An Arithmetical Hierarchy of the Laws of Excluded Middle and Related Principles LICS 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 Science 2.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. Effective Bound Extraction from partially non-constructive proofs: (see Ulrich Kohlenbach’s homepage. 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 Theorem 2.Markov’s Principle 3.  0 1 -Lesser Limited Principle of Omniscience. 4.Excluded Middle for  0 1 -predicates 5.Excluded Middle for  0 1 -predicates

Post’s Theorem Markov Principle  0 1 -L.L.P.O.  0 1 -Ex. Middle  0 1 -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,Q  0 1  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 P  0 1  z:  x.P(x,z)   x.P(x,z) Markov’s Principle is independent from  0 1 -Lesser Limited Principles of Omniscience in HA.

 0 1 -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,Q  0 1  z:  x,y.(P(x,z)  Q(y,z))   x.P(x,z)   y.Q(y,z)

 L.L.P.O and Weak Koenig’s Lemma  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 P  0 1  z:  x.P(x,z)   x.P(x,z)  0 1 -E.M. is, in HA, stronger than  0 1 -LLPO (i.e., than Koenig’s Lemma), but weaker than  0 1 -E.M..

Excluded Middle for  0 1 -predicates “Excluded Middle holds for all positively decidable statements” Equivalently, in HA: for any P  0 1  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  0 n -Ex. Middle  0 n -Ex. Middle Theorem 2. For all n, the only implications provable in HA are:  0 n-1 -Ex. Middle …