Unified Correspondence as a Proof-Theoretic Tool Apostolos Tzimoulis 6 May 2015 Delft University of Technology joint work with: Giuseppe Greco, Minghui Ma, Alessandra Palmigiano, Zhiguang Zhao http://www.appliedlogictudelft.nl Apostolos Tzimoulis Unified Correspondence as a Proof-Theoretic Tool
Motivation Main question: which axioms give rise to analytic rules? Correspondence theory can help in answering this question! Formal connections between correspondence theory and display calculi . Primitive formulas [Kracht ’96] for classical modal logic K generalised to primitive inequalities for general DLE-logics . Apostolos Tzimoulis Unified Correspondence as a Proof-Theoretic Tool
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 operational symbols assemble formulas. Main feature: display property Y ⊢ X > Z X ; Y ⊢ Z Y ; X ⊢ Z X ⊢ Y > Z display property: adjunction at the structural level. Canonical proof of cut elimination Apostolos Tzimoulis Unified Correspondence as a Proof-Theoretic Tool
Canonical Cut elimination Complexity of the cut formula Height of the cut . . . . . 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 Apostolos Tzimoulis Unified Correspondence as a Proof-Theoretic Tool
Proper Display Calculi Theorem (Canonical cut elimination) If a calculus satisfies the properties below, then it enjoys cut elimination. 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. Apostolos Tzimoulis Unified Correspondence as a Proof-Theoretic Tool
DLE-languages and expansions ϕ ::= p | ⊥ | ⊤ | ϕ ∧ ϕ | ϕ ∨ ϕ | f ( ϕ ) | g ( ϕ ) where p ∈ PROP, f ∈ F , g ∈ G . ; > I H K Str. ⊤ ⊥ ∧ ∨ ( > ) ( → ) f g Op. H i K h Str. for ε f ( i ) = ε g ( h ) = 1 ( f ♯ ( g ♭ i ) h ) Op. H i K h Str. for ε f ( i ) = ε g ( h ) = ∂ ( f ♯ ( g ♭ i ) h ) Op. Apostolos Tzimoulis Unified Correspondence as a Proof-Theoretic Tool
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 ) Apostolos Tzimoulis Unified Correspondence as a Proof-Theoretic Tool
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 ) Apostolos Tzimoulis Unified Correspondence as a Proof-Theoretic Tool
Unified correspondence DLE-logics [CP12, CPS14] Mu-calculi Substructural logics [CFPS14, CGP14] [CP14] Regular DLE-logics Display calculi Kripke frames with [GMPTZ14] impossible worlds [PSZ14a] Jónsson-style vs Finite lattices and Sambin-style canonicity monotone ML Canonicity via [PSZ14b] [FPS14] pseudo-correspondence [CPSZ14] Apostolos Tzimoulis Unified Correspondence as a Proof-Theoretic Tool
Algorithmic correspondence for DLE A ckermann L emma B ased A lgorithm engined by the Ackermann lemma. Reduction rules leading to the Ackermann elimination step. Residuation and approximation rules. Soundness on perfect DLEs : approximation: both � -generated by the c. ∨ -primes and � -generated by the c. ∧ -primes; residuation: all the operations are either right or left adjoints or residuals. Perfect DLEs: the natural semantic environment both for ALBA and for display calculi for DLE. Apostolos Tzimoulis Unified Correspondence as a Proof-Theoretic Tool
Primitive inequalities Primitive formulas : [Kracht 1996] p , � ϕ/� ψ/� Left-primitive ϕ := p | ⊤ | ∨ | ∧ | f ( � q ) ψ := p | ⊥ | ∧ | ∨ | g ( � ψ/� ϕ/� p , � q ) Right-primitive Primitive inequalities : ϕ 1 ≤ ϕ 2 with ϕ 1 scattered Left-primitive Right-primitive ψ 1 ≤ ψ 2 with ψ 2 scattered Example: X ⊢ ◦ Z > ◦ Y x ⊢ � q → � p X ⊢ ◦ ( Z > Y ) . � q → � p ≤ � ( q → p ) � � x ⊢ � ( q → p ) Apostolos Tzimoulis Unified Correspondence as a Proof-Theoretic Tool
First Attempt 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 ′∗ s ( � p ,� q ) ≤ s ′ ( � p ,� ϕ ∗ i ( � p ,� i ( � p ,� q ) 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 Apostolos Tzimoulis Unified Correspondence as a Proof-Theoretic Tool
Inductive but not analytic ∀ [ � p ≤ �� p ] iff ∀ [( i ≤ � p & �� p ≤ m ) ⇒ i ≤ m ] ∀ [( i ≤ � j & j ≤ p & �� p ≤ m ) ⇒ i ≤ m ] iff iff ∀ [( i ≤ � j & �� j ≤ m ) ⇒ i ≤ m ] ∀ [ i ≤ � j ⇒ ∀ m [ �� j ≤ m ⇒ i ≤ m ]] iff ∀ [ i ≤ � j ⇒ i ≤ �� j ] iff iff ∀ [ � j ≤ �� j ] Apostolos Tzimoulis Unified Correspondence as a Proof-Theoretic Tool
Analytic inductive inequalities − + ≤ Ske Ske PIA PIA γ γ ′ + p + p Apostolos Tzimoulis Unified Correspondence as a Proof-Theoretic Tool
Type 2: allowing multiple occurrences of var’s in heads of inequalities Let G = ∅ , F = { � , ·} where · binary and of order type ( 1 , 1 ) ∀ [ �� p · � p ≤ � p ] iff ∀ [( j ≤ �� p · � p & � p ≤ m ) ⇒ j ≤ m ] ∀ [( j ≤ �� i · � p & i ≤ p & � p ≤ m ) ⇒ j ≤ m ] iff iff ∀ [( j ≤ �� i · � h & i ≤ p & h ≤ p & � p ≤ m ) ⇒ j ≤ m ] ∀ [( j ≤ �� i · � h & i ∨ h ≤ p & � p ≤ m ) ⇒ j ≤ m ] iff iff ∀ [( j ≤ �� i · � h & � ( i ∨ h ) ≤ m ) ⇒ j ≤ m ] iff ∀ [ j ≤ �� i · � h ⇒ ∀ m [ � ( i ∨ h ) ≤ m ⇒ j ≤ m ]] ∀ [ j ≤ �� i · � h ⇒ j ≤ � ( i ∨ h )] iff iff ∀ [ �� i · � h ≤ � ( i ∨ h )] iff ∀ [ �� p 1 · � p 2 ≤ � p 1 ∨ � p 2 ] (ALBA for primitive) � p 1 ⊢ q � p 2 ⊢ q ◦ X ⊢ Z ◦ Y ⊢ Z · · · � � �� p 1 · � p 2 ⊢ z ◦ ◦ X ⊙ ◦ Y ⊢ Z Apostolos Tzimoulis Unified Correspondence as a Proof-Theoretic Tool
Type 3: allowing PIA-subterms Frege axiom: a first reduction ∀ [ p ⇀ ( q ⇀ r ) ≤ ( p ⇀ q ) ⇀ ( p ⇀ r )] ∀ [( j ≤ p ⇀ ( q ⇀ r ) & ( p ⇀ q ) ⇀ ( p ⇀ r ) ≤ m ) ⇒ j ≤ m ] iff iff ∀ [( j ≤ p ⇀ ( q ⇀ r ) & ( p ⇀ q ) ⇀ ( p ⇀ n ) ≤ m & r ≤ n ) ⇒ j ≤ m ] ∀ [( 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 iff ∀ [( j ≤ i ⇀ ( q ⇀ n ) & h ⇀ ( i ⇀ n ) ≤ m & i • h ≤ q ) ⇒ j ≤ m ] ∀ [( j ≤ i ⇀ (( i • h ) ⇀ n ) & h ⇀ ( i ⇀ n ) ≤ m ) ⇒ j ≤ m ] iff iff ∀ [ j ≤ i ⇀ (( i • h ) ⇀ n ) ⇒ ∀ m [ h ⇀ ( i ⇀ n ) ≤ m ⇒ j ≤ m ]] ∀ [ j ≤ i ⇀ (( i • h ) ⇀ n ) ⇒ j ≤ h ⇀ ( i ⇀ n )] iff iff ∀ [ i ⇀ (( i • h ) ⇀ n ) ≤ h ⇀ ( i ⇀ n )] iff ∀ [ p ⇀ (( p • q ) ⇀ r ) ≤ q ⇀ ( p ⇀ r )] (ALBA for primitive) Apostolos Tzimoulis Unified Correspondence as a Proof-Theoretic Tool
. . . ∀ [ i ⇀ (( i • h ) ⇀ n ) ≤ h ⇀ ( i ⇀ n )] iff ∀ [ p ⇀ (( p • q ) ⇀ r ) ≤ q ⇀ ( p ⇀ r )] (ALBA for primitive) iff by applying the usual procedure, we obtain the following rule: s ⊢ p ⇀ (( p • q ) ⇀ r ) W ⊢ X ≻ (( X � • Y ) ≻ Z ) · · · � � s ⊢ q ⇀ ( p ⇀ r ) W ⊢ Y ≻ ( X ≻ Z ) Apostolos Tzimoulis Unified Correspondence as a Proof-Theoretic Tool
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