useful tool for
investigating this lattice.
The honest α-elementary degrees
Thus, the honest α-elementary is a lattice (for α ≤ ǫ0 and
α ∈ SLim), with canonical degrees
0 < 0′ < 0′′ < 0′′′ < . . .
The Generalised Growth Theorem will be a very useful tool for
investigating this lattice.
We expect that all the theorems we have proved on honest
elementary degrees also will hold for honest α-elementary degrees:
The honest α-elementary degrees
There exists Lown and Highn degrees (for any n ∈ N).
There exists intermediate degrees.
We have jump inversion theorems.
Density results: For any degrees a and b such that a < b, the
exists degrees c1 and c2 strictly between a and b such that
a = c1 ∪ c2 and b = c1 ∩ c2 .
Cupability and capability results.
. . . and so on ....
The honest α-elementary degrees
Open Problem
Is the structure of honest α-elementary degrees isomorphic to the
structure of honest elementary degrees (for any α ∈ SLim) ?
Relations to provability in PA
PART III
The honest ǫ0-elementary degrees and
provability in Peano Arithmetic.
Relations to provability in PA
Notation and basic definitions:
∆0 -statements and Σ1 -statements of the form A(x1 , . . . , xn , y ) will
be called representations. A representation A(x1 , . . . , xn , y ) is a
representation of a function φ(x1 , . . . , xn ) when
i
h
,...,xn ,y
⇔ φ(a1 , . . . , an ) = b
(*)
N |= A sax11,...,a
n ,b
Relations to provability in PA
Notation and basic definitions:
∆0 -statements and Σ1 -statements of the form A(x1 , . . . , xn , y ) will
be called representations. A representation A(x1 , . . . , xn , y ) is a
representation of a function φ(x1 , . . . , xn ) when
i
h
,...,xn ,y
⇔ φ(a1 , . . . , an ) = b
(*)
N |= A sax11,...,a
n ,b
If (*) holds and A is a ∆0 -statement, we will say that A is an
honest representation of φ. (Any honest function has an honest
representation.)
Relations to provability in PA
Notation and basic definitions:
∆0 -statements and Σ1 -statements of the form A(x1 , . . . , xn , y ) will
be called representations. A representation A(x1 , . . . , xn , y ) is a
representation of a function φ(x1 , . . . , xn ) when
i
h
,...,xn ,y
⇔ φ(a1 , . . . , an ) = b
(*)
N |= A sax11,...,a
n ,b
If (*) holds and A is a ∆0 -statement, we will say that A is an
honest representation of φ. (Any honest function has an honest
representation.)
For any representation A(~x , y ), let
tot(A) :≡ ∀~x ∃yA .
Relations to provability in PA
Let φ and ψ be functions.
We define the relation φ ≤PA ψ by
φ ≤PA ψ :⇔ for any representation Aψ of ψ
there exists a representation Bφ of φ such that
PA + tot(Aψ ) ⊢ tot(Bφ ).
Relations to provability in PA
Let φ and ψ be functions.
We define the relation φ ≤PA ψ by
φ ≤PA ψ :⇔ for any representation Aψ of ψ
there exists a representation Bφ of φ such that
PA + tot(Aψ ) ⊢ tot(Bφ ).
When φ ≤PA ψ, we will say that φ is provably total in ψ.
Relations to provability in PA
Theorem (Kristiansen, Weiermann, Schlage-Puchta 2011).
Let f and g be honest functions. Then, we have
f ≤PA g ⇔ f ≤ǫ0 E g .
Relations to provability in PA
Proof: f ≤PA g ⇒ f ≤ǫ0 E g .
Relations to provability in PA
Proof: f ≤PA g ⇒ f ≤ǫ0 E g .
(Claim I) Let h be an honest function, and let Ah be any honest
representation of h. Furthermore, let Bφ be a representation of the
function φ. If PA + tot(Ah ) ⊢ tot(Bφ ), then we have φ(x) ≤ hγ (x)
for some γ < ǫ0 . (End Claim)
Relations to provability in PA
Proof: f ≤PA g ⇒ f ≤ǫ0 E g .
(Claim I) Let h be an honest function, and let Ah be any honest
representation of h. Furthermore, let Bφ be a representation of the
function φ. If PA + tot(Ah ) ⊢ tot(Bφ ), then we have φ(x) ≤ hγ (x)
for some γ < ǫ0 . (End Claim)
The claim is proved by cut elimination in an operator controlled
calculus.
Relations to provability in PA
Proof: f ≤PA g ⇒ f ≤ǫ0 E g .
(Claim I) Let h be an honest function, and let Ah be any honest
representation of h. Furthermore, let Bφ be a representation of the
function φ. If PA + tot(Ah ) ⊢ tot(Bφ ), then we have φ(x) ≤ hγ (x)
for some γ < ǫ0 . (End Claim)
The claim is proved by cut elimination in an operator controlled
calculus.
Assume f ≤PA g . (Then, since g is honest, there exists an honest
representation of g .) By (Claim I), we have f (x) ≤ gα (x) where
α < ǫ0 , and then, by the Growth Theorem, we have f ≤ǫ0 E g .
Relations to provability in PA
Proof: f ≤PA g ⇐ f ≤ǫ0 E g .
Relations to provability in PA
Proof: f ≤PA g ⇐ f ≤ǫ0 E g .
(Claim II) Let ψ be any function ǫ0 -elementary in the honest
function g , and let Ag be any representation of g . Then, there
exists a representation Bψ of ψ such that PA + tot(Ag ) ⊢ tot(Bψ ).
(End Claim)
Relations to provability in PA
Proof: f ≤PA g ⇐ f ≤ǫ0 E g .
(Claim II) Let ψ be any function ǫ0 -elementary in the honest
function g , and let Ag be any representation of g . Then, there
exists a representation Bψ of ψ such that PA + tot(Ag ) ⊢ tot(Bψ ).
(End Claim)
This claim is proved by induction over the build-up of ψ from the
initial functions g , 0, S, Iin , max, 2x by composition, bounded
primitive recursion and α-iteration where α < ǫ0 . (Use that PA is
strong enough to carry out induction over every well-ordering
strictly less that ǫ0 . )
Relations to provability in PA
Proof: f ≤PA g ⇐ f ≤ǫ0 E g .
(Claim II) Let ψ be any function ǫ0 -elementary in the honest
function g , and let Ag be any representation of g . Then, there
exists a representation Bψ of ψ such that PA + tot(Ag ) ⊢ tot(Bψ ).
(End Claim)
This claim is proved by induction over the build-up of ψ from the
initial functions g , 0, S, Iin , max, 2x by composition, bounded
primitive recursion and α-iteration where α < ǫ0 . (Use that PA is
strong enough to carry out induction over every well-ordering
strictly less that ǫ0 . )
It follows straightaway from the claim that f ≤PA g ⇐ f ≤ǫ0 E g .
Relations to provability in PA
Thus, given certain restrictions on the meaning of
PA + Tot(g ) ⊢ Tot(f ) (we must be careful in this respect), we
have
f ≤ǫ0E g ⇔ PA + Tot(g ) ⊢ Tot(f ) .
for any honest functions f and g .
Relations to provability in PA
Thus, given certain restrictions on the meaning of
PA + Tot(g ) ⊢ Tot(f ) (we must be careful in this respect), we
have
f ≤ǫ0E g ⇔ PA + Tot(g ) ⊢ Tot(f ) .
for any honest functions f and g .
In words: f is ǫ0 -elementary in g if, and only if, f is provable total
in g
Relations to provability in PA
Thus, given certain restrictions on the meaning of
PA + Tot(g ) ⊢ Tot(f ) (we must be careful in this respect), we
have
f ≤ǫ0E g ⇔ PA + Tot(g ) ⊢ Tot(f ) .
for any honest functions f and g .
In words: f is ǫ0 -elementary in g if, and only if, f is provable total
in g if, and only if, f is dominated by gα for some α < ǫ0 .
Relations to provability in PA
Hence, each honest ǫ0 -elementary degree corresponds to a
statement independent of PA, and the structure of honest
ǫ0 -elementary degrees yields such statements with interesting
properties.
Relations to provability in PA
Hence, each honest ǫ0 -elementary degree corresponds to a
statement independent of PA, and the structure of honest
ǫ0 -elementary degrees yields such statements with interesting
properties.
For instance, we know that there exists minimal pair, that is, we
have degrees a, b > 0 such that a ∩ b = 0.
This translates to the following theorem:
Relations to provability in PA
Hence, each honest ǫ0 -elementary degree corresponds to a
statement independent of PA, and the structure of honest
ǫ0 -elementary degrees yields such statements with interesting
properties.
For instance, we know that there exists minimal pair, that is, we
have degrees a, b > 0 such that a ∩ b = 0.
This translates to the following theorem:
Theorem. There exist two computable functions f0 and f1 not
provable total in PA such that any function provable total in both
PA + Tot(f0 ) and PA + Tot(f1 ) also will be provable total in PA.
Relations to provability in PA
Beware the definition of ≤PA .
Relations to provability in PA
Beware the definition of ≤PA .
Here comes again:
φ ≤PA ψ :⇔ for any representation Aψ of ψ
there exists a representation Bφ of φ such that
PA + tot(Aψ ) ⊢ tot(Bφ ).
Relations to provability in PA
Beware the definition of ≤PA .
Here comes again:
φ ≤PA ψ :⇔ for any representation Aψ of ψ
there exists a representation Bφ of φ such that
PA + tot(Aψ ) ⊢ tot(Bφ ).
Beware that our degrees are degrees of honest functions.
Relations to provability in PA
A related degree structure is recently introduced by Mingzhong Cai
(Cornell University).
Relations to provability in PA
A related degree structure is recently introduced by Mingzhong Cai
(Cornell University).
Let φ0 , φ1 , φ2 , . . . be a standard enumeration of the computable
functions.
Relations to provability in PA
A related degree structure is recently introduced by Mingzhong Cai
(Cornell University).
Let φ0 , φ1 , φ2 , . . . be a standard enumeration of the computable
functions.
Let
φi ≤p φj ⇔ PA + tot(φj ) ⊢ tot(φi ) .
Relations to provability in PA
A related degree structure is recently introduced by Mingzhong Cai
(Cornell University).
Let φ0 , φ1 , φ2 , . . . be a standard enumeration of the computable
functions.
Let
φi ≤p φj ⇔ PA + tot(φj ) ⊢ tot(φi ) .
The reducibility relation ≤p induce a degree structure on the
Π02 -statements of PA.
Relations to provability in PA
A related degree structure is recently introduced by Mingzhong Cai
(Cornell University).
Let φ0 , φ1 , φ2 , . . . be a standard enumeration of the computable
functions.
Let
φi ≤p φj ⇔ PA + tot(φj ) ⊢ tot(φi ) .
The reducibility relation ≤p induce a degree structure on the
Π02 -statements of PA.
These degrees are degrees of algorithms (and not degrees of
functions), e.g., any degree will contain a Π02 -statement stating
that the trivial function f (x) = 0 is total.
Relations to provability in PA
Cai’s proof methods are very different from ours: They are based
on classical computability-theoretic methods.
Relations to provability in PA
Cai’s proof methods are very different from ours: They are based
on classical computability-theoretic methods.
However, Cai’s structure do also turn out to be a lattice.
Relations to provability in PA
Cai’s proof methods are very different from ours: They are based
on classical computability-theoretic methods.
However, Cai’s structure do also turn out to be a lattice.
Research for the future: Study the relationship between Cai’s
degree structure of algorithms and our structure of honest
functions.
Welcome to the future!
THE FUTURE IS NOW.
Degrees of Total Algorithms
versus
Degrees of Honest Functions
PA-provability degrees of algorithms
PA-provability degrees of algorithms
Assume some standard enumeration of the Turing machines.
PA-provability degrees of algorithms
Assume some standard enumeration of the Turing machines.
Let tot(Φe ) be a Π02 formula of Peano Arithmetic (PA) such that
N |= tot(Φe ) if and only if
the e th Turing machine computes a total function.
PA-provability degrees of algorithms
We define the set of total algorithms, written Atot , by
Atot = {i | N |= tot(Φi )}
PA-provability degrees of algorithms
We define the set of total algorithms, written Atot , by
Atot = {i | N |= tot(Φi )}
and we define the reducibility relation ≤p over the total algorithms
by
i ≤p j : ⇔ PA + tot(Φj ) ⊢ tot(Φi ) .
PA-provability degrees of algorithms
We define the set of total algorithms, written Atot , by
Atot = {i | N |= tot(Φi )}
and we define the reducibility relation ≤p over the total algorithms
by
i ≤p j : ⇔ PA + tot(Φj ) ⊢ tot(Φi ) .
The PA-provability degrees of (total) algorithms are the
equivalence classes induced on Atot by ≤p .
Total algorithms versus honest functions
We define honest associate of the total algorithm e, written ψe , by
ψe (x)
=
max( 2x , max stepe (z) )
z≤x
where stepe (z) is the number of steps the e th algorithm requires
to terminate on input z.
Total algorithms versus honest functions
We define honest associate of the total algorithm e, written ψe , by
ψe (x)
=
max( 2x , max stepe (z) )
z≤x
where stepe (z) is the number of steps the e th algorithm requires
to terminate on input z.
REMARK: ψe is an honest function iff e is a total algorithm.
Total algorithms versus honest functions
The first-order theory PA+ is PA extended with all true
Π1 -statements.
Total algorithms versus honest functions
The first-order theory PA+ is PA extended with all true
Π1 -statements.
We define the reducibility relation ≤p+ over the total algorithms by
i ≤p+ j ⇔ PA+ + tot(Φj ) ⊢ tot(Φi ) .
Total algorithms versus honest functions
The first-order theory PA+ is PA extended with all true
Π1 -statements.
We define the reducibility relation ≤p+ over the total algorithms by
i ≤p+ j ⇔ PA+ + tot(Φj ) ⊢ tot(Φi ) .
The PA+ -provability degrees of (total) algorithms are the
equivalence classes induced on Atot by ≤p+ .
Total algorithms versus honest functions
Growth Theorem for Algorithms (Kristiansen, 2012).
For any i , j ∈ Atot , we have
i ≤p+ j
⇔
ψi (x) ≤ (ψj )α (x) for some α < ǫ0 .
Total algorithms versus honest functions
Growth Theorem for Algorithms (Kristiansen, 2012).
For any i , j ∈ Atot , we have
i ≤p+ j
⇔
ψi (x) ≤ (ψj )α (x) for some α < ǫ0 .
Isomorphism Theorem (Kristiansen, 2012).
The structure of PA+ -degrees of algorithms is isomorphic to the
structure of honest ǫ0 -elementary degrees.
Total algorithms versus honest functions
Corollary.
Let i , j ∈ Atot and recall that ψi and ψj are the honest associates
of respectively i and j. The following statements are equivalent:
Total algorithms versus honest functions
Corollary.
Let i , j ∈ Atot and recall that ψi and ψj are the honest associates
of respectively i and j. The following statements are equivalent:
(i) i ≤p+ j
Total algorithms versus honest functions
Corollary.
Let i , j ∈ Atot and recall that ψi and ψj are the honest associates
of respectively i and j. The following statements are equivalent:
(i) i ≤p+ j
(ii) ψi (x) ≤ (ψj )α (x) for some α < ǫ0
Total algorithms versus honest functions
Corollary.
Let i , j ∈ Atot and recall that ψi and ψj are the honest associates
of respectively i and j. The following statements are equivalent:
(i) i ≤p+ j
(ii) ψi (x) ≤ (ψj )α (x) for some α < ǫ0
(iii) ψi ≤ǫ0 E ψj
Total algorithms versus honest functions
Corollary.
Let i , j ∈ Atot and recall that ψi and ψj are the honest associates
of respectively i and j. The following statements are equivalent:
(i) i ≤p+ j
(ii) ψi (x) ≤ (ψj )α (x) for some α < ǫ0
(iii) ψi ≤ǫ0 E ψj
(iv) For any representation A of ψj there exists a representation B
of ψi such that PA + tot(A) ⊢ tot(B)
Total algorithms versus honest functions
Corollary.
Let i , j ∈ Atot and recall that ψi and ψj are the honest associates
of respectively i and j. The following statements are equivalent:
(i) i ≤p+ j
(ii) ψi (x) ≤ (ψj )α (x) for some α < ǫ0
(iii) ψi ≤ǫ0 E ψj
(iv) For any representation A of ψj there exists a representation B
of ψi such that PA + tot(A) ⊢ tot(B)
(v) For any algorithm j ′ computing ψj there exists an algorithm i ′
computing ψi such that i ′ ≤p j ′ .
Thanks for you attention!
Thanks for you attention!
This talk was based on joint work with
J.-C. Schlage-Puchta (Ghent)
A. Weiermann (Ghent).