Craig interpolation in displayable logics James Brotherston 1 and - PowerPoint PPT Presentation

Craig interpolation in displayable logics James Brotherston 1 and Rajeev Gor´ e 2 1 Imperial College London 2 ANU Canberra TABLEAUX, Universit¨ at Bern, 7 Jul 2011 1/ 14

Craig interpolation Definition A (propositional) logic satisfies Craig interpolation iff for any provable F ⊢ G there exists an interpolant I s.t.: F ⊢ I provable and I ⊢ G provable and V ( I ) ⊆ V ( F ) ∩ V ( G ) ( V ( X ) is the set of propositional variables occurring in X ) 2/ 14

Craig interpolation Definition A (propositional) logic satisfies Craig interpolation iff for any provable F ⊢ G there exists an interpolant I s.t.: F ⊢ I provable and I ⊢ G provable and V ( I ) ⊆ V ( F ) ∩ V ( G ) ( V ( X ) is the set of propositional variables occurring in X ) Applications in: ◮ logic: consistency; compactness; definability 2/ 14

Craig interpolation Definition A (propositional) logic satisfies Craig interpolation iff for any provable F ⊢ G there exists an interpolant I s.t.: F ⊢ I provable and I ⊢ G provable and V ( I ) ⊆ V ( F ) ∩ V ( G ) ( V ( X ) is the set of propositional variables occurring in X ) Applications in: ◮ logic: consistency; compactness; definability ◮ computer science: invariant generation; type inference; model checking; ontology decomposition 2/ 14

Display calculi ◮ are consecution calculi ` a la Gentzen; 3/ 14

Display calculi ◮ are consecution calculi ` a la Gentzen; ◮ Characterisation: any part of a consecution can be “displayed” alone on one side of the ⊢ ; 3/ 14

Display calculi ◮ are consecution calculi ` a la Gentzen; ◮ Characterisation: any part of a consecution can be “displayed” alone on one side of the ⊢ ; ◮ Needs a richer consecution structure than simple sequents; 3/ 14

Display calculi ◮ are consecution calculi ` a la Gentzen; ◮ Characterisation: any part of a consecution can be “displayed” alone on one side of the ⊢ ; ◮ Needs a richer consecution structure than simple sequents; ◮ Cut-elimination is guaranteed when the proof rules satisfy some simple conditions; 3/ 14

Display calculi ◮ are consecution calculi ` a la Gentzen; ◮ Characterisation: any part of a consecution can be “displayed” alone on one side of the ⊢ ; ◮ Needs a richer consecution structure than simple sequents; ◮ Cut-elimination is guaranteed when the proof rules satisfy some simple conditions; ◮ But decidability, interpolation etc. don’t follow directly as they often do in sequent calculi. 3/ 14

Display calculi ◮ are consecution calculi ` a la Gentzen; ◮ Characterisation: any part of a consecution can be “displayed” alone on one side of the ⊢ ; ◮ Needs a richer consecution structure than simple sequents; ◮ Cut-elimination is guaranteed when the proof rules satisfy some simple conditions; ◮ But decidability, interpolation etc. don’t follow directly as they often do in sequent calculi. ◮ We show interpolation for a large class of display calculi. 3/ 14

Display calculus syntax ◮ Formulas given by: F ::= P | ⊤ | ⊥ | ¬ F | F & F | F ∨ F | F → F | . . . 4/ 14

Display calculus syntax ◮ Formulas given by: F ::= P | ⊤ | ⊥ | ¬ F | F & F | F ∨ F | F → F | . . . ◮ Structures given by: X ::= F | ∅ | ♯X | X ; X 4/ 14

Display calculus syntax ◮ Formulas given by: F ::= P | ⊤ | ⊥ | ¬ F | F & F | F ∨ F | F → F | . . . ◮ Structures given by: X ::= F | ∅ | ♯X | X ; X ◮ Consecutions are given by X ⊢ Y for X, Y structures. 4/ 14

Display calculus syntax ◮ Formulas given by: F ::= P | ⊤ | ⊥ | ¬ F | F & F | F ∨ F | F → F | . . . ◮ Structures given by: X ::= F | ∅ | ♯X | X ; X ◮ Consecutions are given by X ⊢ Y for X, Y structures. ◮ Substructures of X ⊢ Y are antecedent or consequent parts (similar to positive / negative occurrences in formulas). 4/ 14

Display-equivalence We have the following display postulates: X ; Y ⊢ Z X ⊢ ♯Y ; Z Y ; X ⊢ Z <> D <> D X ⊢ Y ; Z <> D X ; ♯Y ⊢ Z <> D X ⊢ Z ; Y X ⊢ Y ♯Y ⊢ ♯X ♯♯X ⊢ Y <> D <> D 5/ 14

Display-equivalence We have the following display postulates: X ; Y ⊢ Z X ⊢ ♯Y ; Z Y ; X ⊢ Z <> D <> D X ⊢ Y ; Z <> D X ; ♯Y ⊢ Z <> D X ⊢ Z ; Y X ⊢ Y ♯Y ⊢ ♯X ♯♯X ⊢ Y <> D <> D Display-equivalence ≡ D given by transitive closure of <> D . 5/ 14

Display-equivalence We have the following display postulates: X ; Y ⊢ Z X ⊢ ♯Y ; Z Y ; X ⊢ Z <> D <> D X ⊢ Y ; Z <> D X ; ♯Y ⊢ Z <> D X ⊢ Z ; Y X ⊢ Y ♯Y ⊢ ♯X ♯♯X ⊢ Y <> D <> D Display-equivalence ≡ D given by transitive closure of <> D . Proposition (Display property) For any antecedent part Z of X ⊢ Y there is a W s.t. X ⊢ Y ≡ D Z ⊢ W (and similarly for consequent parts). 5/ 14

Some proof rules Identity rules: X ′ ⊢ Y ′ ( X ⊢ Y ≡ D X ′ ⊢ Y ′ ) ( ≡ D ) (Id) P ⊢ P X ⊢ Y 6/ 14

Some proof rules Identity rules: X ′ ⊢ Y ′ ( X ⊢ Y ≡ D X ′ ⊢ Y ′ ) ( ≡ D ) (Id) P ⊢ P X ⊢ Y Logical rules: X ⊢ F Y ⊢ G F ; G ⊢ X (&R) . . . (&L) X ; Y ⊢ F & G F & G ⊢ X 6/ 14

Some proof rules Identity rules: X ′ ⊢ Y ′ ( X ⊢ Y ≡ D X ′ ⊢ Y ′ ) ( ≡ D ) (Id) P ⊢ P X ⊢ Y Logical rules: X ⊢ F Y ⊢ G F ; G ⊢ X (&R) . . . (&L) X ; Y ⊢ F & G F & G ⊢ X Structural rules: W ; ( X ; Y ) ⊢ Z ∅ ; X ⊢ Y ( α ) ( ∅ C L ) ( W ; X ) ; Y ⊢ Z X ⊢ Y X ⊢ Z X ; X ⊢ Y (W) . . . (C) X ; Y ⊢ Z X ⊢ Y 6/ 14

Interpolation: our approach ◮ Proof-theoretic strategy: given a cut-free proof of X ⊢ Y , we construct its interpolant I . 7/ 14

Interpolation: our approach ◮ Proof-theoretic strategy: given a cut-free proof of X ⊢ Y , we construct its interpolant I . ◮ Induction on proofs: from interpolants for the premises of a rule, construct an interpolant for its conclusion. 7/ 14

Interpolation: our approach ◮ Proof-theoretic strategy: given a cut-free proof of X ⊢ Y , we construct its interpolant I . ◮ Induction on proofs: from interpolants for the premises of a rule, construct an interpolant for its conclusion. ◮ But not enough info to do this for display steps, e.g.: X ; Y ⊢ Z ( ≡ D ) X ⊢ ♯Y ; Z 7/ 14

Local AD-interpolation (LADI) property Let ≡ AD be the least equivalence closed under ≡ D and applications of associativity ( α ) (if present). 8/ 14

Local AD-interpolation (LADI) property Let ≡ AD be the least equivalence closed under ≡ D and applications of associativity ( α ) (if present). Definition A proof rule with conclusion C has the LADI property if, given that for each premise of the rule C i we have interpolants for all i ≡ AD C i , we can construct interpolants for all C ′ ≡ AD C . C ′ 8/ 14

Local AD-interpolation (LADI) property Let ≡ AD be the least equivalence closed under ≡ D and applications of associativity ( α ) (if present). Definition A proof rule with conclusion C has the LADI property if, given that for each premise of the rule C i we have interpolants for all i ≡ AD C i , we can construct interpolants for all C ′ ≡ AD C . C ′ Proposition If the proof rules of a display calculus D all have the LADI property then D enjoys Craig interpolation. 8/ 14

LADI: (&R) X ⊢ F Y ⊢ G (&R) X ; Y ⊢ F & G 9/ 14

LADI: (&R) X ⊢ F Y ⊢ G (&R) X ; Y ⊢ F & G Need interpolant for arbitrary W ⊢ Z ≡ AD X ; Y ⊢ F & G . 9/ 14

LADI: (&R) X ⊢ F Y ⊢ G (&R) X ; Y ⊢ F & G Need interpolant for arbitrary W ⊢ Z ≡ AD X ; Y ⊢ F & G . Case: F & G occurs in Z . 9/ 14

LADI: (&R) X ⊢ F Y ⊢ G (&R) X ; Y ⊢ F & G Need interpolant for arbitrary W ⊢ Z ≡ AD X ; Y ⊢ F & G . Case: F & G occurs in Z . Subcase: W built entirely from parts of X ( W ⊳ X ). 9/ 14

LADI: (&R) X ⊢ F Y ⊢ G (&R) X ; Y ⊢ F & G Need interpolant for arbitrary W ⊢ Z ≡ AD X ; Y ⊢ F & G . Case: F & G occurs in Z . Subcase: W built entirely from parts of X ( W ⊳ X ). By a LEMMA ∃ U. X ⊢ F ≡ AD W ⊢ U . 9/ 14

LADI: (&R) X ⊢ F Y ⊢ G (&R) X ; Y ⊢ F & G Need interpolant for arbitrary W ⊢ Z ≡ AD X ; Y ⊢ F & G . Case: F & G occurs in Z . Subcase: W built entirely from parts of X ( W ⊳ X ). By a LEMMA ∃ U. X ⊢ F ≡ AD W ⊢ U . Claim: interpolant I for W ⊢ U is an interpolant for W ⊢ Z . 9/ 14

LADI: (&R) X ⊢ F Y ⊢ G (&R) X ; Y ⊢ F & G Need interpolant for arbitrary W ⊢ Z ≡ AD X ; Y ⊢ F & G . Case: F & G occurs in Z . Subcase: W built entirely from parts of X ( W ⊳ X ). By a LEMMA ∃ U. X ⊢ F ≡ AD W ⊢ U . Claim: interpolant I for W ⊢ U is an interpolant for W ⊢ Z . Main issue: show I ⊢ Z provable given I ⊢ U provable. 9/ 14

LADI: (&R) By display property we have I ⊢ U ≡ D V ⊢ F . 10/ 14

LADI: (&R) By display property we have I ⊢ U ≡ D V ⊢ F . Next, we have: W ⊢ Z ≡ AD X ⊢ ♯Y ; F & G 10/ 14

LADI: (&R) By display property we have I ⊢ U ≡ D V ⊢ F . Next, we have: W ⊢ Z ≡ AD X ⊢ ♯Y ; F & G X ⊢ F [( ♯Y ; F & G ) /F ] = 10/ 14