An Arithmetical Hierarchy of the Laws of Excluded

Middle and Related PrinciplesLICS 2004, Turku, Finland

Yohji Akama (Tohoku University)

Stefano Berardi (Turin University)

Susumu Hayashi (Kobe University)

Ulrich Kohlenbach (Darmstadt University)

Acknoledgements

Our research was supported by: 1. the Grant in Aid for Scientific Research of

Japan Society of the Promotion of Science2. the McTati Research Project (constructive

methods in Topology, Algebra and Computer Science).

3. the Grant from the Danish Natural Science Research Council.

The subject of this talk

We are concerned with classifying classical principles from a constructive viewpoint.

Some motivations for our research work

• Limit Interpretation for non-constructive proofs: see Susumu Hayashi’s homepage.

http://www.shayashi.jp/PALCM/index-eng.html

• Effective Bound Extraction from partially non-constructive proofs: (see Ulrich Kohlenbach’s homepage.

http://www.mathematik.tu-darmstadt.de/~kohlenbach/novikov.ps.gz

Some Classical Principles we are concerned with

We compare up to provability in HA (Heyting’s Intuitionistic Arithmetic):

1. Post’s Theorem2. Markov’s Principle

3. 01-Lesser Limited Principle of

Omniscience.

4. Excluded Middle for 01-predicates

5. Excluded Middle for 01-predicates

Post’s Theorem

Markov Principle

01-L.L.P.O.

01-Ex. Middle

01-Ex. Middle

Theorem 1. The only implications provable in HA are:

No principle in this picture

is provable in HA

Post’s Theorem

“Any subset of N which both positively and negatively decidable is decidable”

• Equivalently, in HA: for any P,Q01

z: ( x.P(x,z) y.Q(y,z) ) x.P(x,z) x. P(x,z)

• Post’s Theorem is not derivable in HA. It is strictly weaker in HA than any other classical principle we considered.

Markov’s Principle

“Any computation which does not run foverer eventually stops”

• Equivalently, in HA: for any P01

z: x.P(x,z) x.P(x,z)

• Markov’s Principle is independent from 01-

Lesser Limited Principles of Omniscience in HA.

01-Lesser Limited Principles of

Omniscience“If two positively decidable statements are

not both true, then some of them is false”

• Equivalently, in HA: for any P,Q01

z: x,y.(P(x,z) Q(y,z))

x.P(x,z) y.Q(y,z)

01- L.L.P.O and

Weak Koenig’s Lemma0

1- L.L.P.O is equivalent, in HA+Choice, to:

Weak Koenig’s Lemma for recursive trees

“any infinite binary recursive tree has some infinite branch”

Excluded Middle for 0

1-predicates“Excluded Middle holds for all

negatively decidable statements”

• Equivalently, in HA: for any P01

z: x.P(x,z) x.P(x,z)0

1-E.M. is, in HA, stronger than 01-LLPO

(i.e., than Koenig’s Lemma), but weaker than 0

1-E.M..

Excluded Middlefor 0

1-predicates“Excluded Middle holds for all

positively decidable statements”

• Equivalently, in HA: for any P01

z: x.P(x,z) x.P(x,z) 0

1-E.M. is stronger, in HA, than all classical principles we considered until now.

Generalizing to higher degrees

• For each principle there is a degree n version, for degree n formulas.

• For degree n principles we proved the same classification results we proved for the originary principles.

n-Post’s Theorem

n-Markov’s Principle

n-Koenig’s Lemma

0n-Ex. Middle

0n-Ex. Middle

Theorem 2. For all n, the only implications provable

in HA are:

0n-1-Ex. Middle

… …