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Herbrand's Theorem via Hyper anoni al Extensions

Kazushige Terui

RIMS, Kyoto University

24/09/13, Gudauri

Outline

Introdu tion to

Substru tural Logi s

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

2 / 55

1. Introdu tion to substru tural logi s

2. Algebrai proof theory for substru tural logi s

3. Herbrand's theorem via hyper anoni al extensions

Introdu tion to Substru tural Logi s

Introdu tion to

Substru tural

Logi s

What are SLs?

Full Lambek al ulus

Residuated Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

3 / 55

What are substru tural logi s?

Introdu tion to

. What are SLs?

Full Lambek al ulus

Residuated Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

4 / 55

� Logi s that may la k some of stru tural rules

(ex hange/weakening/ ontra tion)

� Axiomati extensions of Full Lambek Cal ulus FL

(= non ommutative intuitionisti linear logi without !)

� Study of universe of logi s

Why is the subje t interesting?

� Common basis for various non lassi al logi s

linear, BI, relevant, fuzzy, superintuitionisti logi s

� Common basis for various ordered algebras

latti e-ordered groups, relation algebras, ideal latti es of

rings, MV algebras, Heyting algebras

� Abundan e of weird logi s/algebras

- allows us to speak of riteria for various properties

Introdu tion to

. What are SLs?

Full Lambek al ulus

Residuated Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

4 / 55

Introdu tion to

. What are SLs?

Full Lambek al ulus

Residuated Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

4 / 55

Introdu tion to

. What are SLs?

Full Lambek al ulus

Residuated Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

4 / 55

Introdu tion to

. What are SLs?

Full Lambek al ulus

Residuated Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

4 / 55

Introdu tion to

. What are SLs?

Full Lambek al ulus

Residuated Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

4 / 55

Full Lambek al ulus FL

Introdu tion to

What are SLs?

Full Lambek

al ulus

Residuated Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

5 / 55

� The base system for substru tural logi s (Ono 90)

� Intuitionisti logi without stru tural rules.

� Formulas:

'; ::= p j ' ^ j ' _ j ' � j 'n j '= j 1 j 0

� 0 is used to de�ne negations:

�a = an0; � a = 0=a:

� Sequents: �) �

(�: sequen e of formulas, �: at most one formula)

Inferen e Rules of FL

Introdu tion to

What are SLs?

Full Lambek

al ulus

Residuated Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

6 / 55

�) A �

; A;�

;�;�

; A;�

) � �

; B;�

; A _B;�

�) A

�) A �) B

�) A ^B

; A;B;�

; A � B;�

(�l)

�) A �) B

�;�) A �B

(�r)

�) A �

; B;�

;�; AnB;�

A;�) B

�) AnB

�) A �

; B;�

; B=A;�;�

�; A) B

�) B=A

�) �

; 1;�

�) 0

Introdu tion to

What are SLs?

Full Lambek

al ulus

Residuated Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

7 / 55

� Given a set � [ fsg of sequents, � `

s if s is derivable

from �.

� We often identify formula ' with sequent ) '.

� We often write '! for 'n and ='.

� A substru tural logi is an axiomati extension of FL.

Some axioms:

(e) ' � ! � ' (ex hange)

(w) '! 1; 0! ' (weakening)

( ) '! ' � ' ( ontra tion)

(in) ::'! ' (involutivity)

(dist) ' ^ (

)! (' ^

) _ (' _

) (distributivity)

(pl) ('! ) _ ( ! ') (prelinearity)

(div) ' ^ ! ' � ('! ) (divisibility)

Introdu tion to

What are SLs?

Full Lambek

al ulus

Residuated Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

7 / 55

s if s is derivable

from �.

Some axioms:

(e) ' � ! � ' (ex hange)

(w) '! 1; 0! ' (weakening)

( ) '! ' � ' ( ontra tion)

(dist) ' ^ (

)! (' ^

) _ (' _

) (distributivity)

Introdu tion to

What are SLs?

Full Lambek

al ulus

Residuated Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

7 / 55

s if s is derivable

from �.

Some axioms:

(e) ' � ! � ' (ex hange)

(w) '! 1; 0! ' (weakening)

( ) '! ' � ' ( ontra tion)

(dist) ' ^ (

)! (' ^

) _ (' _

) (distributivity)

Introdu tion to

What are SLs?

Full Lambek

al ulus

Residuated Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

7 / 55

s if s is derivable

from �.

Some axioms:

(e) ' � ! � ' (ex hange)

(w) '! 1; 0! ' (weakening)

( ) '! ' � ' ( ontra tion)

(dist) ' ^ (

)! (' ^

) _ (' _

) (distributivity)

Residuated Latti es

Introdu tion to

What are SLs?

Full Lambek al ulus

Residuated

Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

9 / 55

Residuated latti e: A = hA;^;_; �; n; =; 1i su h that

� hA;^;_i is a latti e;

� hA; �; 1i is a monoid;

� a � b � , b � an , a � =b.

An FL algebra is a residuated latti e with onstant 0 2 A.

Some identities:

� a � (anb) � b

� (a _ b) � = (a � ) _ (b � )

� a! (b ^ ) = (a! b) ^ (a! )

� (a _ b)! = (a! ) ^ (b! )

� anb = b=a (with (e))

� a � b � a ^ b (with (w))

� a � b � a ^ b (with ( ))

Residuated Latti es

Introdu tion to

What are SLs?

Full Lambek al ulus

Residuated

Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

9 / 55

� a � b � , b � an , a � =b.

Some identities:

� a � (anb) � b

� (a _ b) � = (a � ) _ (b � )

� a! (b ^ ) = (a! b) ^ (a! )

� (a _ b)! = (a! ) ^ (b! )

� a � b � a ^ b (with (w))

� a � b � a ^ b (with ( ))

Residuated Latti es

Introdu tion to

What are SLs?

Full Lambek al ulus

Residuated

Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

9 / 55

� a � b � , b � an , a � =b.

Some identities:

� a � (anb) � b

� (a _ b) � = (a � ) _ (b � )

� a! (b ^ ) = (a! b) ^ (a! )

� (a _ b)! = (a! ) ^ (b! )

� a � b � a ^ b (with (w))

� a � b � a ^ b (with ( ))

Algebraization

Introdu tion to

What are SLs?

Full Lambek al ulus

Residuated

Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

10 / 55

A lass V of algebras (of the same type) is a variety if

V = HSP (V):

� H: homomorphi images

� S: subalgebras

� P : dire t produ ts

Birkho�'s Theorem

V is a variety i� V is equationally de�nable.

FL := the variety of FL algebras.

Algebraization Theorem

The substru tural logi s are in 1-1 orresponden e with the sub-

varieties of FL. If L orresponds to V,

i� 1 � � j=

1 � :

Algebraization

Introdu tion to

What are SLs?

Full Lambek al ulus

Residuated

Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

10 / 55

V = HSP (V):

� S: subalgebras

Birkho�'s Theorem

i� 1 � � j=

1 � :

Algebraization

Introdu tion to

What are SLs?

Full Lambek al ulus

Residuated

Latti es

Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

10 / 55

V = HSP (V):

� S: subalgebras

Birkho�'s Theorem

i� 1 � � j=

1 � :

Example of resear h topi s: omputational omplexity

Introdu tion to

What are SLs?

Full Lambek al ulus

Residuated Latti es

. Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

12 / 55

Let L be a onsistent substru tural logi and onsider the

de ision problem:

Given '; `

Theorem (Hor ik-T. 11)

1. Any L is oNP-hard.

2. If L is �nite-valued, then L is oNP- omplete.

3. If L enjoys the disjun tion property, then L is PSPACE-hard.

(Neither 2. nor 3. is a ne essary ondition.)

oNP and PSPACE seem a natural way to lassify logi s into

\semanti ally easy" and \ omputationally expressive" ones.

Di hotomy Problem

Is there a substru tural logi whi h is neither oNP- omplete nor

PSPACE-hard?

Introdu tion to

What are SLs?

Full Lambek al ulus

Residuated Latti es

. Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

12 / 55

de ision problem:

Given '; `

Di hotomy Problem

PSPACE-hard?

Introdu tion to

What are SLs?

Full Lambek al ulus

Residuated Latti es

. Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

12 / 55

de ision problem:

Given '; `

Di hotomy Problem

PSPACE-hard?

Introdu tion to

What are SLs?

Full Lambek al ulus

Residuated Latti es

. Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

12 / 55

de ision problem:

Given '; `

Di hotomy Problem

PSPACE-hard?

Introdu tion to

What are SLs?

Full Lambek al ulus

Residuated Latti es

. Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

12 / 55

de ision problem:

Given '; `

Di hotomy Problem

PSPACE-hard?

Introdu tion to

What are SLs?

Full Lambek al ulus

Residuated Latti es

. Complexity

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

12 / 55

de ision problem:

Given '; `

Di hotomy Problem

PSPACE-hard?

Algebrai Proof Theory for Substru tural

Logi s

Introdu tion to

Algebrai Proof

Theory for

Substru tural

Logi s

APT for SL

Completions

Ma Neille omp.

Residuated frames

From axioms to rules

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

13 / 55

Substru tural proof theory is dying ...

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

14 / 55

� Full of bureau ra y and ad ho studies,

� Few of appli ations,

� Nevertheless there are some brilliant ideas.

Our aim: to salvage those brilliant ideas from the sea of

bureau ra y.

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

14 / 55

bureau ra y.

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

14 / 55

bureau ra y.

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

14 / 55

bureau ra y.

Algebrai proof theory for SL

Introdu tion to

Algebrai Proof

Theory for

. APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

15 / 55

� Collaborators: Paolo Baldi, Agata Ciabattoni,

Nikolaos Galatos, Rostislav Hor� ��k, Lutz Stra�burger, . . .

� Explore the onne tion between proof theory and ordered

algebra.

{ Uniform proof theory

{ Appli ations to ordered algebra

� Core: ut elimination � algebrai ompletion

� Origin:

{ omputability/redu ibility argument (Tait/Girard)

{ phase semanti ut elimination (Okada 96)

{ algebrai meaning of ut elimination

(Belardinelli-Jipsen-Ono 04)

{ residuated frames (Jipsen-Galatos 13)

� Today I will fo us on ut elimination � algebrai ompletion.

Introdu tion to

Algebrai Proof

Theory for

. APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

15 / 55

algebra.

� Origin:

Introdu tion to

Algebrai Proof

Theory for

. APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

15 / 55

algebra.

� Origin:

Introdu tion to

Algebrai Proof

Theory for

. APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

15 / 55

algebra.

� Origin:

Introdu tion to

Algebrai Proof

Theory for

. APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

15 / 55

algebra.

� Origin:

Introdu tion to

Algebrai Proof

Theory for

. APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

15 / 55

algebra.

� Origin:

Introdu tion to

Algebrai Proof

Theory for

. APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

15 / 55

algebra.

� Origin:

Introdu tion to

Algebrai Proof

Theory for

. APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

15 / 55

algebra.

� Origin:

Completions

Introdu tion to

Algebrai Proof

Theory for

APT for SL

. Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

16 / 55

Let A be an FL algebra.

A ompletion of A is a pair of a omplete FL algebra B and an

embedding e : A ,! B.

We may assume A � B.

We onsider 4 types of ompletion:

� Ma Neille ompletions

(Dedekind, Ma Neille, S hmidt, Banas hewski . . . )

� Canoni al extensions

(Tarski, J�onson, Gehrke, Harding . . . )

� Hyper-Ma Neille ompletions

� Hyper anoni al extensions

Completions

Introdu tion to

Algebrai Proof

Theory for

APT for SL

. Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

16 / 55

A ompletion of A is a pair of a omplete FL algebra B and an

embedding e : A ,! B.

We may assume A � B.

We onsider 4 types of ompletion:

� Ma Neille ompletions

(Dedekind, Ma Neille, S hmidt, Banas hewski . . . )

� Canoni al extensions

(Tarski, J�onson, Gehrke, Harding . . . )

� Hyper-Ma Neille ompletions

� Hyper anoni al extensions

Ma Neille ompletions

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

. Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

17 / 55

Dedekind ompletion: ([0; 1℄

;min;max) ,! ([0; 1℄

;min;max).

What is the distin tive feature of this ompletion?

For every x 2 [0; 1℄

x = supfa 2 [0; 1℄

: a � xg = inffa 2 [0; 1℄

: a � xg:

Let A be a latti e. Its ompletion B is

� join-dense if for every x 2 B, x =

fa 2 A : a � xg.

� meet-dense if for every x 2 B, x =

fa 2 A : a � xg.

Theorem (S hmidt 56, Banas hewski 56)

Every latti e A has a join-dense and meet-dense ompletion A,

unique up to isomorphism, alled the Ma Neille ompletion.

It an be extended to FL algebras too.

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

. Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

17 / 55

;min;max) ,! ([0; 1℄

;min;max).

x = supfa 2 [0; 1℄

: a � xg = inffa 2 [0; 1℄

: a � xg:

fa 2 A : a � xg.

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

. Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

17 / 55

;min;max) ,! ([0; 1℄

;min;max).

x = supfa 2 [0; 1℄

: a � xg = inffa 2 [0; 1℄

: a � xg:

fa 2 A : a � xg.

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

. Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

17 / 55

;min;max) ,! ([0; 1℄

;min;max).

x = supfa 2 [0; 1℄

: a � xg = inffa 2 [0; 1℄

: a � xg:

fa 2 A : a � xg.

Residuated frames

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated

frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

18 / 55

A preframe is W = (W;W

; N; Æ; "; �) su h that

� N �W �W

� (W; Æ; ") is a monoid, � 2W

A residuated frame is a preframe where for every x 2W; z 2W

there are elements x z and z�x 2W

su h that

x Æ y N z () x N z�y () y N x z:

If W is a preframe, then

W := (W; W�W

N; Æ; "; ("; �; "))

N (u; z; v) () uÆxÆv N z

is a residuated frame.

Residuated frames

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated

frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

18 / 55

; N; Æ; "; �) su h that

� N �W �W

su h that

W := (W; W�W

N; Æ; "; ("; �; "))

Residuated frames

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated

frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

18 / 55

; N; Æ; "; �) su h that

� N �W �W

su h that

W := (W; W�W

N; Æ; "; ("; �; "))

Residuated frames

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated

frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

19 / 55

Residuated frames are e�e tive tools to build omplete FL

algeblras.

Given X �W and Z �W

:= fz 2W

: x N z for every x 2 Xg

:= fx 2W : x N z for every z 2 Zg

) forms a Galois onne tion:

X � Z

that indu es a losure operator (X) := X

on }(W ).

Residuated frames

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated

frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

19 / 55

Residuated frames are e�e tive tools to build omplete FL

algeblras.

Given X �W and Z �W

:= fz 2W

: x N z for every x 2 Xg

:= fx 2W : x N z for every z 2 Zg

) forms a Galois onne tion:

X � Z

that indu es a losure operator (X) := X

on }(W ).

Residuated frames

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated

frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

20 / 55

Given W = (W;W

; N; Æ; "; �) and X;Y �W ,

G(W ) := the set of Galois- losed subsets of W

(X = (X) = X

XnY := fy : x Æ y 2 Y for every x 2 Xg

Y=X := fy : y Æ x 2 Y for every x 2 Xg

Y := (X Æ Y )

Y := (X [ Y )

:= (G(W );\;[

; n; =; ("); �

is a omplete FL algebra, alled the omplex algebra of W.

Appli ations of residuated frames

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated

frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

21 / 55

Fm := the set of formulas.

:= the set of formula sequen es.

� Let W

:= (Fm

; Fm [ f;g; N

; Æ; ;; ;) where

� i� �) � is ut-free derivable.

Then W

is an FL algebra su h that

1 � ' implies ) ' is ut-free derivable.

) Algebrai ut elimination.

� Given an FL algebra A, let W

:= (A;A;�

; �; 1; 0).

Then W

is the Ma Neille ompletion of A.

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated

frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

21 / 55

� Let W

:= (Fm

; Fm [ f;g; N

; Æ; ;; ;) where

Then W

:= (A;A;�

; �; 1; 0).

Then W

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated

frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

21 / 55

� Let W

:= (Fm

; Fm [ f;g; N

; Æ; ;; ;) where

Then W

:= (A;A;�

; �; 1; 0).

Then W

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

From axioms to

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

22 / 55

Let L be a SL (axiomati extension of FL). To obtain an analyti

al ulus for L, axioms have to be transformed into stru tural rules:

'! ' � ' ' � '! ' :(' ^ :')

�;�;�;�) �

�;�;�) �

�;�;�) � �;�;�) �

�;�;�;�) �

�;�)

Let V be a subvariety of FL. To show that V is losed under

ompletions, identities have to be transformed into quasi-identities:

x � xx xx � x x ^ :x � 0

xx � z

x � z

x � z y � z

xy � z

xx � 0

x � 0

Fundamental Question

Whi h axioms/identities an be transformed into \good" stru tural

rules/quasi-identities?

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

From axioms to

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

22 / 55

'! ' � ' ' � '! ' :(' ^ :')

�;�;�;�) �

�;�;�) �

�;�;�) � �;�;�) �

�;�;�;�) �

�;�)

x � xx xx � x x ^ :x � 0

xx � z

x � z

x � z y � z

xy � z

xx � 0

x � 0

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

From axioms to

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

22 / 55

'! ' � ' ' � '! ' :(' ^ :')

�;�;�;�) �

�;�;�) �

�;�;�) � �;�;�) �

�;�;�;�) �

�;�)

x � xx xx � x x ^ :x � 0

xx � z

x � z

x � z y � z

xy � z

xx � 0

x � 0

Substru tural hierar hy

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

. Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

23 / 55

�� 6

Classi� ation of axioms

::= the set of variables

j 1 j P

j 0 j N

Some N

axioms:

�! 1, 0! � weakening

�! � � � ontra tion

� � �! � expansion

knotted axioms (n;m � 0)

:(� ^ :�) no- ontradi tion

orresponds to sequent al ulus and Ma Neille ompletions

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

. Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

24 / 55

�� 6

Theorem (Ciabattoni-Galatos-T. 12)

1. Every N

axiom an be transformed into a

set of stru tural rules in sequent al ulus FL.

2. For every set E of N

axioms, the following

are equivalent.

� FL(E) admits a strongly analyti sequent

al ulus ( ut elimination for derivations with

assumptions + subformula property).

� FL(E) is losed under Ma Neille omple-

tions.

� E is a y li (a synta ti riterion).

3. The above three hold whenever (w) 2 E.

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

. Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

24 / 55

�� 6

1. Every N

are equivalent.

tions.

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

. Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

24 / 55

�� 6

1. Every N

are equivalent.

tions.

Let's limb up the hierar hy

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

. Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

25 / 55

�� 6

Classi� ation of axioms

::= the set of variables

j 1 j P

j 0 j N

Some P

axioms:

(�! �) _ (� ! �) prelinearity

� _ :� ex luded middle

:� _ ::� weak ex luded middle

:(� � �) _ (� ^ � ! � � �) weak nilpotent minimum

) bounded width � k

^ � � � ^ �

) bounded size � k

Limitation of sequent al ulus/Ma Neille ompletions

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

. Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

26 / 55

Every stru tural rule in sequent al ulus is either derivable or on-

tradi tory in Int.

�) � �) �

�;�) �

�) �

Theorem (G.Bezhanishvili-Harding 04)

There is no intermediate variety between HA and BA that is losed

under Ma Neille ompletions.

Eg. prelinearity annot be dealt with by sequent

al ulus/Ma Neille ompletions.

Limitation of sequent al ulus/Ma Neille ompletions

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

. Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

26 / 55

Every stru tural rule in sequent al ulus is either derivable or on-

tradi tory in Int.

�) � �) �

�;�) �

�) �

Theorem (G.Bezhanishvili-Harding 04)

There is no intermediate variety between HA and BA that is losed

under Ma Neille ompletions.

Eg. prelinearity annot be dealt with by sequent

al ulus/Ma Neille ompletions.

Hypersequent al ulus (Avron 91)

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

28 / 55

� Hypersequents: �

j � � � j �

(meaning (�

) _ � � � _ (�

� Hypersequent al ulus for FL onsists of

Rules of FL Ext-Weakening Ext-Contra tion

� j �;�) �

� j �) �! �

� j �) �

� j �) � j �) �

� j �) �

� (pl) (�! �) _ (� ! �) is equivalent (in FLew) to

� j �

) � � j �

� j �

) � j �

�) � � ) �

�) � j � ) �

) �! � j ) � ! �

) (�! �) _ (� ! �) j ) (�! �) _ (� ! �)

) (�! �) _ (� ! �)

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

28 / 55

j � � � j �

(meaning (�

) _ � � � _ (�

� j �;�) �

� j �) �! �

� j �) �

� j �) � j �) �

� j �) �

� j �

) � � j �

� j �

) � j �

�) � � ) �

�) � j � ) �

) �! � j ) � ! �

) (�! �) _ (� ! �) j ) (�! �) _ (� ! �)

) (�! �) _ (� ! �)

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

28 / 55

j � � � j �

(meaning (�

) _ � � � _ (�

� j �;�) �

� j �) �! �

� j �) �

� j �) � j �) �

� j �) �

� j �

) � � j �

� j �

) � j �

�) � � ) �

�) � j � ) �

) �! � j ) � ! �

) (�! �) _ (� ! �) j ) (�! �) _ (� ! �)

) (�! �) _ (� ! �)

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

28 / 55

j � � � j �

(meaning (�

) _ � � � _ (�

� j �;�) �

� j �) �! �

� j �) �

� j �) � j �) �

� j �) �

� j �

) � � j �

� j �

) � j �

�) � � ) �

�) � j � ) �

) �! � j ) � ! �

) (�! �) _ (� ! �) j ) (�! �) _ (� ! �)

) (�! �) _ (� ! �)

Hypersequent al ulus

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

axiom is equivalent (in FLew) to a set of stru tural

rules in hypersequent al ulus.

:(� � �) _ (� ^ � ! � � �)

is equivalent to

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j ) (� � �)! 0 j ) � ^ � ! � � �

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j � � � ) 0 j ) � ^ � ! � � �

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j �; � ) 0 j ) � ^ � ! � � �

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j �; � ) j ) � ^ � ! � � �

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j �; � ) j � ^ � ) � � �

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j ) � ^ �

� j �; � ) j ) � � �

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j ) � � j ) �

� j �; � ) j ) � � �

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j ) � � j ) � � j � � � ) Æ

� j �; � ) j ) Æ

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j ) � � j ) � � j �; � ) Æ

� j �; � ) j ) Æ

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j ) � � j ) � � j �; � ) Æ

� j �; � ) j ) Æ

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j ) � � j ) � � j �; � ) Æ

� j �) �

� j �; � ) j ) Æ

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j ) � � j ) � � j �; � ) Æ

� j �) � � j �) �

� j �;�) j ) Æ

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j ) � � j ) � � j �; � ) Æ

� j �) � � j �) � � j �)

� j �;�) j �) Æ

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j ) � � j ) � � j �; � ) Æ

� j �) � � j �) � � j �) � j Æ;�) �

� j �;�) j �;�) �

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j ) � � j ) � � j �; �;�) �

� j �) � � j �) � � j �)

� j �;�) j �;�) �

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j �) � � j �) � � j �; �;�) �

� j �) � � j �) �

� j �;�) j �;�) �

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j �; �;�) � � j �; �;�) �

� j �) � � j �) �

� j �;�) j �;�) �

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j �;�;�) � � j �;�;�) �

� j �;�) j �;�) �

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent

Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

29 / 55

Theorem (CGT 08)

Every P

:(� � �) _ (� ^ � ! � � �)

is equivalent to

� j �;�;�) � � j �;�;�) �

� j �;�) j �;�) �

� Cf. A kermann Lemma-based Algorithm

(Conradie-Palmigiano)

� Implemented by (Ciabattoni-Spendier)

Hyper-Ma Neille ompletions

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

. Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

30 / 55

Let A be an FLew algebra. De�ne

:= (A�A;A�A;N; (�;_); (1; 0); (0; 0))

(a; h) N (b; k) () 1 = (a! b)_h _ k

Theorem

is a ompletion of A, alled the hyper-Ma Neill ompletion.

Theorem (CGT)

For every set E of P

axioms,

� FLew(E) admits a strongly analyti hypersequent al ulus.

� FLew(E) is losed under hyper-Ma Neille ompletions.

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

. Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

30 / 55

:= (A�A;A�A;N; (�;_); (1; 0); (0; 0))

(a; h) N (b; k) () 1 = (a! b)_h _ k

Theorem

Theorem (CGT)

axioms,

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

. Hyper-Ma Neille

Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

30 / 55

:= (A�A;A�A;N; (�;_); (1; 0); (0; 0))

(a; h) N (b; k) () 1 = (a! b)_h _ k

Theorem

Theorem (CGT)

axioms,

Class N

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

. Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

31 / 55

�� 6

Some N

axioms:

� ^ (� _ )! (� ^ �) _ (� ^ ) distributivity

(�! � � �)! � an ellativity

� ^ � ! � � (�! �) divisibility

BL := FLew + (pl) + (div)

L := BL+ (in)

Theorem ( f. Kowalski-Litak 08)

The varieties BL, MV(= V( L)) are not losed under

any ompletions. Hen e the logi s BL and L do not

admit any strongly analyti al ulus.

Limitation of uniform proof theory!

Class N

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

. Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

31 / 55

�� 6

Some N

axioms:

� ^ (� _ )! (� ^ �) _ (� ^ ) distributivity

� ^ � ! � � (�! �) divisibility

L := BL+ (in)

Class N

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

. Class N

Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

31 / 55

�� 6

Some N

axioms:

� ^ (� _ )! (� ^ �) _ (� ^ ) distributivity

� ^ � ! � � (�! �) divisibility

L := BL+ (in)

Summary

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

. Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

32 / 55

�� 6

: sequent al ulus

: hypersequent al ulus

hyper-Ma Neille ompletions

: limitation of uniform proof theory

� Axioms ) rules is important in both proof

theory and algebra.

� The hyper- onstru tion (proof theory) is

useful for ompletions (algebra) too.

� Limitation of proof theory is imposed by al-

gebrai fa ts.

Summary

Introdu tion to

Algebrai Proof

Theory for

APT for SL

Completions

Ma Neille omp.

Residuated frames

Subst. hierar hy

Limitation

Hypersequent al .

Hyper-Ma Neille

Class N

. Summary

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

32 / 55

�� 6

: sequent al ulus

: hypersequent al ulus

hyper-Ma Neille ompletions

: limitation of uniform proof theory

� Axioms ) rules is important in both proof

theory and algebra.

� The hyper- onstru tion (proof theory) is

useful for ompletions (algebra) too.

� Limitation of proof theory is imposed by al-

gebrai fa ts.

Herbrand's theorem via hyper anoni al

extensions

Introdu tion to

Algebrai Proof

Theory for

Herbrand's

theorem via

hyper anoni al

extensions

Predi ate SL

Herbrand property

Canoni al extensions

HP ompa t ompl.

HP for �nite

HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

33 / 55

Predi ate substru tural logi s

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

. Predi ate SL

Herbrand property

HP ompa t ompl.

HP for �nite

HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

34 / 55

Let L be a propositional substru tural logi .

QL := the predi ate extension of L obtained by adding

8x:�(x)! �(t) �(t)! 9x:�(x)

� ! �(x)

� ! 8x:�(x)

�(x)! �

9x:�(x)! � (x not free in �)

Herbrand property

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand

property

HP ompa t ompl.

HP for �nite

HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

35 / 55

De�nition

L satis�es the Herbrand property if for every set of universal

formulas and every quanti�er-free formula '(x),

9x:'(x) ()

) _ � � � _ '(t

for some t

; : : : ; t

:= f (t) : 8x: (x) 2 g.

Herbrand property is related to ompa tness phenomena

(previous talk).

What is the algebrai form of ompa tness?

Herbrand property

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand

property

HP ompa t ompl.

HP for �nite

HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

35 / 55

De�nition

L satis�es the Herbrand property if for every set of universal

formulas and every quanti�er-free formula '(x),

9x:'(x) ()

) _ � � � _ '(t

for some t

; : : : ; t

:= f (t) : 8x: (x) 2 g.

Herbrand property is related to ompa tness phenomena

(previous talk).

What is the algebrai form of ompa tness?

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

Canoni al

extensions

HP ompa t ompl.

HP for �nite

HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

36 / 55

Let C be a Boolean algebra and X

be its Stone spa e. Then

:= (P(X

);\;[;

is a ompletion of C.

Let D be a bounded distributive latti e and Y

be its Priestly

spa e. Then

);\;[)

is a ompletion of D.

Re all that

Ma Neille ompletions = join-dense, meet-dense ompletions

Do we have a similar abstra t hara terization for C

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

Canoni al

extensions

HP ompa t ompl.

HP for �nite

HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

36 / 55

:= (P(X

);\;[;

be its Priestly

spa e. Then

);\;[)

Re all that

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

Canoni al

extensions

HP ompa t ompl.

HP for �nite

HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

36 / 55

:= (P(X

);\;[;

be its Priestly

spa e. Then

);\;[)

Re all that

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

Canoni al

extensions

HP ompa t ompl.

HP for �nite

HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

37 / 55

� dense if for every x 2 B, there exist C

(i 2 I; j 2 J) su h that

� ompa t if for every C;D � A,

for some �nite C

� C and D

� D.

Theorem (Gehrke-Harding 01)

Every latti e A has a dense and ompa t ompletion A

, unique

up to isomorphism, alled the anoni al extension.

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

Canoni al

extensions

HP ompa t ompl.

HP for �nite

HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

37 / 55

for some �nite C

� C and D

� D.

, unique

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

Canoni al

extensions

HP ompa t ompl.

HP for �nite

HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

37 / 55

for some �nite C

� C and D

� D.

, unique

Herbrand property via ompa t ompletions

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t

HP for �nite

HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

38 / 55

A ompletion of A is ompa t if for every C;D � A,

for some �nite C

� C and D

� D.

Theorem

If V(L) is losed under ompa t ompletions, then L satis�es the

Herbrand property.

Herbrand property for �nite-valued logi s

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

. HP for �nite

HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

39 / 55

Let V be a variety of monotone latti e expansions.

If V is generated by a �nite algebra, then V is losed under anon-

i al extensions.

Corollary

Every �nite-valued substru tural logi satis�es the Herbrand prop-

It a tually applies to a mu h wider range of �nite-valued logi s.

The GH theorem is an algebrai ounterpart of the uniform

midsequent theorem for �nite-valued logi s

(Baaz-Ferm�uller-Za h 94).

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

. HP for �nite

HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

39 / 55

i al extensions.

Corollary

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

. HP for �nite

HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

39 / 55

i al extensions.

Corollary

Herbrand property for N

logi s

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

HP for �nite

. HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

40 / 55

Theorem (Gehrke-Harding-Venema 05)

Let V be a variety of bounded monotone latti e expansions. If

V is losed under Ma Neille ompletions, it is also losed under

anoni al extensions.

Ma Neille ompletions preserve (in) ::�! �.

Canoni al extensions preserve (dist)

(� _ �) ^ $ (� ^ ) _ (� ^ ).

Corollary

Every substru tural logi axiomatized by

� a y li N

axioms

� and/or (in), (dist)

satis�es the Herbrand property.

logi s

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

HP for �nite

. HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

40 / 55

(� _ �) ^ $ (� ^ ) _ (� ^ ).

Corollary

� a y li N

axioms

logi s

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

HP for �nite

. HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

40 / 55

(� _ �) ^ $ (� ^ ) _ (� ^ ).

Corollary

� a y li N

axioms

logi s

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

HP for �nite

. HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

41 / 55

The GHV theorem states:

Ma Neille ompletions =) Canoni al extensions:

It onforms to the proof theoreti intuition:

Cut elimination =) Herbrand's theorem:

What about P

logi s?

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

HP for �nite

. HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

42 / 55

�� 6

Some P

axioms:

^ � � � ^ �

We want ompa t ompletions that preserve P

axioms.

) Hyper anoni al extensions.

What about P

logi s?

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

HP for �nite

. HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

42 / 55

�� 6

Some P

axioms:

^ � � � ^ �

We want ompa t ompletions that preserve P

axioms.

) Hyper anoni al extensions.

Reminder: Ma Neille and hyper-Ma Neille ompletions

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

HP for �nite

. HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

43 / 55

Ma Neille ompletion of A is W

:= (A;A;�; �; 1; 0):

Assuming A is FLew, hyper-Ma Neille ompletion is W

:= (A�A;A�A;N; � � � )

(a; h) N (b; k) () 1 = (a! b)_h _ k:

Reminder: Ma Neille and hyper-Ma Neille ompletions

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

HP for �nite

. HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

43 / 55

Ma Neille ompletion of A is W

:= (A;A;�; �; 1; 0):

Assuming A is FLew, hyper-Ma Neille ompletion is W

:= (A�A;A�A;N; � � � )

(a; h) N (b; k) () 1 = (a! b)_h _ k:

Canoni al and hyper anoni al extensions

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

HP for �nite

. HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

44 / 55

Canoni al extension of A is W

; N; Æ; "1; #0);

:= the �lters of A

:= the ideals of A

f N i () f \ i 6= ;

Assuming A is FLew, hyper anoni al extension of A is W

; N; � � � )

(f; j) N (i; k) () 1 2 (f ! i)_j _ k

Canoni al and hyper anoni al extensions

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

HP for �nite

. HP for N2

HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

44 / 55

Canoni al extension of A is W

; N; Æ; "1; #0);

:= the �lters of A

:= the ideals of A

f N i () f \ i 6= ;

Assuming A is FLew, hyper anoni al extension of A is W

; N; � � � )

(f; j) N (i; k) () 1 2 (f ! i)_j _ k

Herbrand property for P

logi s

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

HP for �nite

HP for N2

. HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

45 / 55

Theorem

Hyper anoni al extensions are ompa t ompletions. They pre-

serve all P

identities.

Corollary

Every substru tural logi over FLew axiomatized by P

axioms

It applies to MTL, G, LQ and many more uniformly.

Herbrand property for P

logi s

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

HP for �nite

HP for N2

. HP for P3

Class N

Herbrand's theorem

for 98

Further topi s

45 / 55

Theorem

Hyper anoni al extensions are ompa t ompletions. They pre-

serve all P

identities.

Corollary

Every substru tural logi over FLew axiomatized by P

axioms

It applies to MTL, G, LQ and many more uniformly.

Class N

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

HP for �nite

HP for N2

HP for P3

. Class N

Herbrand's theorem

for 98

Further topi s

46 / 55

�� 6

Re all that MV(= V( L)) is not losed under any

ompletions.

Theorem (Baaz-Met alfe 08)

L does not satisfy the Herbrand property, al-

though it does satisfy an \approximate" Her-

brand theorem.

Herbrand's theorem for 98-formulas

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

HP for �nite

HP for N2

HP for P3

Class N

Herbrand's

theorem for 98

Further topi s

47 / 55

Herbrand's theorem for 98-formulas:

� ` 9x8y:'(x; y) () � ` '(t

) _ � � � _ '(t

where t

does not ontain y

; : : : ; y

The general form requires the onstant domain axiom ( d):

8x:(�(x) _ �)$ (8x:�(x)) _ �:

Its algebrai ounterpart is meet in�nite distributivity:

_ y) = (

) _ y:

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

HP for �nite

HP for N2

HP for P3

Class N

Herbrand's

theorem for 98

Further topi s

48 / 55

� If A is distributive, then A

satis�es (mid).

� If A is an MTL algebra, then A

satis�es (mid).

Theorem

Let L be a substru tural logi . Herbrand's theorem for 98-

formulas holds for QL( d) if

� either L is axiomatized by distributivity and some N

axioms,

� or L is axiomatized by (e), (w), (pl) and and some P

axioms.

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Predi ate SL

Herbrand property

HP ompa t ompl.

HP for �nite

HP for N2

HP for P3

Class N

Herbrand's

theorem for 98

Further topi s

48 / 55

� If A is distributive, then A

satis�es (mid).

� If A is an MTL algebra, then A

satis�es (mid).

Theorem

Let L be a substru tural logi . Herbrand's theorem for 98-

formulas holds for QL( d) if

� either L is axiomatized by distributivity and some N

axioms,

� or L is axiomatized by (e), (w), (pl) and and some P

axioms.

Further topi s

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

. Further topi s

Extra ting info

Interpolation

Density rule elim.

Con lusion

49 / 55

Extra ting algebrai information from proof theory

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

. Extra ting info

Interpolation

Density rule elim.

Con lusion

50 / 55

� Apparently there is no ni e duality between FL algebras and

residuated frames.

� Residuated frames are lose to syntax so that one an

en ode synta ti information into frames.

� By en oding proof theoreti arguments into frames and

taking the omplex algebra, one an obtain an algebrai

onstru tion.

Case study: Interpolation ) Amalgamation

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

Extra ting info

. Interpolation

Density rule elim.

Con lusion

51 / 55

Let X;Y be sets of propositional variables.

Maehara's lemma

�;�) � with � � Fm(X) and �;� � Fm(Y ),

there is � 2 Fm(X \ Y ) su h that

�)� and `

�;�) �:

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

Extra ting info

. Interpolation

Density rule elim.

Con lusion

52 / 55

Let A;B;C be FLe algebras.

Suppose that A is a subalgebra of both B and C.

��

De�ne

:= (B � C;B [ C;N; � � � )

(b; ) N

() 9i 2 A: b �

i and i �

(b; ) N b

() 9i 2 A: �

i and ib �

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

Extra ting info

. Interpolation

Density rule elim.

Con lusion

53 / 55

Then the omplex algebra gives rise to an amalgam.

��

(interpolation)

= amalgamation

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

Extra ting info

. Interpolation

Density rule elim.

Con lusion

53 / 55

Then the omplex algebra gives rise to an amalgam.

��

(interpolation)

= amalgamation

Another su ess: density rule elimination ) densi� ation

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

Extra ting info

Interpolation

. Density rule elim.

Con lusion

54 / 55

Likewise, the next talk by Hor� ��k is an out ome of:

(density rule elimination)

= densi� ation

The slogan is:

(proof theoreti argument)

= algebrai onstru tion

This way we an salvage ni e proof theoreti ideas and bring

them to ordered algebras.

Another su ess: density rule elimination ) densi� ation

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

Extra ting info

Interpolation

. Density rule elim.

Con lusion

54 / 55

Likewise, the next talk by Hor� ��k is an out ome of:

(density rule elimination)

= densi� ation

The slogan is:

= algebrai onstru tion

This way we an salvage ni e proof theoreti ideas and bring

them to ordered algebras.

Con lusion

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

Extra ting info

Interpolation

Density rule elim.

. Con lusion

55 / 55

We have explored the onne tion between proof theoreti

arugments and algebrai ompletions based on the substru tural

hierar hy:

sequent al . (N

) hypersequent al . (P

ut elimination Ma Neille ompl. hyper-Ma Neille ompl.

Herbran's theorem anoni al ext. hyper anoni al ext.

It is residuated frames that onne t the two:

= algebrai onstru tion.

Substru tural proof theory is full of burea ra y, but hopefully

there are still something good to be salvaged.

Con lusion

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

Extra ting info

Interpolation

Density rule elim.

. Con lusion

55 / 55

hierar hy:

sequent al . (N

Con lusion

Introdu tion to

Algebrai Proof

Theory for

Herbrand's theorem

via hyper anoni al

extensions

Further topi s

Extra ting info

Interpolation

Density rule elim.

. Con lusion

55 / 55

hierar hy:

sequent al . (N

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