from display calculi to decision procedures for full ...apt13.unibe.ch/slides/gore.pdf · hyland...

From Display Calculi to Decision Procedures forFull Intuitionistic Linear Logic

Ranald Clouston, Jeremy Dawson, Rajeev Gore, Alwen Tiu

Logic and Computation GroupResearch School of Computer ScienceThe Australian National University

[email protected]

December 14, 2013

Published in Proc. 22nd EACSL Annual Conference

on Computer Science Logic, Turin September 2013 as

“Annotation-free sequent calculi for Full Intuitionistic Linear Logic”

[email protected]

Overview

What is FILL?

Existing sequent calculi

A Display Calculus for FILL

Nested Sequent Calculus for FILL

Separation

Decidability and Complexity

Further Work

Full Intuitionistic Linear Logic

LL: modal substructural logic without weakening and withoutcontraction ⊗,⊕, 1, 0,(,∧,∨,>,⊥, !, ? Girard 1987

MALL: ⊗,⊕, 1, 0,(,∧,∨,>,⊥ drop exponentials

MILL: ⊗, 1,( intuitionistic

MLL: ⊗, 1,(,⊕, 0 classical ((A ( 0) ( 0) ( A

FILL: ⊗, 1,(,⊕, 0 Hyland and de Paiva 1993

Categorial Semantics for FILL

(⊗, 1,() is a symmetric monoidal closed structure

A⊗ B ( C iff A ( (B ( C ) iff B ( (A ( C )

(A⊗ 1) ( A and A ( (A⊗ 1)

(⊕, 0) is a symmetric monoidal structure

(A⊕ B) ( (B ⊕ A)

(A⊕ 0) ( A and A ( (A⊕ 0)

interaction via either of

weak distributivity ((A⊗ B)⊕ C ) ( (A⊗ (B ⊕ C ))

Grishin(b) ((A ( B)⊕ C ) ( (A ( (B ⊕ C ))

Collapse to (classical) MLL: if we add converse of Grishin(b)Grishin(a) (A ( (B ⊕ C )) ( ((A ( B)⊕ C )

Proof Theory of Full Intuitionistic Linear Logic

LL: substructural logic without weakening and without contraction

MILL ⊗ ( 1 intuitionistic cut-elimination

Γ1 ` A Γ2 ` BΓ1, Γ2 ` A⊗ B

Γ,A ` B

Γ ` A ( B

MLL ⊗ ( 1 ⊕ ⊥ classical cut-elimination

Γ1,A ` B,∆1 Γ2,A ` B,∆2

Γ1, Γ2 ` ∆1,A⊗ B,∆2

Γ ` A,B,∆

Γ ` A⊕ B,∆

Γ,A ` B,∆

Γ ` A ( B,∆

FILL ⊗ ( 1 ⊕ ⊥ intuitionistic cut-elimination fails

Γ1,A ` B,∆1 Γ2,A ` B,∆2

Γ1, Γ2 ` ∆1,A⊗ B,∆2

Γ ` A,B,∆

Γ ` A⊕ B,∆

Γ,A ` B

Γ ` A ( B,∆

Problem and a solutions via annotated derivations

Remember: we need comma on the right to accommodate ⊕

Problem and existing solutions:

multiple conclusions single conclusion existing solutions

Γ,A ` B,∆

Γ ` A ( B,∆

Γ,A ` B

Γ ` A ( B,∆

Γ,A ` B,∆(†)

Γ ` A ( B,∆

unsound no cut-elimination cut-elimination

†: side-conditions which ensure that A is “independent” of ∆

Hyland, de Paiva 1993: type assignments to ensure that thevariable typed by A not appear free in the terms typed by ∆

Further problems and solutions using annotations

Hyland and de Paiva 1993:Γ,A ` B,∆

(†)Γ ` A ( B,∆

†: side-conditions which ensure that A is “independent” of ∆

Bierman 1996: (a⊕ b)⊕ c ` a, ((b ⊕ c) ( d)⊕ (e ( (d ⊕ e))has no cut-free derivation in the Hyland and de Paiva calculus

Bierman, Bellin 1996: refined annotations to regain cut-elimination

Brauner and de Paiva 1997: annotate rules with a binary relationbetween antecedent formulae and succedent formulae, whicheffectively trace variable occurrence

What do derivations look like ?

Hyland , de Paiva and Biermann

(( R) legal as v and (w ⊕ x ( y)⊕ z share no free variables.

v : a ` v : a w : b ` w : bv ⊕ w : a⊕ b ` v : a,w : b x : c ` x : c

(v ⊕ w)⊕ x : (a⊕ b)⊕ c ` v : a,w : b, x : c

(v ⊕ w)⊕ x : (a⊕ b)⊕ c ` v : a,w ⊕ x : b ⊕ c y : d ` y : d

(v ⊕ w)⊕ x : (a⊕ b)⊕ c,w ⊕ x ( y : b ⊕ c ( d ` v : a, y : d z : e ` z : e

(v ⊕ w)⊕ x : (a⊕ b)⊕ c, (w ⊕ x ( y)⊕ z : (b ⊕ c ( d)⊕ e ` v : a, y : d , z : e

(v ⊕ w)⊕ x : (a⊕ b)⊕ c, (w ⊕ x ( y)⊕ z : (b ⊕ c ( d)⊕ e ` v : a, y ⊕ z : d ⊕ e

(v ⊕ w)⊕ x : (a⊕ b)⊕ c ` v : a, λ(w ⊕ x ( y)⊕ z(b⊕c(d)⊕e .(y ⊕ z) : (b ⊕ c ( d)⊕ e ( d ⊕ e

But the type annotations have no computational content

Bellin-style Proof

(( R) is legal because r is not free inlet t be u ⊕ - in let u be v ⊕ - in v

v : a ` v : a w : b ` w : b

u : a ⊕ b ` let u be v ⊕ - in v : a, let u be -⊕ w in w : b x : c ` x : c

t : (a ⊕ b)⊕ c ` let t be u ⊕ - in let u be v ⊕ - in v : a, let t be u ⊕ - in let u be -⊕ w in w : b, let t be -⊕ x in x : c

t : (a ⊕ b)⊕ c ` let t be u ⊕ - in let u be v ⊕ - in v : a, (let t be u ⊕ - in let u be -⊕ w in w)⊕ (let t be -⊕ x in x) : b ⊕ c y : d ` y : d

t : (a ⊕ b)⊕ c, s : b ⊕ c ( d ` let t be u ⊕ - in let u be v ⊕ - in v : a, (s(let t be u ⊕ - in let u be -⊕ w in w)⊕ (let t be -⊕ x in x)) : d z : e ` z : e

t : (a ⊕ b)⊕ c, r : (b ⊕ c ( d)⊕ e ` let t be u ⊕ - in let u be v ⊕ - in v : a, let r be s ⊕ - in (s(let t be u ⊕ - in let u be -⊕ w in w)⊕ (let t be -⊕ x in x)) : d, let s be -⊕ z in z : e

t : (a ⊕ b)⊕ c, r : (b ⊕ c ( d)⊕ e ` let t be u ⊕ - in let u be v ⊕ - in v : a, (let r be s ⊕ - in (s(let t be u ⊕ - in let u be -⊕ w in w)⊕ (let t be -⊕ x in x)))⊕ (let s be -⊕ z in z) : d ⊕ e

t : (a ⊕ b)⊕ c ` let t be u ⊕ - in let u be v ⊕ - in v : a, λr (b⊕c(d)⊕e .(let r be s ⊕ - in (s(let t be u ⊕ - in let u be -⊕ w in w)⊕ (let t be -⊕ x in x)))⊕ (let s be -⊕ z in z) : (b ⊕ c ( d)⊕ e ( d ⊕ e

Again, the type annotations are not given any computationalcontent and this derivation does not even fit on the page!

Brauner and de Paiva-style proof

(( R) legal because (b ⊕ c ( d)⊕ e is not related to a

(a, a)a ` a

(b, b)b ` b

(a⊕ b, a), (a⊕ b, b)a⊕ b ` a, b

(c, c)c ` c

((a⊕ b)⊕ c, a), ((a⊕ b)⊕ c, b), ((a⊕ b)⊕ c, c)(a⊕ b)⊕ c ` a, b, c

((a⊕ b)⊕ c, a), ((a⊕ b)⊕ c, b ⊕ c)(a⊕ b)⊕ c ` a, b ⊕ c

(d , d)d ` d

((a⊕ b)⊕ c, a), ((a⊕ b)⊕ c, d), (b ⊕ c ( d , d)(a⊕ b)⊕ c, b ⊕ c ( d ` a, d

(e, e)e ` e

((a⊕ b)⊕ c, a), ((a⊕ b)⊕ c, d), ((b ⊕ c ( d)⊕ e, d), ((b ⊕ c ( d)⊕ e, e)(a⊕ b)⊕ c, (b ⊕ c ( d)⊕ e ` a, d , e

((a⊕ b)⊕ c, a), ((a⊕ b)⊕ c, d ⊕ e), ((b ⊕ c ( d)⊕ e, d ⊕ e)(a⊕ b)⊕ c, (b ⊕ c ( d)⊕ e ` a, d ⊕ e

((a⊕ b)⊕ c, a), ((a⊕ b)⊕ c, (b ⊕ c ( d)⊕ e ( d ⊕ e)(a⊕ b)⊕ c ` a, (b ⊕ c ( d)⊕ e ( d ⊕ e

Pure annotational device to track variable occurrences

Cut-free derivation in our display calculus

a ` a b ` ba⊕ b ` a, b c ` c

(a⊕ b)⊕ c ` a, b, c

(a⊕ b)⊕ c < a ` b, c

(a⊕ b)⊕ c < a ` b ⊕ c d ` d

b ⊕ c ( d ` ((a⊕ b)⊕ c < a) > d e ` e

(b ⊕ c ( d)⊕ e ` (((a⊕ b)⊕ c < a) > d), e

(b ⊕ c ( d)⊕ e ` ((a⊕ b)⊕ c < a) > d , e

(b ⊕ c ( d)⊕ e, ((a⊕ b)⊕ c < a) ` d , e

(b ⊕ c ( d)⊕ e, ((a⊕ b)⊕ c < a) ` d ⊕ e

(a⊕ b)⊕ c < a ` (b ⊕ c ( d)⊕ e > d ⊕ e

(a⊕ b)⊕ c < a ` (b ⊕ c ( d)⊕ e ( d ⊕ e

(a⊕ b)⊕ c ` a, (b ⊕ c ( d)⊕ e ( d ⊕ e

No annotations, but many extra structural connectives

Cut-free derivation in the deep nested calculus

a⇒ a, (· ⇒ ·)· ⇒ (b ⇒ b)

b ⇒ (· ⇒ b)

a⊕ b ⇒ a, (· ⇒ b)

· ⇒ (c ⇒ c)

c ⇒ (· ⇒ c)

(a⊕ b)⊕ c ⇒ a, (· ⇒ b, c)

(a⊕ b)⊕ c ⇒ a, (· ⇒ b ⊕ c) · ⇒ (d ⇒ d)

(a⊕ b)⊕ c ⇒ a, (b ⊕ c ( d ⇒ d) · ⇒ (e ⇒ e)

(a⊕ b)⊕ c ⇒ a, ((b ⊕ c ( d)⊕ e ⇒ d , e)

(a⊕ b)⊕ c ⇒ a, ((b ⊕ c ( d)⊕ e ⇒ d ⊕ e)

(a⊕ b)⊕ c ⇒ a, (b ⊕ c ( d)⊕ e ( d ⊕ e

No annotations, only commas as structural connective, butsequents are nested (· · · ⇒ · · · ) · · · ⇒ · · · (· · · ⇒ · · · )

Display calculus for (an extension of) FILL

Structural Constant and Binary Connectives: Φ , < >

Antecedent Structure: Xa Ya ::= A | Φ | Xa,Ya | Xa < Ys

Succcedent Structure: Xs Ys ::= A | Φ | Xs ,Ys | Xa > Ys

Sequent: Xa ` Ys (drop subscripts to avoid clutter)

Display Postulates: reversible structural rules

Xa ` Ya > Zs

Xa,Ya ` Zs

Ya ` Xa > Zs

Za < Ys ` Xs

Za ` Xs ,Ys

Za < Xs ` Ys

Display Property: For every antecedent (succedent) part Z of thesequent X ` Y , there is a sequent Z ` Y ′ (resp. X ′ ` Z )obtainable from X ` Y using only the display postulates,thereby displaying the Z as the whole of one side

Logical rules: introduced formula is always displayed

(id) p ` pX ` A A ` Y(cut)

X ` Y

Φ ` X(1 `)1 ` X

(` 1) Φ ` 1

(0 `) 0 ` ΦX ` Φ(` 0)X ` 0

A,B ` X(⊗ `)

A⊗ B ` XX ` A Y ` B(` ⊗)X ,Y ` A⊗ B

A ` X B ` Y(⊕ `)A⊕ B ` X ,Y

X ` A,B(` ⊕)

X ` A⊕ B

X ` A B ` Y((`)A ( B ` X > Y

X ` A > B(`()

X ` A ( B

A < B ` X(−< `)

A−<B ` XX ` A B ` Y(` −<)X < Y ` A−<B

read upwards, one rule is a “rewrite” while other “constrains”

Structural rules: no occurrences of formula meta-variables

all sub-structural properties captured in a modular way

X ,Φ ` Y(Φ `)

X ` Y

X ` Φ,Y(` Φ)

X ` Y

W , (X ,Y ) ` Z(Ass `)

(W ,X ),Y ` Z

W ` (X ,Y ),Z(` Ass)

W ` X , (Y ,Z )

X ,Y ` Z(Com `)

Y ,X ` Z

Z ` Y ,X(` Com)

Z ` X ,Y

W , (X < Y ) ` Z(Grnb `)

(W ,X ) < Y ` Z

W ` (X > Y ),Z(` Grnb)

W ` X > (Y ,Z )

((A ( B)⊕ C ) ( (A ( (B ⊕ C ))

Example derivation in our display calculus

a ` a b ` b(⊕ `)

a ⊕ b ` a, b c ` c(⊕ `)

(a⊕ b) ⊕ c ` (a, b), c(ass)

(a⊕ b)⊕ c ` a, (b, c)(drp)

(a⊕ b)⊕ c < a ` b, c(` ⊕)

(a⊕ b)⊕ c < a ` b ⊕ c d ` d((`)

b ⊕ c ( d ` ((a⊕ b)⊕ c < a) > d e ` e(⊕ `)

(b ⊕ c ( d) ⊕ e ` (((a⊕ b)⊕ c < a) > d), e(` Grnb)

(b ⊕ c ( d)⊕ e ` ((a⊕ b)⊕ c < a) > (d , e)(rp)

(b ⊕ c ( d)⊕ e,((a⊕ b)⊕ c < a) ` d , e(` ⊕)

(b ⊕ c ( d)⊕ e, ((a⊕ b)⊕ c < a) ` d ⊕ e(rp)

(a⊕ b)⊕ c < a ` (b ⊕ c ( d)⊕ e > (d ⊕ e)(`()

(a⊕ b)⊕ c < a ` (b ⊕ c ( d)⊕ e ( (d ⊕ e)(drp)

(a⊕ b)⊕ c ` a,(b ⊕ c ( d)⊕ e ( (d ⊕ e)

But we implicitly created an occurrence of −< via <

Categorial semantics for bi-intuitionistic linear logic BiILL

(⊗, 1,() is a symmetric monoidal closed structure

A⊗ B ( C iff A ( (B ( C ) iff B ( (A ( C )

(A⊗ 1) ( A and A ( (A⊗ 1)

(−<,⊕, 0) is a symmetric monoidal co-closed structure

A ( (B ⊕ C ) iff (A−<B) ( C iff (A−<C ) ( B

(A⊕ 0) ( A and A ( (A⊕ 0)

interaction via either of

Grishin(b) ((A ( B)⊕ C ) ( (A ( (B ⊕ C ))

dualGrishin(b) ((A⊗ B)−<C ) ( (A⊗ (B−<C ))

Collapse to (classical) MLL: if we add converse of either

Soundness, completeness and cut-elimination

Thm: The sequent X ` Y is derivable iff the formula-translationτa(X ) ( τs(Y ) is BiILL-valid

Proof: the display calculus proof rules and the arrows of the freeBiILL-category are inter-definable.

Thm: If X ` Y is derivable then it is cut-free derivable.

Proof: The rules obey conditions C1-C8 given by Belnap (1982),hence the calculus enjoys cut-admissibility

Method: Essentially Gentzen’s original method

Left-principal cuts: turned into left and right principal cutsRight-principal cuts: turned into left and right principal cutsLeft&right principal cuts: replaced by multiple “smaller” cutsNeed: substitution principles guaranteed by display property

and explicit structural rulesTermination: double induction on grade and size of cut-formula

Proof search via display calculi

Given end-sequent X ` Y how to find a derivation?

Apply (invertible) “rewrite” rules immediately

For “constraint” rules, use structural rules to match structure

Backward Proof-search for display calculus

a ` a b ` ba ⊕ b ` a, b c ` c

(a⊕ b) ⊕ c ` (a, b), c

(a⊕ b)⊕ c ` a, (b, c)(?)

(a⊕ b)⊕ c < a ` b, c(` ⊕)

(a⊕ b)⊕ c < a ` b ⊕ c d ` d((`)

b ⊕ c ( d ` ((a⊕ b)⊕ c < a) > d e ` e(⊕ `)

(b ⊕ c ( d) ⊕ e ` (((a⊕ b)⊕ c < a) > d), e(?)

(b ⊕ c ( d)⊕ e ` ((a⊕ b)⊕ c < a) > (d , e)(?)

(b ⊕ c ( d)⊕ e,((a⊕ b)⊕ c < a) ` d , e(` ⊕)

(b ⊕ c ( d)⊕ e, ((a⊕ b)⊕ c < a) ` d ⊕ e(?)

(a⊕ b)⊕ c < a ` (b ⊕ c ( d)⊕ e > (d ⊕ e)(`()

(a⊕ b)⊕ c < a ` (b ⊕ c ( d)⊕ e ( (d ⊕ e)(?)

(a⊕ b)⊕ c ` a,(b ⊕ c ( d)⊕ e ( (d ⊕ e)

Proof search via display calculi

Given end-sequent X ` Y how to find a derivation?

Apply (invertible) “rewrite” rules immediately

For “constraint” rules, use structural rules to match structure

But there are just too many possibilities for this to be efficient

Need to compile “display”, “undisplay” and “structural rules” intothe logical rules as much as possible (maintaining cut-elimination)

Cut-free derivation in the deep nested calculus

a⇒ a, (· ⇒ ·)· ⇒ (b ⇒ b)

b ⇒ (· ⇒ b)

a⊕ b ⇒ a, (· ⇒ b)

· ⇒ (c ⇒ c)

c ⇒ (· ⇒ c)

(a⊕ b)⊕ c ⇒ a, (· ⇒ b, c)

(a⊕ b)⊕ c ⇒ a, (· ⇒ b ⊕ c) · ⇒ (d ⇒ d)

(a⊕ b)⊕ c ⇒ a, (b ⊕ c ( d ⇒ d) · ⇒ (e ⇒ e)

(a⊕ b)⊕ c ⇒ a, ((b ⊕ c ( d)⊕ e ⇒ d , e)

(a⊕ b)⊕ c ⇒ a, ((b ⊕ c ( d)⊕ e ⇒ d ⊕ e)

(a⊕ b)⊕ c ⇒ a, (b ⊕ c ( d)⊕ e ( d ⊕ e

No annotations, only commas as structural connective, butsequents are nested

Nested sequent calculi: Kashima 1994 (for tense logics)

Nested sequent: where comma is associative and commutative

S T ::= S1, . . . ,Sk ,A1, . . . ,Am ⇒ B1, . . . ,Bn,T1, . . . ,Tl

Context: nested sequent with exactly one hole [ ]

Positive/Negative: if [ ] occurs immediately to right/left of ⇒

Turn Rules: reversible rules using multisets of nested sequentsand formulae

S2 ⇒ (S1 ⇒ T )

S1,S2 ⇒ T(S ⇒ T2)⇒ T1S ⇒ (T1, T2)

Xa ` Ya > Zs

Xa,Ya ` Zs

Ya ` Xa > Zs

Za < Ys ` Xs

Za ` Xs ,Ys

Za < Xs ` Ys

Nested sequent calculi: Kashima 1994 (for tense logics)

Nested sequent: where comma is associative and commutative

S T ::= S1, . . . ,Sk ,A1, . . . ,Am ⇒ B1, . . . ,Bn,T1, . . . ,Tl

Context: nested sequent with exactly one hole [ ]

Positive/Negative: if [ ] occurs immediately to right/left of ⇒Turn Rules: reversible rules using multisets of nested sequents

and formulae

S2 ⇒ (S1 ⇒ T )

S1,S2 ⇒ T(S ⇒ T2)⇒ T1S ⇒ (T1, T2)

Display Property: For every positive (negative) context X [ ] andevery S, there exists T such that T ⇒ S (respectively S ⇒ T )is derivable from X [S] using only the structural turn rulesabove. Thus S is “displayed” in T ⇒ S (resp. S ⇒ T ).

Polarity: note that ⇒ always points to the positive contexts

Shallow nested sequent calculus for BiILL

Logical rules:

p ⇒ p idS ⇒ S ′,A A, T ⇒ T ′

S, T ⇒ S ′, T ′ cut

0⇒ · 0lS ⇒ TS ⇒ T , 0 0r

S ⇒ TS, 1⇒ T 1l · ⇒ 1

1r

S,A,B ⇒ TS,A⊗ B ⇒ T ⊗l

S ⇒ A, T S ′ ⇒ B, T ′

S,S ′ ⇒ A⊗ B, T , T ′ ⊗r

S,A⇒ T S ′,B ⇒ T ′

S,S ′,A⊕ B ⇒ T , T ′ ⊕lS ⇒ A,B, TS ⇒ A⊕ B, T ⊕r

S ⇒ A, T S ′,B ⇒ T ′

S,S ′,A ( B ⇒ T , T ′ (lS ⇒ T , (A⇒ B)

S ⇒ T ,A ( B(r

S, (A⇒ B)⇒ TS,A−<B ⇒ T −<l

S ⇒ A, T S ′,B ⇒ T ′

S,S ′ ⇒ A−<B, T , T ′ −<r

Shallow nested sequent calculus for BiILL

Structural Rules: Grishin (b) analogues

T , (S ⇒ S ′)⇒ T ′

(S, T ⇒ S ′)⇒ T ′gl

S ⇒ (S ′ ⇒ T ′), TS ⇒ (S ′ ⇒ T ′, T )

gr

W , (X < Y ) ` Z(Grnb `)

(W ,X ) < Y ` Z

W ` (X > Y ),Z(` Grnb)

W ` X > (Y ,Z )

Thm: Every formula has a cut-free nested shallow sequentderivation iff it has cut-free display calculus derivation

Thm: if a nested sequent is derivable it is cut-free derivable

Proof: essentially the same as for display calculus

Prof search: how to absorb the turn and gl and gr rules ?

Deep nested sequents: just apply the rules inside contextsX [ ] and U and V are hollow.

X [U , p ⇒ p,V]idd similarly for units (no cut rule)

X [S,A,B ⇒ T ]

X [S,A⊗ B ⇒ T ]⊗d

l

X1[S1 ⇒ A, T1] X2[S2 ⇒ B, T2]

X [S ⇒ A⊗ B, T ]⊗d

r

X1[S1 ⇒ A, T1] X2[S2,B ⇒ T2]

X [S,A ( B ⇒ T ](d

l

X [S ⇒ T , (A⇒ B)]

X [S ⇒ T ,A ( B](d

r

X1[S1,A⇒ T1] X2[S2,B ⇒ T2]

X [S,A⊕ B ⇒ T ]⊕d

l

X [S ⇒ A,B, T ]

X [S ⇒ A⊕ B, T ]⊕d

r

X [S, (A⇒ B)⇒ T ]

X [S,A−<B ⇒ T ]−<d

l

X1[S1 ⇒ A, T1] X2[S2,B ⇒ T2]

X [S ⇒ A−<B, T ]−<d

r

Hollow: X [] contains no formulae (⇒-tree of empty nodes)

Merge: X [ ] ∈ X1[ ] • X2[ ] and S ∈ S1 • S2 and T ∈ T1 • T2

Deep nested sequents: just apply the rules inside contexts

Propagation rules: allow formulae to be moved in a context

X [S ⇒ (S ′,A⇒ T ′), T ]

X [S,A⇒ (S ′ ⇒ T ′), T ]pl1

X [S ′, (S ⇒ A, T )⇒ T ′]

X [S ′, (S ⇒ T )⇒ A, T ′]pr1

X [S, (S ′ ⇒ T ′),A⇒ T ]

X [S, (S ′,A⇒ T ′)⇒ T ]pl2

X [S ⇒ A, (S ′ ⇒ T ′), T ]

X [S ⇒ (S ′ ⇒ A, T ′), T ]pr2

Note: these are not just reversible rules expanded into two!

Note: the rules do not need to track polarity

Thm: the turn rules and rules gl and gr are (cut-free) admissible

Thm: if a nested sequent is (cut-free) derivable in the deepcalculus then it is cut-free derivable in the shallow calculus

Thm: if a nested sequent is cut-free derivable in the shallowcalculus then it is (cut-free) derivable in the deep calculus

Cor: the deep and shallow nested calculi derive the same sequents

From BiILL back to FILL

a ` a b ` ba ⊕ b ` a, b c ` c

(a⊕ b) ⊕ c ` (a, b), c

(a⊕ b)⊕ c ` a, (b, c)(?)

(a⊕ b)⊕ c < a ` b, c(` ⊕)

(a⊕ b)⊕ c < a ` b ⊕ c d ` d((`)

b ⊕ c ( d ` ((a⊕ b)⊕ c < a) > d e ` e(⊕ `)

(b ⊕ c ( d) ⊕ e ` (((a⊕ b)⊕ c < a) > d), e(?)

(b ⊕ c ( d)⊕ e ` ((a⊕ b)⊕ c < a) > (d , e)(?)

(b ⊕ c ( d)⊕ e,((a⊕ b)⊕ c < a) ` d , e(` ⊕)

(b ⊕ c ( d)⊕ e, ((a⊕ b)⊕ c < a) ` d ⊕ e(?)

(a⊕ b)⊕ c < a ` (b ⊕ c ( d)⊕ e > (d ⊕ e)(`()

(a⊕ b)⊕ c < a ` (b ⊕ c ( d)⊕ e ( (d ⊕ e)(?)

(a⊕ b)⊕ c ` a,(b ⊕ c ( d)⊕ e ( (d ⊕ e)


Problem: all our calculi are for BiILL ... are they sound for FILL?

Display calculus: must create antecedent < structures in itsderivation of FILL-formulae in order to display and undisplay

Question: is BiILL a conservative extension of FILL?

we were not able to find a categorial proof for BiILL

Compare: to tense logic Kt say where there is a simple semanticproof that Kt is a conservative extension of K (same frames)


Nested FILL-sequent: nested sequent that has no nesting ofsequents on the left of ⇒ and no occurrences of −<

Why? because then it is free of any traces of −< since only the−<d

l rule creates such nestings

FILLdn: remove −<dl , −<d

r , pl2 and pr1

Separation Thm: nested FILL-sequents are (cut-free) derivable inFILLdn iff they are derivable in the original deep calculus

Proof: if the end-sequent contains only FILL-formulae thenbackward proof search will never create the offending nesting

Thm: every rule of FILLdn preserves FILL-validity downwards

Cor: FILLdn is sound and cut-free complete for FILL-validity

Cor: BiILL is a conservative extension of FILL

Complexity of deciding FILL

Backward proof search: need to try all ways to partition a contextand apply all propagation rules in all ways, but this terminates

Subformula property: every formula in the premises is asubformula of a formula in the conclusion

Complexity: FILL is in NP

Proof: guess a cut-free derivation and check it in time polynomialin size of the given formula. Possible since each formula isintroduced exactly once in any derivation

Complexity: our calculus handles the problem of decidingconstants-only multiplicative linear logic (COMLL) since thennf-formulae of this logic contain no propositional atoms andno implication connectives and it is known to be NP-hard

Thm: deciding FILL-validity is NP-complete

Our Methodology

Start with a categorially formulated logic L

Give a display calculus for L

X Easily shown to be complete.X Easily shown to support a cut-elimination procedure.X Embeds L into an extension so conservativity, and hence

soundness, not clear.X Not well suited for proof search.

Refine to a nested sequent calculus with deep inference.

X Conservativity, and hence soundness, now clear.X Supports proof search and complexity analysis.

Question: how far can we push this methodology?

We know: it works for the exponentials ! and ?

But: BiILL plus additives not conservative over FILL plusadditivies e.g > ` p,> is BiILL-valid but not FILL-valid

Happy (belated) birthday Gerhard!

Belnap’s Eight Conditions a la Kracht

(C1) Each formula variable occurring in some premise of a rule ρ is asubformula of some formula in the conclusion of ρ.

(C2) Congruent parameters is a relation between parameters of theidentical structure variable occurring in the premise and conclusion

(C3) Each parameter is congruent to at most one structure variable inthe conclusion. Equivalently, no two structure variables in theconclusion are congruent to each other.

(C4) Congruent parameters are either all antecedent or all succedentparts of their respective sequent.

(C5) A formula in the conclusion of a rule ρ is either the entireantecedent or the entire succedent. Such a formula is called aprincipal formula of ρ.

(C6/7) Each rule is closed under simultaneous substitution of arbitrarystructures for congruent parameters.

Belnap’s Eight Conditions a la Kracht

(C8) If there are rules ρ and σ with respective conclusions X ` A andA ` Y with formula A principal in both inferences (in the sense of C5)and if cut is applied to yield X ` Y , then either X ` Y is identical toeither X ` A or A ` Y ; or it is possible to pass from the premises of ρand σ to X ` Y by means of inferences falling under cut where thecut-formula always is a proper subformula of A.

X ` C > DX ` C ( D

U ` C D ` ZC ( D ` U > Z

cutX ` U > Z

U ` C

X ` C > DX ,C ` D D ` Z

cutX ,C ` Z

C ` X > Zcut

U ` X > ZX ,U ` Z

X ` U > Z

Devil’s Advocate Questions

Can’t we do all of this with labelled sequent calculi?

Γ,R(x , y , z), x : A ` z : B,∆x and y do not occur in the conclusion

Γ ` y : (A ( B),∆

Almost certainly we can, but the decidability cannot be read off aswe are able to do, and I can’t see the “Grishin” condition

Where are the Grishin rules in the deep calculus?In the logical rules and propagation rules

X1[S1,A⇒ T1] X2[S2,B ⇒ T2]

X [S,A⊕ B ⇒ T ]⊕d

l

X [S ⇒ A, (S ′ ⇒ T ′), T ]

X [S ⇒ (S ′ ⇒ A, T ′), T ]pr2

Isn’t “merge” just another way of hiding annotations? No, it is avery natural definition ... superpose two identical ⇒-trees

from display calculi to decision procedures for full ...apt13.unibe.ch/slides/gore.pdf · hyland...

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