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 of 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 Parametric stage . . . . . π 1 . π 2 Z ⊢ ◦ A A ⊢ Y Z ⊢ � A � A ⊢ ◦ Y Cut Z ⊢ ◦ Y A B ⇓ . . . π 1 . . . π 2 Z ⊢ ◦ A Display • Z ⊢ A A ⊢ Y Cut • Z ⊢ Y Display Z ⊢ ◦ Y

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 one 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 ; X ⊢ Y X ; Z ⊢ Y X ⊢ Y X ⊢ Y ; Z Z ; X ⊢ Y X ⊢ Y Non-Examples X ⊢ Y ; Z X ; Z ⊢ Y X ⊢ Y X ; X ⊢ Y X < Z ⊢ Y X ⊢ 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 . ; > I H K Str. ( > ) ( → ) ⊤ ⊥ ∧ ∨ f g Log. H i K h Str. for ε f ( i ) = ε g ( h ) = 1 ◮ ( f ♯ ( g ♭ i ) h ) Log. H i K h Str. for ε f ( i ) = ε g ( h ) = ∂ ◮ ( f ♯ ( g ♭ i ) h ) log.

Introduction rules for f ∈ F and g ∈ G X ⊢ K ( A 1 , . . . , A n g ) H ( A 1 , . . . , A n f ) ⊢ X g R f L f ( A 1 , . . . , A n f ) ⊢ X X ⊢ g ( A 1 , . . . , A n g ) � � ε f ( i ) = 1 ε f ( j ) = ∂ X i ⊢ A i A j ⊢ X j | f R H ( X 1 , . . . , X n f ) ⊢ f ( A 1 , . . . , A n f ) � � ε g ( i ) = 1 ε g ( j ) = ∂ A i ⊢ X i X j ⊢ A j | g L g ( A 1 , . . . , A n g ) ⊢ K ( X 1 , . . . , X n g )

Display postulates for f ∈ F and g ∈ G ◮ If ε f ( i ) = ε g ( h ) = 1 Y ⊢ K ( X 1 . . . , X h , . . . X n g ) H ( X 1 , . . . , X i , . . . , X n f ) ⊢ Y X i ⊢ H i ( X 1 , . . . , Y , . . . , X n f ) K h ( X 1 , . . . , Y , . . . , X n g ) ⊢ X h ◮ If ε f ( i ) = ε g ( h ) = ∂ H ( X 1 , . . . , X i , . . . , X n f ) ⊢ Y Y ⊢ K ( X 1 , . . . , X h , . . . , X n g ) H i ( X 1 , . . . , Y , . . . , X n f ) ⊢ X i X h ⊢ K h ( X 1 , . . . , Y , . . . , X n g )

Primitive inequalities Primitive formulas : [Kracht 1996] p , � ϕ/� ψ/� ϕ := p | ⊤ | ϕ ∨ ϕ | ϕ ∧ ϕ | f ( � q ) Left-primitive ψ := p | ⊥ | ψ ∧ ψ | ψ ∨ ψ | g ( � ψ/� ϕ/� Right-primitive p , � q ) Primitive inequalities : Left-primitive: ϕ 1 ≤ ϕ 2 with ϕ 1 scattered (i.e. each p occurs at most once) ψ 1 ≤ ψ 2 with ψ 2 scattered Right-primitive: Example: F := { � } , G := { � , →} ◦ > Str. � � → Log. x ⊢ � q → � p X ⊢ ◦ Z > ◦ Y � q → � p ≤ � ( q → p ) X ⊢ ◦ ( Z > Y ) . � � x ⊢ � ( q → p )

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: � � q ) ≤ ϕ ′∗ q ) ≤ s ′ ( � & ϕ ∗ s ( � p ,� p ,� q ) i ( � p ,� i ( � p ,� q ) | i ∈ I � ALBA � ALBA on primitives � � � � i ( � ∗ ( � i ( � ∗ ( � & & ϕ ∗ i , � m ) ≤ ϕ ′ i , � ϕ ∗ i , � m ) ≤ ϕ ′ i , � m ) | i ∈ I = m ) | i ∈ I 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 ] ∀ [( i ≤ � j & �� j ≤ m ) ⇒ i ≤ m ] iff iff ∀ [ i ≤ � j ⇒ ∀ m [ �� j ≤ m ⇒ i ≤ m ]] ∀ [ i ≤ � j ⇒ i ≤ �� j ] iff iff ∀ [ � j ≤ �� j ]

Analytic inductive inequalities − + ≤ + f , − g , ∧ , ∨ − g , + f , ∧ , ∨ + g , − f ... + g , − f ... + p + p

The Church-Rosser inequality Let F = { � } and G = { � } . ∀ [ �� p ≤ �� p ]

The Church-Rosser inequality Let F = { � } and G = { � } . ∀ [ �� p ≤ �� p ] ∀ [ � �� p ≤ � p ] iff iff ∀ [ i ≤ � �� p & � p ≤ m ⇒ i ≤ m ] ∀ [ i ≤ � � j & j ≤ � p & � p ≤ m ⇒ i ≤ m ] iff iff ∀ [ i ≤ � � j & � j ≤ p & � p ≤ m ⇒ i ≤ m ] ∀ [ i ≤ � � j & � � j ≤ m ⇒ i ≤ m ] iff ∀ [ � � j ≤ � � j ] iff iff ∀ [ � � p ≤ � � p ] (ALBA for primitive) � � p ⊢ z ◦ • X ⊢ Z · · · � � � � p ⊢ 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 ∀ [( r 1 ≤ p & q ≤ r 2 & r 3 ≤ q & p ≤ r 4 ) ⇒ ⊤ ≤ ( r 1 ⇀ r 2 ) ∨ ( r 3 ⇀ r 4 )] ∀ [( r 1 ≤ r 4 & q ≤ r 2 & r 3 ≤ q ) ⇒ ⊤ ≤ ( r 1 ⇀ r 2 ) ∨ ( r 3 ⇀ r 4 )] iff ∀ [( r 1 ≤ r 4 & r 3 ≤ r 2 ) ⇒ ⊤ ≤ ( r 1 ⇀ r 2 ) ∨ ( r 3 ⇀ r 4 )] iff 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 ]