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

Craig interpolation in displayable logics

James Brotherston1 and Rajeev Gor´ e2

1Imperial College London 2ANU Canberra

TABLEAUX, Universit¨ at Bern, 7 Jul 2011

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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)

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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

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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

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Display calculi

◮ are consecution calculi `

a la Gentzen;

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Display calculi

◮ are consecution calculi `

a la Gentzen;

◮ Characterisation: any part of a consecution can be

“displayed” alone on one side of the ⊢;

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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;

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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;

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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.

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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.

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Display calculus syntax

◮ Formulas given by:

F ::= P | ⊤ | ⊥ | ¬F | F&F | F ∨ F | F → F | . . .

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Display calculus syntax

◮ Formulas given by:

F ::= P | ⊤ | ⊥ | ¬F | F&F | F ∨ F | F → F | . . .

◮ Structures given by:

X ::= F | ∅ | ♯X | X ; X

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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.

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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).

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Display-equivalence

We have the following display postulates: X ; Y ⊢ Z <>D X ⊢ ♯Y ; Z <>D Y ; X ⊢ Z X ⊢ Y ; Z <>D X ; ♯Y ⊢ Z <>D X ⊢ Z ; Y X ⊢ Y <>D ♯Y ⊢ ♯X <>D ♯♯X ⊢ Y

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Display-equivalence

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

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Display-equivalence

We have the following display postulates: X ; Y ⊢ Z <>D X ⊢ ♯Y ; Z <>D Y ; X ⊢ Z X ⊢ Y ; Z <>D X ; ♯Y ⊢ Z <>D X ⊢ Z ; Y X ⊢ Y <>D ♯Y ⊢ ♯X <>D ♯♯X ⊢ Y 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).

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Some proof rules

Identity rules: (Id) P ⊢ P X′ ⊢ Y ′ (X ⊢ Y ≡D X′ ⊢ Y ′) (≡D) X ⊢ Y

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Some proof rules

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

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Some proof rules

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

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Interpolation: our approach

◮ Proof-theoretic strategy: given a cut-free proof of X ⊢ Y ,

we construct its interpolant I.

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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.

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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

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Local AD-interpolation (LADI) property

Let ≡AD be the least equivalence closed under ≡D and applications of associativity (α) (if present).

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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 Ci we have interpolants for all C′

i ≡AD Ci, we can construct interpolants for all C′ ≡AD C.

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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 Ci we have interpolants for all C′

i ≡AD Ci, we can construct interpolants for all C′ ≡AD C.

Proposition

If the proof rules of a display calculus D all have the LADI property then D enjoys Craig interpolation.

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LADI: (&R)

X ⊢ F Y ⊢ G (&R) X ; Y ⊢ F&G

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LADI: (&R)

X ⊢ F Y ⊢ G (&R) X ; Y ⊢ F&G Need interpolant for arbitrary W ⊢ Z ≡AD X; Y ⊢ F&G.

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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.

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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).

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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.

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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.

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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.

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LADI: (&R)

By display property we have I ⊢ U ≡D V ⊢ F.

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LADI: (&R)

By display property we have I ⊢ U ≡D V ⊢ F. Next, we have: W ⊢ Z ≡AD X ⊢ ♯Y ; F&G

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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]

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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] ≡AD W ⊢ U[(♯Y ; F&G)/F] by an easy LEMMA

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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] ≡AD W ⊢ U[(♯Y ; F&G)/F] by an easy LEMMA Thus by a substitutivity LEMMA we obtain: I ⊢ Z ≡AD I ⊢ U[(♯Y ; F&G)/F]

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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] ≡AD W ⊢ U[(♯Y ; F&G)/F] by an easy LEMMA Thus by a substitutivity LEMMA we obtain: I ⊢ Z ≡AD I ⊢ U[(♯Y ; F&G)/F] ≡AD V ⊢ F[(♯Y ; F&G)/F]

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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] ≡AD W ⊢ U[(♯Y ; F&G)/F] by an easy LEMMA Thus by a substitutivity LEMMA we obtain: I ⊢ Z ≡AD I ⊢ U[(♯Y ; F&G)/F] ≡AD V ⊢ F[(♯Y ; F&G)/F] ≡AD V ; Y ⊢ F&G

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LADI: contraction

Consider the following instance of contraction: (X1; X2); (X1; X2) ⊢ Y (C) X1; X2 ⊢ Y

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LADI: contraction

Consider the following instance of contraction: (X1; X2); (X1; X2) ⊢ Y (C) X1; X2 ⊢ Y In particular we need an interpolant for X1 ⊢ ♯X2; Y .

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LADI: contraction

Consider the following instance of contraction: (X1; X2); (X1; X2) ⊢ Y (C) X1; X2 ⊢ Y In particular we need an interpolant for X1 ⊢ ♯X2; Y . If we have associativity the premise rearranges to X1; X1 ⊢ ♯(X2; X2); Y whose interpolant will work for X1 ⊢ ♯X2; Y as well.

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LADI: contraction

Consider the following instance of contraction: (X1; X2); (X1; X2) ⊢ Y (C) X1; X2 ⊢ Y In particular we need an interpolant for X1 ⊢ ♯X2; Y . If we have associativity the premise rearranges to X1; X1 ⊢ ♯(X2; X2); Y whose interpolant will work for X1 ⊢ ♯X2; Y as well. If not, about the best we can do is: X1 ⊢ ♯X2; (♯(X1; X2); Y ) whose interpolant is far too weak to work for X1 ⊢ ♯X2; Y .

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Summary of results

D(+) (α) (∅CL) (∅CR) (∅WL) (∅WR) (C) (W) LADI of the proof rule(s) at a node holds in a calculus with all

• f the proof rules at its ancestor nodes. Thus:

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Summary of results

D(+) (α) (∅CL) (∅CR) (∅WL) (∅WR) (C) (W) LADI of the proof rule(s) at a node holds in a calculus with all

• f the proof rules at its ancestor nodes. Thus:

Theorem

Any display calculus satisfying the constraints in the above diagram has Craig interpolation. (This includes MLL, MALL and classical logic.)

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Future work

• 1. Machine formalisation of results; an Isabelle mechanisation,

led by Jeremy Dawson (ANU), is currently under way.

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Future work

• 1. Machine formalisation of results; an Isabelle mechanisation,

led by Jeremy Dawson (ANU), is currently under way.

• 2. More logics:

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Future work

• 1. Machine formalisation of results; an Isabelle mechanisation,

led by Jeremy Dawson (ANU), is currently under way.

• 2. More logics:

◮ non-commutative logics; 13/ 14

Future work

• 1. Machine formalisation of results; an Isabelle mechanisation,

led by Jeremy Dawson (ANU), is currently under way.

• 2. More logics:

◮ non-commutative logics; ◮ multiple-family display calculi (bunched & relevant logics); 13/ 14

Future work

• 1. Machine formalisation of results; an Isabelle mechanisation,

led by Jeremy Dawson (ANU), is currently under way.

• 2. More logics:

◮ non-commutative logics; ◮ multiple-family display calculi (bunched & relevant logics); ◮ modalities, quantifiers, linear exponentials . . . 13/ 14

Further reading

Nuel D. Belnap, Jr. Display logic. In Journal of Philosophical Logic, vol. 11, 1982. Greg Restall. Displaying and deciding substructural logics 1: Logics with contraposition. In Journal of Philosophical Logic, vol. 27, 1998. Dirk Roorda. Interpolation in fragments of classical linear logic. In Journal of Symbolic Logic 59(2), 1994.

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