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Interpolation Using Sequents and Their Generalizations Roman Kuznets Embedded Computing Systems / Theory and Logic TU Wien PhDs in Logic X May 2, 2018 Roman Kuznets Interpolation using sequents and beyond 1 / 54 Craig Interpolation

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SLIDE 1

Interpolation Using Sequents and Their Generalizations

Roman Kuznets Embedded Computing Systems / Theory and Logic TU Wien PhDs in Logic X May 2, 2018

Roman Kuznets Interpolation using sequents and beyond 1 / 54

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SLIDE 2

Craig Interpolation

Definition A logic L has the Craig Interpolation Property (CIP) if, whenever L ⊢ A → B, there exists an interpolant C such that C only uses that common language of A and B L ⊢ A → C L ⊢ C → B These three statements are abbreviated as A → B ← − C Motivation logical (connections to Beth definability, Robinson joint consistency) mathematical (connections to algebra) applied (applications in computer science)

Roman Kuznets Interpolation using sequents and beyond 2 / 54

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SLIDE 3

Craig Interpolation: Example

Example (Classical propositional logic CPL) CPL ⊢ p ∧ ¬q → (q → r) One of the interpolants is ¬q. Indeed, CPL ⊢ p ∧ ¬q → ¬q and CPL ⊢ ¬q → (q → r) In fact, in this case ¬q is a Lyndon interpolant: not only q but its polarity is common among p ∧ ¬q and q → r.

Roman Kuznets Interpolation using sequents and beyond 3 / 54

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SLIDE 4

The Role of Constants

What are interpolants of q → (p → p) and ¬(q → q) → p? Definition Craig Interpolation A logic without constants in the language has CIP if the interpolation property holds for every valid A → B such that neither B nor ¬A is valid by itself.

Roman Kuznets Interpolation using sequents and beyond 4 / 54

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SLIDE 5

Craig Interpolation: Proof Methods

Craig, 1957: Sequent calculus for predicate logic modulo propositional one. Proof theoretic method is the most robust constructive way of proving interpolation. Constructive means that the interpolant C is constructed by an algorithm, given a “derivation of A → B”

Roman Kuznets Interpolation using sequents and beyond 5 / 54

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SLIDE 6

Craig Interpolation: First Proper Proof-Theoretic Proofs

Maehara(1960, in Japanese/German) Maehara and Takeuti (1961) for predicate infinite language Sch¨ utte (1962, in German) for intuitionistic predicate logic Fitting (1983) for several modal logics

Roman Kuznets Interpolation using sequents and beyond 6 / 54

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SLIDE 7

Craig or Sequent Interpolation: naively

From Formulas to Sequents... C interpolates A → B if A → C and C → B and CL ↓ C interpolates A → B if A ⇒ C and C ⇒ B and CL ↓ C interpolates Γ ⇒ ∆ if Γ ⇒ C and C ⇒ ∆ and CL

CL is the common language condition

Example (Problem) p ⇒ p ⇒ p → p

Roman Kuznets Interpolation using sequents and beyond 7 / 54

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SLIDE 8

Craig or Sequent Interpolation: naively

From Formulas to Sequents... C interpolates A → B if A → C and C → B and CL ↓ C interpolates A → B if A ⇒ C and C ⇒ B and CL ↓ C interpolates Γ ⇒ ∆ if Γ ⇒ C and C ⇒ ∆ and CL

CL is the common language condition

...naively Already Maehara noted that formulas move between the antecedent Γ and succedent ∆. These moves make preserving the CL problematic.

Roman Kuznets Interpolation using sequents and beyond 8 / 54

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SLIDE 9

Craig Interpolation: First Proper Proof-Theoretic Proofs

Maehara(1960, in Japanese/German) Maehara and Takeuti (1961) for predicate infinite language Sch¨ utte (1962, in German) for intuitionistic predicate logic Fitting (1983) for several modal logics

Roman Kuznets Interpolation using sequents and beyond 9 / 54

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SLIDE 10

These Are Not the Sides You’re Looking for

Omitting the CL as always present, C interpolates Γ ⇒ ∆ if Γ ⇒ C and C ⇒ ∆ ↓ C interpolates Γ; Π ⇒ Σ; ∆ if C = ⇒ A for some A ∈ Γ

  • r

B for some B ∈ Σ and C = ⇒ A for some A ∈ Π

  • r

B for some B ∈ ∆ ↓ C interpolates Γ; Π ⇒ Σ; ∆ if C = ⇒ Γ ⇒ Σ and C = ⇒ Π ⇒ ∆

Roman Kuznets Interpolation using sequents and beyond 10 / 54

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SLIDE 11

Proof-theoretic Method

It is not entirely correct to say that Craig interpolation is proved by induction on the sequent derivation. In reality, one finds a suitable generalization of CIP and proves that by induction on the split version of the sequent derivation.

Roman Kuznets Interpolation using sequents and beyond 11 / 54

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SLIDE 12

Splitting Sequent Rules

Γ, A, B ⇒ ∆ Γ, A ∧ B ⇒ ∆ ւ ց Γ, A, B; Π ⇒ Σ; ∆ Γ, A ∧ B; Π ⇒ Σ; ∆ Γ; A, B, Π ⇒ Σ; ∆ Γ; A ∧ B, Π ⇒ Σ; ∆ Γ ⇒ A, ∆ Γ ⇒ B, ∆ Γ ⇒ A ∧ B, ∆ ւ ց

Γ; Π ⇒ Σ, A; ∆ Γ; Π ⇒ Σ, B; ∆ Γ; Π ⇒ Σ, A ∧ B; ∆ Γ; Π ⇒ Σ; A, ∆ Γ; Π ⇒ Σ; B, ∆ Γ; Π ⇒ Σ; A ∧ B, ∆

Roman Kuznets Interpolation using sequents and beyond 12 / 54

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SLIDE 13

Splitting Initial Sequents

Γ; A, Π ⇒ Σ, A; ∆ Γ, A; Π ⇒ Σ, A; ∆ տ ր Γ, A ⇒ A, ∆ ւ ց Γ; A, Π ⇒ Σ; A, ∆ Γ, A; Π ⇒ Σ; A, ∆

Roman Kuznets Interpolation using sequents and beyond 13 / 54

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SLIDE 14

General Plan of Finding an Interpolant

Given a theorem A → B

1 Find a sequent derivation of A ⇒ B 2 Split the endsequent to be A; ⇒; B 3 Propagate this split from the endsequent to the leaves turning

the derivation into a split derivation

4 Find interpolants for each split initial sequent in a leaf 5 Propagate interpolants from the leaves to the endsequent

using rules for transforming interpolants for each rule

6 Process the interpolant for the endsequent A; ⇒; B into the

best form suitable for A → B

Roman Kuznets Interpolation using sequents and beyond 14 / 54

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SLIDE 15

Interpolating Initial Sequents

Γ; A, Π ⇒ Σ, A; ∆ ← − ¬A Γ, A; Π ⇒ Σ, A; ∆ ← − ⊥ տ ր ւ ց Γ; A, Π ⇒ Σ; A, ∆ ← − ⊤ Γ, A; Π ⇒ Σ; A, ∆ ← − A

Roman Kuznets Interpolation using sequents and beyond 15 / 54

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SLIDE 16

Typical Interpolant Transformation

Γ, A, B; Π ⇒ Σ; ∆ ← − C Γ, A ∧ B; Π ⇒ Σ; ∆ ← − C Γ; A, B, Π ⇒ Σ; ∆ ← − C Γ; A ∧ B, Π ⇒ Σ; ∆ ← − C

Roman Kuznets Interpolation using sequents and beyond 16 / 54

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SLIDE 17

Cutting to the Chase

Γ; Π ⇒ Σ; A, ∆ ← − C 1 Γ; A, Π ⇒ Σ; ∆ ← − C 2 Γ; Π ⇒ Σ; ∆ ← − ? Analyticity Cut rule is problematic because it violates the subformula property.

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SLIDE 18

Interpolant Transformation for Two-Premise Rules

Γ; Π ⇒ Σ; A, ∆ ← − C1 Γ; Π ⇒ Σ; B, ∆ ← − C2 Γ; Π ⇒ Σ; A ∧ B, ∆ ← − C1 ∧ C2 Γ; Π ⇒ Σ, A; ∆ ← − C1 Γ; Π ⇒ Σ, B; ∆ ← − C2 Γ; Π ⇒ Σ, A ∧ B; ∆ ← − C1 ∨ C2

Roman Kuznets Interpolation using sequents and beyond 18 / 54

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SLIDE 19

Interpolant Transformations for Structural Rules

Γ; Π ⇒ Σ; ∆ ← − C Γ, Γ′; Π, Π′ ⇒ Σ, Σ′; ∆, ∆′ ← − C Γ; Π ⇒ Σ; A, A, ∆ ← − C Γ; Π ⇒ Σ; A, ∆ ← − C Γ; Π ⇒ Σ; A, B, ∆ ← − C Γ; Π ⇒ Σ; B, A, ∆ ← − C

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SLIDE 20

First Result

Theorem CPL has Craig and Lyndon interpolation.

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SLIDE 21

Going Modal

Γ ⇒ B Γ ⇒ B ւ ց Γ; Π ⇒ B; Γ; Π ⇒ B; Γ; Π ⇒; B Γ; Π ⇒; B Γ; Π ⇒ B; ← − C Γ; Π ⇒ B; ← − C Γ; Π ⇒; B ← − C Γ; Π ⇒; B ← − C

Roman Kuznets Interpolation using sequents and beyond 21 / 54

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SLIDE 22

Kripke Models

Definition A Kripke model M = (W , R, V ) is a triple, where W = ∅ R ⊆ W × W V : Prop → 2W M, w p iff w ∈ V (p) M, w ⊥ M, w ¬A iff M, w A M, w A ∧ B iff M, w A and M, w B M, w A ∨ B iff M, w A or M, w B M, w A → B iff M, w A or M, w B M, w A iff M, v A for all v s.t. wRv M, w A iff M, v A for some v s.t. wRv

Roman Kuznets Interpolation using sequents and beyond 22 / 54

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SLIDE 23

Going Modal

Γ ⇒ B Γ ⇒ B ւ ց Γ; Π ⇒ B; Γ; Π ⇒ B; Γ; Π ⇒; B Γ; Π ⇒; B Γ; Π ⇒ B; ← − C Γ; Π ⇒ B; ← − C Γ; Π ⇒; B ← − C Γ; Π ⇒; B ← − C

Roman Kuznets Interpolation using sequents and beyond 23 / 54

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Un-sequent-ial Modal Logics

Relatively few modal logics have cut-free sequent systems. Solutions Abandon the ship proof theory Use cut analytically (shh! it’s not actually a problem) Use sequents but for a computable pre-image of your logic Extend sequent formalism

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SLIDE 25

The Reach of Sequents

Modal cube from the Stanford Encyclopedia of Philosophy CIP and LIP for normal modal logics axiomatized by any combination of axioms d, t, b, 4, and 5.

  • S4
  • S5
  • T
  • TB
  • D4
  • D45
  • D5
  • D
  • DB
  • K4
  • K45
  • B5
  • K5
  • K
  • KB

6 via sequents 6 via analytic cuts using advanced calculi

Roman Kuznets Interpolation using sequents and beyond 25 / 54

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SLIDE 26

Finding Hypersequent Generalizations of CIP

Hypersequents (Mints, 1968; Pottinger, 1983; Avron, 1987) Modal logics: ıh(Γ1 ⇒ ∆1 | · · · | Γn ⇒ ∆n) =

n

  • i=1

ıs(Γi ⇒ ∆i)

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SLIDE 27

Intermission: Gone But Not Forgotten

Before Avron minted the term hypersequent, Mints called them corteges (archaic Russian word for a “tuple”): Г. Е. Минц (G. E. Minc), О некоторых исчислениях модальной логики, Труды МИАН СССР, 98:88–111, 1968. http://mi.mathnet.ru/eng/tm/v98/p88

Roman Kuznets Interpolation using sequents and beyond 27 / 54

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SLIDE 28

Finding Hypersequent Generalizations of CIP

Hypersequents (Mints, 1968; Pottinger, 1983; Avron, 1987) Modal logics: ıh(Γ1 ⇒ ∆1 | · · · | Γn ⇒ ∆n) =

n

  • i=1

ıs(Γi ⇒ ∆i) This is not a syntactic implication. The interpretation is false at a Kripke world w iff (∃w1) . . . (∃wn)

  • (∀i)wRwi∧

(∀i)(∀A ∈ Γi)wi A ∧ (∀j)(∀B ∈ ∆j)wj B

  • r

(∃w1) . . . (∃wn)

  • (∀i)wRwi ∧ (∀i)(wi Γi ⇒ ∆i
  • Roman Kuznets

Interpolation using sequents and beyond 28 / 54

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SLIDE 29

I Can’t Find No Satisfaction for Hypersequents

Unlike sequents, where both validity and satisfiability are defined, there seemed to be no definition for a hypersequent to be true. Definition (negation of falsehood for the translation) Let w(·) : {1, . . . , n} → W be a function to worlds of a Kripke model (W , R, V ) accessible from some world w ∈ W , i.e., wRwi: w(·) Γ1 ⇒ ∆1 | · · · | Γn ⇒ ∆n ⇐ ⇒

n

  • i=1
  • A∈Γi

wi A ∨

  • B∈∆i

wi B

  • In words, either some antecedent formula is false or

some succedent formula is true (in their respective worlds) Theorem This definition of truth for classical modal hypersequents yields the standard notion of validity for them via a classical disjunction

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SLIDE 30

Disjunction Is the New Implication Either Way You Split

Splitting a hypersequent w(·) Γ1; Γ′

1 ⇒ ∆1; ∆′ 1 | · · · | Γn; Γ′ n ⇒ ∆n; ∆′ n

⇐ ⇒ w(·) Γ1 ⇒ ∆1 | · · · | Γn ⇒ ∆n = ⇒ w(·) Γ′

1 ⇒ ∆′ 1 | · · · | Γ′ n ⇒ ∆′ n

That’s the implication to interpolate

Roman Kuznets Interpolation using sequents and beyond 30 / 54

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SLIDE 31

Semantically Inspired Interpolation Statement

Have Implication, Will Interpolate is an interpolant of split hypersequent Γ1; Γ′

1 ⇒ ∆1; ∆′ 1 | · · · | Γn; Γ′ n ⇒ ∆n; ∆′ n iff

1 w(·)

= ⇒ w(·) Γ1 ⇒ ∆1 | · · · | Γn ⇒ ∆n

2 w(·)

= ⇒ w(·) Γ′

1 ⇒ ∆′ 1 | · · · | Γ′ n ⇒ ∆′ n

3 common language conditions.

Breaking the orthodoxy In general, this interpolation statement cannot be expressed as a

  • hypersequent. However, like in the sequent case, it coincides with

Craig interpolation for single-component hypersequents A; ⇒ ; B.

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The Nature of Interpolant

One last hurdle Since hypersequent formulas are evaluated at various worlds of the model, so should be parts of the interpolant. For instance, p; ⇒ ; p | ⇒ and ⇒ | p; ⇒ ; p are interpolated by p, evaluated at w1 and w2 respectively. These interpolants cannot be mere formulas. We write them as 1: p and 2: p and call uniformulas. To create a Boolean logic on uniformulas, we use external conjunctions and disjunctions . Note that external negation is definable via De Morgan

Roman Kuznets Interpolation using sequents and beyond 32 / 54

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SLIDE 33

Truth for Interpolants

w(·) i : C iff wi C w(·) 1 2 iff w(·) 1 and w(·) 2 w(·) 1 2 iff w(·) 1 or w(·) 2

Roman Kuznets Interpolation using sequents and beyond 33 / 54

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SLIDE 34

Interpolant Transformations for Hypersequent Rules

A rule for S5 Γ1, A; Γ′

1 ⇒ ∆1; ∆′ 1 | · · · | Γn, A; Γ′ n ⇒ ∆n; ∆′ n ←

− Γ1; Γ′

1 ⇒ ∆1; ∆′ 1 | · · · | Γn, A; Γ′ n ⇒ ∆n; ∆′ n ←

−

Roman Kuznets Interpolation using sequents and beyond 34 / 54

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SLIDE 35

Interpolant Transformations for Hypersequent Rules II

A more interesting rule for S5 Γ1; Γ′

1 ⇒ ∆1; ∆′ 1 | · · · | Γn; Γ′ n ⇒ ∆n; ∆′ n |; ⇒; A ←

− Γ1; Γ′

1 ⇒ ∆1; ∆′ 1 | · · · | Γn; Γ′ n ⇒ ∆n; A, ∆′ n ←

− ′ where =

m

  • i=1
  • (n + 1): Di

n

  • j=1 j : Cij
  • ⇓

=

m

  • i=1
  • n: Di

n

  • j=1 j : Cij
  • Roman Kuznets

Interpolation using sequents and beyond 35 / 54

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SLIDE 36

Some Hyper Results

Theorem (K, 2016) S5, i.e., logic of total frames, has CIP and LIP (via hypersequents) Theorem (K, 2018) S4.2, i.e., logic of confluent, reflexive, and transitive frames, has CIP and LIP (via hypersequents) Theorem (Maksimova, 1982) S4.3, i.e., the logic of linear, reflexive, and transitive frames, does not have CIP. A problematic hypersequent rule is G | Σ, Γ ⇒ Π G | Θ, ∆ ⇒ Λ G | Σ, ∆ ⇒ Π | Θ, Γ ⇒ Λ where G = Γ1 ⇒ ∆1 | · · · | Γn ⇒ ∆n

Roman Kuznets Interpolation using sequents and beyond 36 / 54

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SLIDE 37

The Strange Case of G¨

  • del Logic

Propositional G¨

  • del Logic G of linear intuitionistic Kripke frames

S4.3 G (A → B) ∨ (B → A) (A → B) ∨ (B → A) Theorem (Maksimova, 1977) G¨

  • del logic has CIP.

Hypersequent rule G | Γ, Γ′ ⇒ ∆ G | Λ, Λ′ ⇒ ∆′ G | Γ, Λ′ ⇒ ∆ | Λ, Γ′ ⇒ ∆′ Example (Bad split) p; ⇒ p; ; q ⇒; q ; q ⇒ p; | p; ⇒; q

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SLIDE 38

Structural Solutions

Linear Sequents for G¨

  • del Logic (K–Lellmann, 2018)

G/ /Γ ⇒ ∆/ /A ⇒ B G/ /Γ ⇒ ∆, A → B

G/ /Γ ⇒ ∆/ /A ⇒ B/ /Σ ⇒ Π/ /H G/ /Γ ⇒ ∆/ /Σ ⇒ Π, A → B/ /H G/ /Γ ⇒ ∆, A → B/ /Σ ⇒ Π/ /H

G/ /Γ, A ⇒ ∆/ /Σ, A ⇒ Π/ /H G/ /Γ, A ⇒ ∆/ /Σ ⇒ Π/ /H Theorem (K–Lellmann, 2018) G¨

  • del logic has CIP and LIP (new) (via linear sequents).

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SLIDE 39

What I Tried to Sweep under the Intermediate Rug

Problem Can ; p ⇒ p; be interpolated in intuitionistic logic? An interpolant C would have the properties: w C = ⇒ w p w C = ⇒ w p From the first one, it follows that ⊢ ¬p → C. From the second one, it follows that ⊢ C → ¬p. The only potential interpolant is ¬p, but it violates the 1st property. Retrieving Boolean logic on interpolants Add classical external negation: A with the definition w A ⇐ ⇒ w A. Objection But A cannot be represented by an intuitionistic formula.

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SLIDE 40

Truth for Interpolants: Intuitionistic edition

w(·) i : C iff wi C w(·) i : C iff wi C w(·) 1 2 iff w(·) 1 and w(·) 2 w(·) 1 2 iff w(·) 1 or w(·) 2

Roman Kuznets Interpolation using sequents and beyond 40 / 54

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SLIDE 41

Removing Semantic Connectives: Classical Case

Interpolants of w :A; ⇒ ; w :B w :D1 w :D2

  • w :(D1 ∧ D2)

w :D1 w :D2

  • w :(D1 ∨ D2)

Eventually, interpolant is transformed into w :C, and w :C interpolates w :A; ⇒ ; w :B iff C interpolates A → B.

Roman Kuznets Interpolation using sequents and beyond 41 / 54

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SLIDE 42

Removing Semantic Connectives: Intuitionistic Case

Interpolants of w :A; ⇒ ; w :B w :D1 w :D2

  • w :(D1 ∧ D2)

w :D1 w :D2

  • w :(D1 ∨ D2)

w :D1 w :D2

  • w :(D1 ∨ D2)

w :D1 w :D2

  • w :(D1 ∧ D2)

Eventually, interpolant is transformed into a DNF =

n

  • i=1

(w :Ci w :Di) and interpolates w :A; ⇒ ; w :B implies

n

  • i=1

(Ci → Di) interpolates A → B.

Roman Kuznets Interpolation using sequents and beyond 42 / 54

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SLIDE 43

Labelled Sequents

Labelled sequents (Gabbay, 1993; Vigan`

  • , 2000; Negri, 2005)

Labelled sequent has the form u1Ro1, . . . , ukRok, w1 :A1, . . . , wn :An ⇒ v1 :B1, . . . , vm :Bm and is true for [ [·] ] : Labels → W iff (∀i)[ [ui ] ]R [ [oi ] ] = ⇒

  • n
  • j=1

[ [wj ] ] Aj ∨

m

  • l=1

[ [vl ] ] Bl

  • uiRoi are relational atoms.

wj :Aj and vl :Bl are labelled formulas. Method adaptation Do not split relational atoms. Use labelled labels as multiformula labels.

Roman Kuznets Interpolation using sequents and beyond 43 / 54

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SLIDE 44

Sample Labelled Rule, Split and Interpolated

A labelled rule for K4 Rel, wRu, uRv, wRv, Γ; Π ⇒ Σ; ∆ ← − Rel, wRu, uRv, Γ; Π ⇒ Σ; ∆ ← −

Roman Kuznets Interpolation using sequents and beyond 44 / 54

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SLIDE 45

Labelled Results: cr` eme de la cr` eme

Horn clauses P1 ∧ . . . ∧ Pm → Q P1 ∧ . . . ∧ Pm → ⊥ where Pi and Q are relational atoms of the form wRv Theorem (Marx, 1995, for CIP; K, via labelled sequents, 2016) Any logic complete with respect to a class of Kripke frames described by a combination of Horn clauses has CIP and LIP.

Roman Kuznets Interpolation using sequents and beyond 45 / 54

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SLIDE 46

Labelled Results: concrete examples

Interpolable Kripke-frame properties (also with a shift) Reflexivity ⊤ → wRw uRw → wRw Transitivity wRo ∧ oRr → wRr uRw ∧ wRo ∧ oRr → wRr Euclideanness wRo ∧ wRr → oRr uRw ∧ wRo ∧ wRr → oRr Symmetry wRo → oRw uRw ∧ wRo → oRw shift uRw above can be replaced with an arbitrary conjunction

  • f relational atoms

Irreflexive discreteness wRv → ⊥ (1, m)-transitivity w0Rw1 ∧ . . . ∧ wm−1Rwm → w0Rwm Corollary Logics with LIP: K, T, KB, K4, K5, S4, Verum, S5, K4B, K41,m.

Roman Kuznets Interpolation using sequents and beyond 46 / 54

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SLIDE 47

Labelled Results: more

Horn clauses with equality (e.g., functionality) Some Horn clauses with existential quantifiers (e.g., seriality) Most Geach logics (wRhv ∧ wRju → ∃y(vRiy ∧ uRky) h, i, j, k ≥ 1, y / ∈ {w, v, u})

Roman Kuznets Interpolation using sequents and beyond 47 / 54

slide-48
SLIDE 48

Labelled Results: meta-constructivity

Final result Algorithm with input: list of first-order Kripke-frame conditions

  • utput: algorithm for constructing an interpolant with

input: a labelled-sequent derivation of w : A; ⇒ ; w : B

  • utput: an interpolant C of A → B

Roman Kuznets Interpolation using sequents and beyond 48 / 54

slide-49
SLIDE 49

Main Lessons

don’t be dogmatic semantics is your friend the most expressive calculus is not always the best one matching the sequent structure to the model structure can help Roman Kuznets roman@logic.at https://sites.google.com/site/kuznets/publications

Roman Kuznets Interpolation using sequents and beyond 49 / 54

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SLIDE 50

Bibliography

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Logic 52(4):939–951. 1987.

  • M. Baaz, A. Ciabattoni, C. G. Ferm¨
  • uller. Hypersequent calculi

for G¨

  • del logics—a survey. Journal of Logic and Computation

13(6):835–861. 2003.

  • W. Craig. Three uses of the Herbrand–Gentzen theorem in

relating model theory and proof theory. Journal of Symbolic Logic 22(3):269–285. 1957.

  • M. Fitting. Proof Methods for Modal and Intuitionistic Logics.
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2015.

  • D. M. Gabbay, L. Maksimova. Interpolation and Definability:

Modal and Intuitionistic Logic. Clarendon Press, 2005.

Roman Kuznets Interpolation using sequents and beyond 50 / 54

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SLIDE 51

Bibliography

  • J. Garson, Modal logic. In: The Stanford Encyclopedia of

Philosophy.

  • R. Kuznets. Craig interpolation via hypersequents. In:

Concepts of Proof in Mathematics, Philosophy, and Computer

  • Science. Ontos Mathematical Logic 6:193–214. De Gruyter,

2016.

  • R. Kuznets. Proving Craig and Lyndon interpolation using

labelled sequent calculi. In: Proceedings of JELIA 2016. Lecture Notes in Computer Science 10012:320–335. Springer, 2016.

  • R. Kuznets. Interpolation method for multicomponent sequent
  • calculi. In: Proceedings of LFCS 2016. Lecture Notes in

Computer Science 9537:202–218. Springer, 2016.

  • R. Kuznets. Multicomponent Proof-theoretic Method for

Proving Interpolation Properties. Annals of Pure and Applied Logic, in press. 2018.

Roman Kuznets Interpolation using sequents and beyond 51 / 54

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SLIDE 52

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