Combinatorics of Interpolation in Gödel Logic - TU Wien · GODEL¨ LOGIC ONGOING WORK Combinatorics of Interpolation in Godel¨ Logic Simone Bova [email protected] Department of

GODEL LOGIC ONGOING WORK

Combinatorics of Interpolation in Godel Logic

Simone [email protected]

Department of Computer ScienceUniversity of Milan (Milan, Italy)

TACL 2009July 7-11, 2009, Amsterdam (Netherlands)


Outline

Godel LogicGodel LogicFree Godel AlgebraInterpolation Properties

Ongoing WorkBasic LogicDeductive InterpolationConstruction Sketch


Outline




Godel Logic

Godel (propositional) logic, G, is:• intuitionistic logic plus ((x→ y) ∨ (y→ x)),

the intermediate logic of linear Kripke frames;• Hajek’s basic logic plus (x→ (x� x)),

the many-valued logic of “minimum and its residual”,

[0, 1] = ([0, 1],∧ = � = min,∨ = max, x→ y,⊥ = 0,> = 1)

where x→ y equals 1 if x ≤ y and y otherwise.


Free Godel Algebra

Definition (Godel Algebras)Godel algebras are algebras in the variety generated by [0, 1].

FactThe free X-generated Godel algebra, GX, is (isomorphic to)the Lindenbaum algebra of the X-variate fragment of Godel logic.

GX “supports” the investigation of (finite) consequencerelations and interpolation properties in Godel logic.

GX has a nice combinatorial representation (X finite).


Free Godel Algebra






Free Godel Algebra






Free X-generated Godel Algebra | Construction

Step 1: Construction of forest FX:

t = 0: the subsets of X are the maximal elements in FX at t = 0;t = i + 1: if R is maximal in FX at t = i, then there exists S st

S covers R and S is maximal in FX at t = i + 1 iff:X =

⋃T≤R T and S = {1}, or

X 6=⋃

T≤R T and ∅ 6= S ⊆ X \⋃

T≤R T.

Ex.: F{x,y,z} at t = 0.

0xyz 0xy 0xz 0yz 0x 0y 0z 0







X 6=⋃

T≤R T and ∅ 6= S ⊆ X \⋃

T≤R T.

Ex.: F{x,y,z} at t = 1.

0xyz

1

0xy

z

0xz

y

0yz

x yz y z

0x

xz x z

0y

xy x y

0z

xyz xy xz yz x y z

0







X 6=⋃

T≤R T and ∅ 6= S ⊆ X \⋃

T≤R T.

Ex.: F{x,y,z} at t = 2.

0xyz

1

0xy

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1

0xz

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1

0yz

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1

yz

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1

xy

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xz

y

yz

x yz y z

x

xz x z

y

xy x y

z

0







X 6=⋃

T≤R T and ∅ 6= S ⊆ X \⋃

T≤R T.

Ex.: F{x,y,z} at t = 3.

0xyz

1

0xy

z

1

0xz

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1

0yz

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1

yz

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0







X 6=⋃

T≤R T and ∅ 6= S ⊆ X \⋃

T≤R T.

Ex.: F{x,y,z} at t ≥ 4.

0xyz

1

0xy

z

1

0xz

y

1

0yz

x

1

yz

1

y

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1

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Step 2: The generator x ∈ X is the maximal antichain in FX“mentioning x” over each maximal chain in FX.

Ex.: X = {x, y, z}. For each maximal chain in F{x,y,z},the generator x picks the point containing x.

0xyz

1

0xy

z

1

0xz

y

1

0yz

x

1

yz

1

y

z

1

z

y

1

0x

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0y

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0



Step 2: The generator x ∈ X is the maximal antichain in FX“mentioning x” over each maximal chain in FX.

Ex.: X = {x, y, z}. For each maximal chain in F{x,y,z},the generator y picks the point containing y.

0xyz

1

0xy

z

1

0xz

y

1

0yz

x

1

yz

1

y

z

1

z

y

1

0x

xz

1

x

z

1

z

x

1

0y

xy

1

x

y

1

y

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1

0z

xyz

1

xy

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1

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1

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1

z

0



Step 3: The op’s over maximal antichains in FX are defined“maxchainwise” by the corr. operations in [0, 1].

Ex.: X = {x, y, z}. As ⊥ = 0 in [0, 1], ⊥ picks the pointcontaining 0 for each maximal chain in F{x,y,z}.

0xyz

1

0xy

z

1

0xz

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1

0yz

x

1

yz

1

y

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1

z

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0x

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Step 3: Equip the maximal antichains in FX with operationsdefined “maxchainwise” by the corr. generic operations.

Ex.: X = {x, y, z}. As > = 1 in [0, 1], > picks the pointcontaining 1 for each maximal chain in F{x,y,z}.

0xyz

1

0xy

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1

0xz

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Ex.: X = {x, y, z}. As x ∧ y = min{x, y} in [0, 1],x ∧ y picks the minimum of x and y.

0xyz

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0xy

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Ex.: X = {x, y, z}. As x ∨ y = max{x, y} in [0, 1],x ∨ y picks the maximum of x and y.

0xyz

1

0xy

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0xz

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Ex.: X = {x, y, z}. As x→ y is 1 if x ≤ y and y ow in [0, 1],x→ y picks 1 if x ≤ y and y ow.

0xyz

1

0xy

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1

0xz

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Interpolation Properties | Craig (CIP)

X ∩ Y = Z and X ∪ Y = W.

DefinitionG has the CIP iff, for all rX and tY st `G r→ t,there exists sZ st `G r→ s and `G s→ t.

Theorem ([BV99])G has the CIP.

CorollaryFor all AX ∈ GX and CY ∈ GY st AW ≤ CW in GW,there exists BZ ∈ GZ st AW ≤ BW ≤ CW in GW.


CIP | Sampling with X = {x, z} and Y = {y, z}

A{x,z} ∈ G{x,z} and C{y,z} ∈ G{y,z}

are such that A{x,y,z} ≤ C{x,y,z} in G{x,y,z}. Hence, . . .




are such that A{x,y,z} ≤ C{x,y,z} in G{x,y,z}. Hence, . . .



. . . there exists B{z} ∈ G{z}

such that A{x,y,z} ≤ B{x,y,z} ≤ C{x,y,z} in G{x,y,z}.










Interpolation Properties | Deductive (DIP)

X ∩ Y = Z and X ∪ Y = W.

DefinitionG has the DIP iff, for all rX and tY st r `G t,there exists sZ st r `G s and s `G t.

Theorem ([KO09])G has the DIP.

CorollaryFor all AX ∈ GX and CY ∈ GY st AW ∩ > ⊆ CW ∩ > in GW,there is BZ ∈ GZ st AW ∩ > ⊆ BW ∩ > ⊆ CW ∩ > in GW.


DIP | Sampling with X = {x, z} and Y = {y, z}


are st A{x,y,z} ∩ > ⊆ C{x,y,z} ∩ > in G{x,y,z}. Hence, . . .




are st A{x,y,z} ∩ > ⊆ C{x,y,z} ∩ > in G{x,y,z}. Hence, . . .




st A{x,y,z} ∩ > ⊆ B{x,y,z} ∩ > ⊆ C{x,y,z} ∩ > in G{x,y,z}.










Outline




Basic Logic

(Hajek’s) basic (propositional) logic, BL, is:• the many-valued logic of all continuous triangular norms

and their residuals;• the substructural logic of commutative bounded integral

divisible prelinear residuated lattices.


Deductive Interpolation

Fact ([M06])BL has the DIP (not the CIP).

Problem 1: Give a constructive proof.Problem 2: Give a complexity bound.

Fact (Functional Representation [B08])The elements of the Lindenbaum algebra of BL are suitable realfunctions, “described by combining the geometry of Łukasiewicz logicand the combinatorics of Godel logic”.

Goal: Solve Problem 1 and Problem 2 in this setting.




Problem 1: Give a constructive proof.

Problem 2: Give a complexity bound.























X ∩ Y = Z and X ∪ Y = W.

Fact ([M06])For all rX and tY st r `BL t, there exists sZ st r `BL s and s `BL t.

Factr `BL t iff r−1

W (1) ⊆ t−1W (1) ⊆ [0, 1]W.

CorollaryFor all rX and tY st r−1

W (1) ⊆ t−1W (1) ⊆ [0, 1]W,

there exists sZ st r−1W (1) ⊆ s−1

W (1) ⊆ t−1W (1) ⊆ [0, 1]W.

Idea: Exploiting the functional representation of rX and sZ,construct a “strongest possible” interpolant sZ, that is,a sZ with smallest possible

[0, 1]X ⊇ s−1X (1) ⊇ r−1

X (1).



X ∩ Y = Z and X ∪ Y = W.

Fact ([M06])For all rX and tY st r `BL t, there exists sZ st r `BL s and s `BL t.

Factr `BL t iff r−1

W (1) ⊆ t−1W (1) ⊆ [0, 1]W.

CorollaryFor all rX and tY st r−1

W (1) ⊆ t−1W (1) ⊆ [0, 1]W,

there exists sZ st r−1W (1) ⊆ s−1

W (1) ⊆ t−1W (1) ⊆ [0, 1]W.

Idea: Exploiting the functional representation of rX and sZ,construct a “strongest possible” interpolant sZ, that is,a sZ with smallest possible

[0, 1]X ⊇ s−1X (1) ⊇ r−1

X (1).


Construction | Sampling with X = {x, z} and Y = {y, z}.

Idea: Exploiting the functional representation of r{x,z} and s{z},construct a s{z} having the smallest s−1

{z}(1) ⊇ r−1{x,z}(1).

8f< Æ

8g1..<

Æ

8h1..< 8i< Æ

8j1..<

Æ

8k1..<

Æ

The given r{x,z} : [0, 1]{x,z} → [0, 1] decomposesinto finitely many “Łukasiewicz functions”

over a Godel skeleton,satisfying certain constraints.




{z}(1) ⊇ r−1{x,z}(1).

8f< Æ

8g1..<

Æ

8h1..< 8i< Æ

8j1..<

Æ

8k1..<

Æ






{z}(1) ⊇ r−1{x,z}(1).

0xz 0x

z

0z

x xz x

z

z

x

0






{z}(1) ⊇ r−1{x,z}(1).

8f< Æ

8g1..<

Æ

8h1..< 8i< Æ

8j1..<

Æ

8k1..<

Æ






{z}(1) ⊇ r−1{x,z}(1).

8f< Æ

8g1..<

8l<

8h1..< 8m<

Æ

Æ

8j1..<

Æ

8k1..<

Æ

The target s{z} : [0, 1]{z} → [0, 1] decomposesinto finitely many “Łukasiewicz functions”





{z}(1) ⊇ r−1{x,z}(1).

8f< Æ

8g1..<

0z

8h1..< z

0

Æ

8j1..<

Æ

8k1..<

Æ






{z}(1) ⊇ r−1{x,z}(1).

8f< Æ

8g1..<

8l<

8h1..< 8m<

Æ

Æ

8j1..<

Æ

8k1..<

Æ






{z}(1) ⊇ r−1{x,z}(1).

8f< Æ

8g1..<

8l<

8h1..< 8m<

Æ

Æ

8j1..<

Æ

8k1..<

Æ 8f< Æ

8g1..<

Æ

8h1..< 8i< Æ

8j1..<

Æ

8k1..<

Æ

s−1{z}(1) is componentwise constrained by r−1

{x,z}(1),following the Godel skeleton.




{z}(1) ⊇ r−1{x,z}(1).

8f< Æ

8g1..<

0z

8h1..< z

0

Æ

8j1..<

Æ

8k1..<

Æ 0xz 0x

z

0z

x xz x

z

z

x

0






{z}(1) ⊇ r−1{x,z}(1).

8f< Æ

8g1..<

0z

8h1..< z

0

Æ

8j1..<

Æ

8k1..<

Æ 0z 0

z

0z

z

z

z

0




References

F. Montagna.Interpolation and Beth’s Property in Propositional Many-Valued Logics: A Semantic Investigation.Ann. Pure Appl. Logic, 141:148–179, 2006.

S. Bova.BL-Functions and Free BL-Algebra.PhD Thesis, University of Siena, 2008.

P. Hajek.Metamathematics of Fuzzy Logic.Kluwer, Dordrecht, 1998.

M. Busaniche and D. Mundici.Geometry of Robinson Consistency in Łukasiewicz Logic.Ann. Pure Appl. Logic, 147:1–22, 2007.

H. Kihara and H. Ono.Interpolation Properties, Beth Definability Properties and Amalgamation Properties for Substructural Logics.J. Logic Comput., 2009.

M. Baaz and H. Veith.Interpolation in Fuzzy Logic.Arch. Math. Logic, 38:461–489, 1999.

N. Galatos, P. Jipsen, T. Kowalski, and H. Ono.Residuated Lattices: An Algebraic Glimpse at Substructural Logics.Elsevier, 2007.


Thanks!

Combinatorics of Interpolation in Gödel Logic - TU Wien · GODEL¨ LOGIC ONGOING WORK Combinatorics of Interpolation in Godel¨ Logic Simone Bova [email protected] Department of

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