gödel's koan and gentzen's second consistency proof · luiz carlos pereira, daniel...

Godel’s Koan and Gentzen’s Second ConsistencyProof

Luiz Carlos Pereira1 Daniel Durante2

Edward Hermann Haeusler3

1Department of PhilosophyPUC-Rio/UERJ

2Department of PhilosophyUFRN

3Department of Computer SciencePUC-Rio

Logic Colloquium, 2018Udine

Luiz Carlos Pereira, Daniel Durante , Edward Hermann Haeusler Godel’s Koan and Gentzen’s Second Consistency Proof

Godel’s Koan

“A koan is a story, dialogue, question, or statement, which is used in Zenpractice to provoke the ”great doubt” and test a student’s progress inZen practice.” (Wikipedia)


Problem 26 - TLCA list of open problemsSubmitted by Henk BarendregtDate: 2014Statement: Assign (in an ‘easy’ way) ordinals to terms of the simplytyped lambda calculus such that reduction of the term yields a smallerordinal.Problem Origin: First posed by Kurt Godel.Construct an easy assignment of (possibly transfinite) ordinals to termsof the simply typed lambda calculus, i.e., a mapF : Λ→ ⇒ {α : α is an ordinal} such that

∀M,N ∈ Λ→ [M →β N ⇒ F[M] < F[N]].


“As the problem is formulated it contains an element of vagueness as it ispresented as the problem of finding a simple or easy ordinal assignmentfor strong normalization of the beta-reduction of simply typed lambdacalculus. Whether a proof is sufficiently easy to categorize as a solution isthus a matter of opinion.” (Annika Kanckos, Logic Colloquium,Stockholm, 2017)


Previous Results

• Howard [1968]

• de Vrijer [1987]

• Durante [1999]

• Beckmann [2001]

• Sanz [2006]

And quite recently, Annika Kanckos [2017]


Some possible solutions

• Worst reduction sequences

• ∗ − derivations (disastrous derivations)

• Minp-graphs (Cruz, Haeusler, and Gordeev)

• Gentzen reductions (Pereira and Haeusler)


Worst reduction sequences

1 Define the concept of worst reduction sequence

2 Prove that, for any derivation Π, the worst reduction sequence for Πis finite.

3 Define lp[Π] as the length of the worst reduction sequence for Π.

4 Define for any derivation Π the measure on[Π] as: on[Π] = lp[Π].

5 Show that if Π reduces to Π′, then on[Π′] < on[Π].

Problem: The measure on depends on a normalization strategy


Disastrous derivations

General description of the method

Main idea - to associate to a given derivation Π aderivation Π∗ such that all possible maximumformulas that may arise in reduction sequencesstarting with Π occur in Π∗.


1 Applications of → −Int that do not discharge assumptions produce“vacuous reductions”, while those that discharge assumptionsproduce “multiplicative reductions”.

2 A derivation that can only produce “vacuous reductions” is called∗ − derivations.

3 Any reduction applied to a ∗ − derivation produces a decrease in itslength.

4 Define a method to check if a derivation is a ∗ − derivation (this isdone by means of α− segments.


5 Associate to a given derivation Π a ∗ − derivation Π∗. Thisderivation Π∗ will contain all possible maximum formulas of Π[occurring as pair of formula occurrences, the α pairs].

6 We can now “count” the number of such pairs (of all possiblemaximum formulas of Π).

7 This number will be the natural ordinal of Π, on(Π).

8 Clearly, if Π reduces to Π′, then on(Π′) < on(Π)


Basic definions1 Definition: A derivation Π is said to be a star-derivation iff∀Π′ such that Π reduces to Π′, l(Π′) < l(Π).

2 The notion of α− segment will allow us to discoverwhether a derivation Π is a star derivation or not.


3 A segment in a derivation Π is a sequence A1, A2, ..., An ofconsecutive formula occurrences in a thread in Π.

4 Let S be the segment A1, ..., An. The center of S, denoted by c(S),is the rational number given by: c(S) = (n+ 1)/2. If c(S) is aninteger, then Ac(S) is called the central occurrence of S.

5 Let S = A1, ..., An be a segment in a derivation Π of centraloccurrence Ai. We say that S is an α− segment of level 1 in Π if,for all j such that 1 ≤ j ≤ i = c(S), Aj (an occurrence in the firsthalf of S) and An−j+1 (an occurrence of the same formula in thesecond half of S symmetric to Aj), where Aj is the consequence ofan introduction rule and An−j+1 is the major premise of anelimination rule..


α− segment of level 1

Consider the following derivation:


α− segment of level 1

The segment

(A∧B), ((A∧B)∧C), (((A∧B)∧C)∧D), ((A∧B)∧C), (A∧B)

is an α− segment of level 1. The formula ((A ∧B) ∧ C) is acandidate to be a maximum formula.


α− segment


α− segment of arbitrary level



An α− segment of level 2

S = (A ∧B)2, ((A ∧B) ∧ C)3, (((A ∧B) ∧ C) ∧D)4, ((A ∧B) ∧ C)5,(A ∧B)6, (C → (A ∧B))7, (D → (C → (A ∧B)))8,(C → (A ∧B))9, (A ∧B)10.


This derivation reduces to:


The moral is: sequences of segments separated byoccurrences of the same formula may also determinepairs of formula occurrences that reductions mayturn into maximum formulas!


Definition

We say that an occurrence of a formula A in a derivation Π is heavy inΠ iff A is the major premiss of an elimination rule and belongsto some α− segment in Π.

Remark: If A is heavy in Π and is not a maximum formula in Π, then Ais a candidate to be a maximum formula in Π.


Non-multiplicative reductions

Some → −reductions reduces the size of derivations. These are callednon-multiplicative reductions.

Π1

A

Π2

B(A→ B)

BΠ3

Reduces to:

Π2

[B]

Π3


Multiplicative occurrences



Definition

A derivation Π is said to be a star-derivation iff ∀Π′ such that Π reducesto Π′, l(Π′) < l(Π).

Theorem

Let Π be a derivation in I→. Then, there is a unique ∗ − derivationΠ∗associated to Π.


Definition

Let Π be a derivation in I→. The weight of Π, w(Π), is defined as thenumber of heavy formula occurrences in Π.

Definition

Let Π be a derivation in I→ and let Π∗ be the unique ∗ − derivationassociated to Π. The natural ordinal of Π, no(Π), is defined as: on(Π) =w(Π∗)


Theorem

Let Π be a derivation in I→. If Π reduces to Π′, then on(Π′) < on(Π).


Minp-graphs

Main idea: to use proof-graphs to represent proofs in the implicationalfragment of minimal logic!


Minp-graphs


Theorem

Every standard tree-like natural deduction Π in the implicationalfragment of minimal logic has a unique (up to graph-isomorphism)[F-minimal] Mimp-like representation G[Π]

Definition

Let G be a Minp-graph. We define Nmax(G) as the number of maximalformulas in G.

Theorem

Let G be a Minp-graph. If G reduces to G′, then Nmax(G′) <Nmax(G).

Important: The measure does not depend on anynormalization strategy


Gentzen’s second (published) consistency

proof 1938 - The New Version


The New Version

Gentzen‘s New Version can be roughly described asfollows:

1 Consider an LK like formulation of arithmetic.

2 Define an assignment Ord of ordinals < ε0 to proofs in the system.

3 Define a set of reduction operations OP.

4 Show that if there is a proof Π in the system of the empty sequent“⇒”, then there is always an operation op ∈ OP such op[Π] is aproof of “⇒” and Ord[op[Π]] < Ord[Π].

5 The result immediately follows by transfinite induction up to ε0.


The New Version and cut-elimination

In 1982 it was showed how the techniques and methodology used byGentzen in NV could be used to obtain cut-elimination results for sequentcalculi for classical first order logic (LK) and for intuitionistic first orderlogic (LJ). In fact, it was shown, somewhat surprisingly, how theassignment used by Gentzen produces an interesting measure for theestimation of the length of normal proofs in these calculi. Prawitz provedthe same result for Natural Deduction in 2015.


Gentzen’s New version and strong cut-elimination

More interesting: we can use the reductionsof the New Version to obtain a strongcut-elimination result for the propositionalpart of LK.


Operational Reductions

Let Π Be:

Π1

Γ1 ⇒ ∆1, A

Π2

Γ1 ⇒ ∆1, B

Γ1 ⇒ ∆1, A ∧BΠ3

Γ⇒ ∆, A ∧B

Σ1

A,Θ1 ⇒ Ψ1

A ∧B,Θ1 ⇒ Ψ1

Σ2

A ∧B,Θ⇒ Ψ

Γ,Θ⇒ ∆,Ψ

Σ3

Γ3 ⇒ Θ3

Σ4


The derivation Π reduces to the following derivation Π′:

Π1

Γ1 ⇒ ∆1, A

Γ1 ⇒ ∆1, A,A ∧ B

Π′3

Γ ⇒ ∆, A, A ∧ B

Σ1

A, Θ1 ⇒ Ψ1

A ∧ B, Θ1 ⇒ Ψ1

Σ2

A ∧ B, Θ ⇒ Ψ

Γ, Θ ⇒ ∆, A, Ψ

Σ′′3

Γ3 ⇒ Θ3, A

Π1

Γ1 ⇒ ∆1, A

Π2

Γ1 ⇒ ∆1, B

Γ1 ⇒ ∆1, A ∧ B

Π3

Γ ⇒ ∆, A ∧ B

Σ1

A, Θ1 ⇒ Ψ1

A ∧ B,A, Θ1 ⇒ Ψ1

Σ′2

A ∧ B,A, Θ ⇒ Ψ

Γ, A, Θ ⇒ ∆, Ψ

Σ′3

A, Γ3 ⇒ Θ3

Γ3, Γ3 ⇒ Θ3, Θ3

.

.

.

Γ3 ⇒ Θ3

Σ4


Theorem

(Strong Cut-Elimination): Every reduction sequence for a derivation Π inLKP is finite.

Proof.

By induction over the value G3(Π) associated with a derivation Π. Weshow that if Π reduces to Π′ then, G3(Π) < G3(Π′).


Can we extend this strong cut-elimination result to the implicationalfragment of LJ? No, not directly!!

Gentzen’s reduction takes us out of LJ!!

Possible solution:

The system FIL


THE SYSTEM FIL

AxiomA(n) ` A{n}

Γ ` A/S,∆′ A(n),Γ′ ` ∆Cut

Γ,Γ′ ` ∆′,∆∗

Γ, A(n), B(m),Γ′ ` ∆ExL

Γ, B(m), A(n),Γ′ ` ∆

Γ ` A/S,B/S′,∆ExR

Γ ` B/S′, A/S,∆

Γ ` ∆WL

Γ, A(n) ` ∆∗Γ ` ∆

WRΓ ` A/{ },∆

Γ, A(n), A(m) ` ∆ConL

Γ, A(k) ` ∆∗Γ ` A/S,A/S′∆

ComRΓ ` A/S ∪ S′,∆

Γ ` A/S,∆ Γ′, B(n) ` ∆′→LΓ,Γ′, A→ b(n) ` ∆,∆′

Γ, A(n) ` B/S,∆ →RΓ ` (A→ B)/S − {n},∆


Similar problems appear in FIL: reductions may takeus out of FIL!!Solution: define FIL-notations


Conclusions

• Minp-graphs appear as an interesting possibility - the measure doesnot depend on specific normalization strategies!

• The ∗ − derivation strategy does not depend on a normalizationstrategy, but the method used to associate a ∗ − derivation Π∗ to aderivation Π is too close to a normalization strategy.

• We can define FIL-notations (analogous to Scarpellini’s “almostintuitionistic derivations”).

• We can define an (natural number) assignment G that establishesthe desired result for FIL-notations.

• We can map FIL-derivations into FIL-notations.


gödel's koan and gentzen's second consistency proof · luiz carlos pereira, daniel...

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