Cut Elimination and Second-Order Quantifier Elimination Alessandra - - PowerPoint PPT Presentation

Cut Elimination and Second-Order Quantifier Elimination

Alessandra Palmigiano

7 December 2017 http://www.appliedlogictudelft.nl

Main questions

◮ Which logics have nice sequent calculi (are properly

displayable)?

◮ If L is properly displayable, which of its axiomatic extensions

are too? Algorithmic correspondence theory can help in answering these questions.

Setting

Expansions of distributive lattice logic with normal modal operators

• f arbitrary arity and polarity type (normal DLE-logics).

Main results

◮ Syntactic characterization of properly displayable axiomatic

extensions of basic normal DLE-logics.

◮ Algorithmic computation of structural rules corresponding to

these additional axioms.

Analytic calculi and cut elimination

Main requirement for analyticity

Proofs must proceed by stepwise decomposition, without any insertion of information not contained in the conclusion: A ⊢ A B ⊢ B A, A → B ⊢ B A ∧ (A → B) ⊢ B

Core violating rule

X ⊢ A A ⊢ Y Cut X ⊢ Y

Cut elimination theorem (Gentzen, 1935)

If X ⊢ Y is derivable, then a derivation of X ⊢ Y exists in which Cut is never applied.

However, proofs of cut elimination theorem are:

◮ lengthy, ◮ error prone, ◮ non-modular.

Proper Display Calculi

Natural generalization of sequent calculi. Sequents X ⊢ Y, where X, Y are structures: A, A; B, ... X > Y, ... structural symbols assemble and disassemble structures; logical symbols assemble formulas. Main feature: display property Y ⊢ X > Z X; Y ⊢ Z Y; X ⊢ Z X ⊢ Y > Z This machinery ensures the existence of a Canonical proof of cut elimination

Canonical cut elimination

Theorem (Belnap 1982, Wansing 1997)

If a calculus satisfies the properties below, then it enjoys cut elimination and subformula property.

◮ C1: structures can disappear, formulas are forever; ◮ tree-traceable formula-occurrences, via suitably defined

congruence:

◮ C2: same shape, C3: non-proliferation, C4: same position;

◮ C5: principal = displayed; ◮ C6, C7: rules are closed under uniform substitution of

congruent parameters;

◮ C8: reduction strategy exists when cut formulas are both

principal.

Canonical cut elimination: proof strategy

Principal stage

. . . π1

Z ⊢ ◦A Z ⊢ A

. . . π2

A ⊢ Y

A ⊢ ◦Y Cut

Z ⊢ ◦Y

⇓ . . . π1

Z ⊢ ◦A Display •Z ⊢ A

. . . π2

A ⊢ Y Cut

• Z ⊢ Y

Display Z ⊢ ◦Y Parametric stage

A B

Analytic structural rules

Those structural rules the shape of which supports the canonical cut elimination strategy:

• C1. structural variables in the premises appear also in the

conclusion;

• C2. congruent parametric parts have the same shape;
• C3. structural variables in the premises are congruent to at most
• ne variable in the conclusion (non-proliferation);
• C4. congruent parametric parts have the same position (either

antecedent or consequent); C6,7. rules are closed under uniform substitution of congruent parameters;

Examples

X ⊢ Y X ⊢ Y ; Z X ; X ⊢ Y X ⊢ Y X ; Z ⊢ Y Z ; X ⊢ Y

Non-Examples

X ⊢ Y ; Z X ⊢ Y X ⊢ Y X ; X ⊢ Y X ; Z ⊢ Y X < Z ⊢ Y

Normal DLE-logics

DLE: Distributive Lattice Expansions: (distributive) lattice signature + operations of any finite arity. Additional operations partitioned in families f ∈ F and g ∈ G. Normality: In each coordinate,

◮ f-type operations preserve finite joins or reverse finite meets; ◮ g-type operations preserve finite meets or reverse finite joins.

Examples

◮ Distributive Modal Logic: F := {, } and G := {, } ◮ Bi-intuitionistic modal logic: F := {, > } and G := {, →} ◮ Full Lambek calculus: F := {◦} and G := {/, \} ◮ Lambek-Grishin calculus: F := {◦, /⊕, \⊕} and G := {⊕, /◦, \◦} ◮ . . .

Relational/complex algebra semantics

◮ f-type operations have residuals f♯ i in each coordinate i; ◮ g-type operations have residuals g♭ h in each coordinate h.

Proper display calculi for basic normal DLE-logics

ϕ ::= p | ⊥ | ⊤ | ϕ ∧ ϕ | ϕ ∨ ϕ | f(ϕ) | g(ϕ)

where p ∈ PROP, f ∈ F , g ∈ G.

Str.

I

; >

H K

Log.

⊤ ⊥ ∧ ∨ ( > ) (→)

f g

◮ Str.

Hi Kh

Log.

(f♯

i )

(g♭

h)

for εf(i) = εg(h) = 1

◮ Str.

Hi Kh

log.

(f♯

i )

(g♭

h)

for εf(i) = εg(h) = ∂

Introduction rules for f ∈ F and g ∈ G

H(A1, . . . , Anf) ⊢ X

fL

f(A1, . . . , Anf) ⊢ X X ⊢ K(A1, . . . , Ang)

gR

X ⊢ g(A1, . . . , Ang)

• Xi ⊢ Ai

Aj ⊢ Xj

| εf(i) = 1 εf(j) = ∂

• fR

H(X1, . . . , Xnf) ⊢ f(A1, . . . , Anf)

• Ai ⊢ Xi

Xj ⊢ Aj

| εg(i) = 1 εg(j) = ∂

• gL

g(A1, . . . , Ang) ⊢ K(X1, . . . , Xng)

Display postulates for f ∈ F and g ∈ G

◮ If εf(i) = εg(h) = 1

H (X1, . . . , Xi, . . . , Xnf) ⊢ Y Xi ⊢ Hi (X1, . . . , Y, . . . , Xnf) Y ⊢ K (X1 . . . , Xh, . . . Xng) Kh (X1, . . . , Y, . . . , Xng) ⊢ Xh

◮ If εf(i) = εg(h) = ∂

H (X1, . . . , Xi, . . . , Xnf) ⊢ Y Hi (X1, . . . , Y, . . . , Xnf) ⊢ Xi Y ⊢ K (X1, . . . , Xh, . . . , Xng) Xh ⊢ Kh (X1, . . . , Y, . . . , Xng)

Primitive inequalities

Primitive formulas: [Kracht 1996] Left-primitive

ϕ := p | ⊤ | ϕ ∨ ϕ | ϕ ∧ ϕ | f( ϕ/

p,

ψ/

q) Right-primitive

ψ := p | ⊥ | ψ ∧ ψ | ψ ∨ ψ | g( ψ/

p,

ϕ/

q) Primitive inequalities: Left-primitive:

ϕ1 ≤ ϕ2 with ϕ1 scattered

(i.e. each p occurs at most once) Right-primitive:

ψ1 ≤ ψ2 with ψ2 scattered

Example: F := {}, G := {, →}

Str.

• >

Log.

• →

q → p ≤ (q → p)

• x ⊢ q → p

x ⊢ (q → p)

• X ⊢ ◦Z > ◦Y

X ⊢ ◦(Z > Y).

Strategy

Crucial observation: same structural connectives for the basic and for the expanded DLE. Main strategy: transform non-primitive DLE inequalities into (conjunctions of) primitive DLE inequalities in the expanded language: s( p, q) ≤ s′( p, q)

&

• ϕ∗

i (

p, q) ≤ ϕ′∗

i (

p, q) | i ∈ I

• ALBA

ALBA on primitives

&

• ϕ∗

i (

i, m) ≤ ϕ′

i ∗(

i, m) | i ∈ I

• =

&

• ϕ∗

i (

i, m) ≤ ϕ′

i ∗(

i, m) | i ∈ I

Inductive but not analytic

∀[p ≤ p]

Inductive but not analytic

∀[p ≤ p]

iff

∀[(i ≤ p & p ≤ m) ⇒ i ≤ m]

iff

∀[(i ≤ j & j ≤ p & p ≤ m) ⇒ i ≤ m]

iff

∀[(i ≤ j & j ≤ m) ⇒ i ≤ m]

iff

∀[i ≤ j ⇒ ∀m[j ≤ m ⇒ i ≤ m]]

iff

∀[i ≤ j ⇒ i ≤ j]

iff

∀[j ≤ j]

Analytic inductive inequalities

+

+f, −g, ∧, ∨ +p +g, −f... ≤

−

−g, +f, ∧, ∨ +p +g, −f...

The Church-Rosser inequality

Let F = {} and G = {}.

∀[p ≤ p]

The Church-Rosser inequality

Let F = {} and G = {}.

∀[p ≤ p]

iff

∀[p ≤ p]

iff

∀[i ≤ p & p ≤ m ⇒ i ≤ m]

iff

∀[i ≤ j & j ≤ p & p ≤ m ⇒ i ≤ m]

iff

∀[i ≤ j & j ≤ p & p ≤ m ⇒ i ≤ m]

iff

∀[i ≤ j & j ≤ m ⇒ i ≤ m]

iff

∀[j ≤ j]

iff

∀[p ≤ p] (ALBA for primitive) · · ·

• p ⊢ z

p ⊢ z

• • X ⊢ Z
• ◦ X ⊢ Z

ALBA-guided uniform strategy to derive the axiom from the rule

p ⊢ p

p ⊢ ◦p

• p ⊢ p
• • p ⊢ p
• ◦ p ⊢ p
• p ⊢ ◦p
• p ⊢ p

p ⊢ p

The prelinearity axiom

Let F = ∅, G = {⇀} with ⇀ binary and of order-type (∂, 1).

∀[⊤ ≤ (p ⇀ q) ∨ (q ⇀ p)]

The prelinearity axiom

Let F = ∅, G = {⇀} with ⇀ binary and of order-type (∂, 1).

∀[⊤ ≤ (p ⇀ q) ∨ (q ⇀ p)]

iff

∀[(r1 ≤ p & q ≤ r2 & r3 ≤ q & p ≤ r4) ⇒ ⊤ ≤ (r1 ⇀ r2) ∨ (r3 ⇀ r4)]

iff

∀[(r1 ≤ r4 & q ≤ r2 & r3 ≤ q) ⇒ ⊤ ≤ (r1 ⇀ r2) ∨ (r3 ⇀ r4)]

iff

∀[(r1 ≤ r4 & r3 ≤ r2) ⇒ ⊤ ≤ (r1 ⇀ r2) ∨ (r3 ⇀ r4)]

X ⊢ W Z ⊢ Y I ⊢ (X ≻ Y) ; (Z ≻ W) p ⊢ p q ⊢ q

⊢ (p ≻ q) ; (q ≻ p) ⊢ (p ⇀ q) ∨ (q ⇀ p)

Example

Let G = ∅, F = {, ·} where · binary and of order type (1, 1)

∀[p · p ≤ p]

Example

Let G = ∅, F = {, ·} where · binary and of order type (1, 1)

∀[p · p ≤ p]

iff

∀[(j ≤ p · p & p ≤ m) ⇒ j ≤ m]

iff

∀[(j ≤ i · p & i ≤ p & p ≤ m) ⇒ j ≤ m]

iff

∀[(j ≤ i · h & i ≤ p & h ≤ p & p ≤ m) ⇒ j ≤ m]

iff

∀[(j ≤ i · h & i ∨ h ≤ p & p ≤ m) ⇒ j ≤ m]

iff

∀[(j ≤ i · h & (i ∨ h) ≤ m) ⇒ j ≤ m]

iff

∀[j ≤ i · h ⇒ ∀m[(i ∨ h) ≤ m ⇒ j ≤ m]]

iff

∀[j ≤ i · h ⇒ j ≤ (i ∨ h)]

iff

∀[i · h ≤ (i ∨ h)]

iff

∀[p1 · p2 ≤ p1 ∨ p2] (ALBA for primitive) · · ·

• p1 ⊢ q

p2 ⊢ q p1 · p2 ⊢ z

• X ⊢ Z
• Y ⊢ Z
• ◦ X ⊙ ◦Y ⊢ Z

Frege axiom: a first reduction

∀[p ⇀ (q ⇀ r) ≤ (p ⇀ q) ⇀ (p ⇀ r)]

Frege axiom: a first reduction

∀[p ⇀ (q ⇀ r) ≤ (p ⇀ q) ⇀ (p ⇀ r)]

iff

∀[(j ≤ p ⇀ (q ⇀ r) & (p ⇀ q) ⇀ (p ⇀ r) ≤ m) ⇒ j ≤ m]

iff

∀[(j ≤ p ⇀ (q ⇀ r) & (p ⇀ q) ⇀ (p ⇀ n) ≤ m & r ≤ n) ⇒ j ≤ m]

iff

∀[(j ≤ p ⇀ (q ⇀ n) & (p ⇀ q) ⇀ (p ⇀ n) ≤ m) ⇒ j ≤ m]

iff

∀[(j ≤ p ⇀ (q ⇀ n) & (p ⇀ q) ⇀ (i ⇀ n) ≤ m & i ≤ p) ⇒ j ≤ m]

iff

∀[(j ≤ i ⇀ (q ⇀ n) & (i ⇀ q) ⇀ (i ⇀ n) ≤ m) ⇒ j ≤ m]

iff

∀[(j ≤ i ⇀ (q ⇀ n) & h ⇀ (i ⇀ n) ≤ m & h ≤ i ⇀ q) ⇒ j ≤ m]

iff

∀[(j ≤ i ⇀ (q ⇀ n) & h ⇀ (i ⇀ n) ≤ m & i • h ≤ q) ⇒ j ≤ m]

iff

∀[(j ≤ i ⇀ ((i • h) ⇀ n) & h ⇀ (i ⇀ n) ≤ m) ⇒ j ≤ m]

iff

∀[j ≤ i ⇀ ((i • h) ⇀ n) ⇒ ∀m[h ⇀ (i ⇀ n) ≤ m ⇒ j ≤ m]]

iff

∀[j ≤ i ⇀ ((i • h) ⇀ n) ⇒ j ≤ h ⇀ (i ⇀ n)]

iff

∀[i ⇀ ((i • h) ⇀ n) ≤ h ⇀ (i ⇀ n)]

iff

∀[p ⇀ ((p • q) ⇀ r) ≤ q ⇀ (p ⇀ r)] (ALBA for primitive)

. . .

iff

∀[i ⇀ ((i • h) ⇀ n) ≤ h ⇀ (i ⇀ n)]

iff

∀[p ⇀ ((p • q) ⇀ r) ≤ q ⇀ (p ⇀ r)] (ALBA for primitive)

by applying the usual procedure, we obtain the following rule:

· · ·

• s ⊢ p ⇀ ((p • q) ⇀ r)

s ⊢ q ⇀ (p ⇀ r)

• W ⊢ X ≻ ((X
• Y) ≻ Z)

W ⊢ Y ≻ (X ≻ Z)

Frege axiom: a second reduction

∀[p ⇀ (q ⇀ r) ≤ (p ⇀ q) ⇀ (p ⇀ r)]

Frege axiom: a second reduction

∀[p ⇀ (q ⇀ r) ≤ (p ⇀ q) ⇀ (p ⇀ r)]

iff

∀[(p ⇀ (q ⇀ r)) • (p ⇀ q) ≤ p ⇀ r]

iff

∀[((p ⇀ (q ⇀ r)) • (p ⇀ q)) • p ≤ r]

iff

∀[i ≤ ((p ⇀ (q ⇀ r)) • (p ⇀ q) • p & r ≤ m ⇒ i ≤ m]

iff

∀[i ≤ (h • k) • j & h≤ p ⇀ (q ⇀ r) &

k≤ p ⇀ q & j ≤ p & r ≤ m ⇒ i ≤ m] iff

∀[i ≤ (h • k) • j & (h•p) • q ≤ r &

k•p ≤ q & j ≤ p & r ≤ m ⇒ i ≤ m] iff

∀[i ≤ (h • k) • j & (h•j) • q ≤ r & k • j ≤ q & r ≤ m ⇒ i ≤ m]

iff

∀[i ≤ (h • k) • j & (h • j) • (k • j) ≤ r & r ≤ m ⇒ i ≤ m]

iff

∀[i ≤ (h • k) • j & (h • j) • (k • j) ≤ m ⇒ i ≤ m]

iff

∀[(h • k) • j ≤ (h • j) • (k • j)]

iff

∀[(r • q) • p ≤ (r • p) • (q • p)] (ALBA for primitive)

. . .

iff

∀[(h • k) • j ≤ (h ◦ j) • (k • j)]

iff

∀[(r • q) • p ≤ (r • p) • (q • p)] (ALBA for primitive)

by applying the usual procedure, we obtain the following rule:

· · ·

• (r • p) • (q • p) ⊢ s

(r • q) • p ⊢ s

• (Z
• X)
• (Y
• X) ⊢ W

(Z

• Y)
• X ⊢ W

Properties of rules and calculi guaranteed by ALBA

◮ The analytic structural rule ALBA-corresponding to a given

inequality is sound on the class of algebras/frames defined by that inequality.

◮ ALBA runs on analytic inductive inequalities encode

instructions for the cut-free derivations of the same inequality using the analytic structural rule(s) corresponding to it. Hence the resulting calculus is syntactically complete w.r.t. the corresponding Hilbert-style logic.

◮ Analytic inductive inequalities are canonical. Hence, the

resulting calculus is a conservative extension of the corresponding Hilbert-style logic.

◮ Cut elimination and subformula property are guarantee by

the general theory of proper display calculi.

Conclusions

◮ There are surprising connections between algorithmic

correspondence theory and structural proof theory, seminally

• bserved by Kracht.

◮ The same algorithm ALBA originally introduced to compute

the first order correspondent of DLE-formulas and inequalities can be used to compute the analytic structural rule(s) corresponding to analytic inductive inequalities.

◮ Analytic structural rules have been identified as exactly those

supporting the canonical strategy for cut elimination for proper display calculi.

◮ Analytic inductive inequalities exactly correspond to analytic

structural rules.

◮ ALBA guarantees that the resulting analytic calculus is sound,

complete and conservative.