Craig interpolation in displayable logics
James Brotherston1 and Rajeev Goré2
1 Imperial
College London
2 ANU
Canberra
TABLEAUX, Universität 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:
I
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:
I
logic: consistency; compactness; definability
I
computer science: invariant generation; type inference;
model checking; ontology decomposition
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Display calculi
I
are consecution calculi à la Gentzen;
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Display calculi
I
are consecution calculi à la Gentzen;
I
Characterisation: any part of a consecution can be
“displayed” alone on one side of the `;
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Display calculi
I
are consecution calculi à la Gentzen;
I
Characterisation: any part of a consecution can be
“displayed” alone on one side of the `;
I
Needs a richer consecution structure than simple sequents;
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Display calculi
I
are consecution calculi à la Gentzen;
I
Characterisation: any part of a consecution can be
“displayed” alone on one side of the `;
I
Needs a richer consecution structure than simple sequents;
I
Cut-elimination is guaranteed when the proof rules satisfy
some simple conditions;
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Display calculi
I
are consecution calculi à la Gentzen;
I
Characterisation: any part of a consecution can be
“displayed” alone on one side of the `;
I
Needs a richer consecution structure than simple sequents;
I
Cut-elimination is guaranteed when the proof rules satisfy
some simple conditions;
I
But decidability, interpolation etc. don’t follow directly as
they often do in sequent calculi.
3/ 14
Display calculi
I
are consecution calculi à la Gentzen;
I
Characterisation: any part of a consecution can be
“displayed” alone on one side of the `;
I
Needs a richer consecution structure than simple sequents;
I
Cut-elimination is guaranteed when the proof rules satisfy
some simple conditions;
I
But decidability, interpolation etc. don’t follow directly as
they often do in sequent calculi.
I
We show interpolation for a large class of display calculi.
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Display calculus syntax
I
Formulas given by:
F ::= P | > | ⊥ | ¬F | F &F | F ∨ F | F → F | . . .
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Display calculus syntax
I
Formulas given by:
F ::= P | > | ⊥ | ¬F | F &F | F ∨ F | F → F | . . .
I
Structures given by:
X ::= F | ∅ | ]X | X ; X
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Display calculus syntax
I
Formulas given by:
F ::= P | > | ⊥ | ¬F | F &F | F ∨ F | F → F | . . .
I
Structures given by:
X ::= F | ∅ | ]X | X ; X
I
Consecutions are given by X ` Y for X, Y structures.
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Display calculus syntax
I
Formulas given by:
F ::= P | > | ⊥ | ¬F | F &F | F ∨ F | F → F | . . .
I
Structures given by:
X ::= F | ∅ | ]X | X ; X
I
Consecutions are given by X ` Y for X, Y structures.
I
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:
P `P
(Id)
X0 ` Y 0
X `Y
(X ` Y ≡D X 0 ` Y 0 ) (≡D )
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Some proof rules
Identity rules:
(Id)
P `P
Logical rules:
F ;G`X
F &G ` X
X0 ` Y 0
X `Y
(X ` Y ≡D X 0 ` Y 0 ) (≡D )
X`F Y `G
(&L)
X ; Y ` F &G
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(&R)
...
Some proof rules
Identity rules:
X0 ` Y 0
(Id)
P `P
Logical rules:
F ;G`X
F &G ` X
Structural rules:
X `Y
(X ` Y ≡D X 0 ` Y 0 ) (≡D )
X`F Y `G
(&L)
W ; (X ; Y ) ` Z
(W ; X) ; Y ` Z
X `Z
X ;Y `Z
X ; Y ` F &G
(&R)
∅;X `Y
(α)
X`Y
X ;X `Y
(W)
X`Y
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...
(∅CL )
(C)
...
Interpolation: our approach
I
Proof-theoretic strategy: given a cut-free proof of X ` Y ,
we construct its interpolant I.
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Interpolation: our approach
I
Proof-theoretic strategy: given a cut-free proof of X ` Y ,
we construct its interpolant I.
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
I
Proof-theoretic strategy: given a cut-free proof of X ` Y ,
we construct its interpolant I.
I
Induction on proofs: from interpolants for the premises of a
rule, construct an interpolant for its conclusion.
I
But not enough info to do this for display steps, e.g.:
X;Y `Z
X ` ]Y ; Z
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(≡D )
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
Ci0 ≡AD Ci , we can construct interpolants for all C 0 ≡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
Ci0 ≡AD Ci , we can construct interpolants for all C 0 ≡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
X ; Y ` F &G
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(&R)
LADI: (&R)
X `F Y `G
X ; Y ` F &G
(&R)
Need interpolant for arbitrary W ` Z ≡AD X; Y ` F &G.
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LADI: (&R)
X `F Y `G
X ; Y ` F &G
(&R)
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
X ; Y ` F &G
(&R)
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 C X).
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LADI: (&R)
X `F Y `G
X ; Y ` F &G
(&R)
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 C X).
By a LEMMA ∃U. X ` F ≡AD W ` U .
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LADI: (&R)
X `F Y `G
X ; Y ` F &G
(&R)
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 C 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
X ; Y ` F &G
(&R)
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 C 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 ]
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by an easy LEMMA
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
X1 ; X2 ` Y
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(C)
LADI: contraction
Consider the following instance of contraction:
(X1 ; X2 ); (X1 ; X2 ) ` Y
X1 ; X2 ` Y
(C)
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
X1 ; X2 ` Y
(C)
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
X1 ; X2 ` Y
(C)
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
(W)
(∅CL )
(∅CR )
(∅WL )
(+)
D0
(∅WR )
(C)
(α)
LADI of the proof rule(s) at a node holds in a calculus with all
of the proof rules at its ancestor nodes. Thus:
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Summary of results
(W)
(∅CL )
(∅CR )
(∅WL )
(∅WR )
(+)
D0
(C)
(α)
LADI of the proof rule(s) at a node holds in a calculus with all
of 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:
I
non-commutative 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:
I
I
non-commutative logics;
multiple-family display calculi (bunched & relevant 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:
I
I
I
non-commutative logics;
multiple-family display calculi (bunched & relevant logics);
modalities, quantifiers, linear exponentials . . .
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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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