proof theory for many valued logics – some applicationsbrunella.gerla/manyval/...proof theory for...
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Proof theory for many valued logics– some applications
Agata Ciabattoni
Vienna University of Technology
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The beginning of the story
(Cost Action "Many-valued Logics for CS Applications")
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The beginning of the story
(Cost Action "Many-valued Logics for CS Applications")
Proof theory for many valued logics – some applications – p.2/33
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This talk
Many-valued logics have semantic origins. But as logicsthey also have something to do with proofs.
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This talk
Many-valued logics have semantic origins. But as logicsthey also have something to do with proofs.
My aim is to
introduce some recent developments in proof theoryfor non-classical logics (especially many-valued logics)
use proof theory for uniform (and automated) proofs ofstandard completeness
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Analytic Calculi
"Praedicatum Inest Subjecto"
Calculi in which proof search proceeds by step-wisedecomposition of the formulas to be proved
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Analytic Calculi
"Praedicatum Inest Subjecto"
Calculi in which proof search proceeds by step-wisedecomposition of the formulas to be proved
Sequent, hypersequent calculi, labelled calculi, many-
placed sequents, sequents-of-relations, display logic, CoS
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Introducing such calculi
Semantic-based approach
(Baaz, Fermüller, Montagna, ...)
Syntactic approach
(Avron, Baaz, Baldi, Galatos, Terui, Metcalfe, Spendier, ...)
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Introducing such calculi
Semantic-based approach
E.g. decidability, complexity of validity ..(Baaz, Fermüller, Montagna, ...)
Syntactic approach
(Avron, Baaz, Baldi, Galatos, Terui, Metcalfe, Spendier, ...)
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Introducing such calculi
Semantic-based approach
E.g. decidability, complexity of validity ..(Baaz, Fermüller, Montagna, ...)
Syntactic approach : uniform and systematic
E.g.
Herbrand theorem
order theoretic completions
standard completeness
...
(Avron, Baaz, Baldi, Galatos, Terui, Metcalfe, Spendier, ...)
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Semantic-based Approach
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Semantic-based Approach
Finite-valued logics: (Baaz, Zach...)
S1 | . . . | Sn (Ex. A | B | C)
MULTLOG
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Semantic-based Approach
Finite-valued logics: (Baaz, Zach...)
S1 | . . . | Sn (Ex. A | B | C)
Projective logics: (Baaz and Fermüller)
¤M (x1, ..., xn) =
t1 if ∧ ∨Rjk
......
tm if ∧ ∨Rpq
Ri1(F11 , . . . , F 1
r1) | . . . | Rik(F
k1 , . . . , F k
rk) (Ex. A ≤ B | A < C)
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Semantic-based Approach
Finite-valued logics: (Baaz, Zach...)
S1 | . . . | Sn (Ex. A | B | C)
Projective logics: (Baaz and Fermüller)
¤M (x1, ..., xn) =
t1 if ∧ ∨Rjk
......
tm if ∧ ∨Rpq
Ri1(F11 , . . . , F 1
r1) | . . . | Rik(F
k1 , . . . , F k
rk) (Ex. A ≤ B | A < C)
Semi-Projective logics: (- and Montagna, new)Ex.: Nilpotent Minimum logic, n-contractive BL...
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Syntactic Approach
From Hilbert calculi to analytic calculi.
Hilbert calculi consist of:
many axioms
few rules (MP, generalization,...)
Pro : easy to define logicsContra : not suitable for
finding proofs
analyzing proofs
establishing properties of logics
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Sequent Calculi
SequentsA1, . . . , An ⇒ B
Intuitively a sequent is understood as “the conjunction ofA1, . . . , An implies B.
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Sequent Calculi
SequentsA1, . . . , An ⇒ B
Intuitively a sequent is understood as “the conjunction ofA1, . . . , An implies B.AxiomsE.g., A ⇒ A
Rules
Logical
(cut)Γ ⇒ A A, ∆ ⇒ Π
Γ, ∆ ⇒ ΠCut
Structural
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The system FLe
FLe = commutative Lambek calculus (= intuitionistic LinearLogic or Monoidal Logic)
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The system FLe
A,B, Γ ⇒ ΠA ⊗ B, Γ ⇒ Π
⊗lΓ ⇒ A ∆ ⇒ BΓ, ∆ ⇒ A ⊗ B
⊗r
Γ ⇒ A B, ∆ ⇒ ΠΓ, A → B, ∆ ⇒ Π
→ lA, Γ ⇒ B
Γ ⇒ A → B→ r
A, Γ ⇒ Π B, Γ ⇒ ΠA ∨ B, Γ ⇒ Π
∨lΓ ⇒ Ai
Γ ⇒ A1 ∨ A2
∨r0 ⇒ 0l
Ai, Γ ⇒ Π
A1 & A2, Γ ⇒ Π&l
Γ ⇒ A Γ ⇒ BΓ ⇒ A & B
&rΓ ⇒ ⊤
⊤r
Γ ⇒Γ ⇒ 0
0r⇒ 1
1r ⊥, Γ ⇒ Π⊥l
Γ ⇒ Π1, Γ ⇒ Π
1l
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Commutative Residuated Lattices
A (bounded pointed) commutative residuated lattice is
P = 〈P,&,∨,⊗,→,⊤,0,1,⊥〉
1. 〈P,&,∨,⊤,0〉 is a lattice with ⊤ greatest and ⊥ least
2. 〈P,⊗,1〉 is a commutative monoid.
3. For any x, y, z ∈ P , x ⊗ y ≤ z ⇐⇒ y ≤ x → z
4. 0 ∈ P .
Many-valued logics= FLe + axioms
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Commutative Residuated Lattices
A (bounded pointed) commutative residuated lattice is
P = 〈P,&,∨,⊗,→,⊤,0,1,⊥〉
1. 〈P,&,∨,⊤,0〉 is a lattice with ⊤ greatest and ⊥ least
2. 〈P,⊗,1〉 is a commutative monoid.
3. For any x, y, z ∈ P , x ⊗ y ≤ z ⇐⇒ y ≤ x → z
4. 0 ∈ P .
Many-valued logics= FLe + axioms
Cut elimination is not preserved when axioms are added
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Commutative Residuated Lattices
A (bounded pointed) commutative residuated lattice is
P = 〈P,&,∨,⊗,→,⊤,0,1,⊥〉
1. 〈P,&,∨,⊤,0〉 is a lattice with ⊤ greatest and ⊥ least
2. 〈P,⊗,1〉 is a commutative monoid.
3. For any x, y, z ∈ P , x ⊗ y ≤ z ⇐⇒ y ≤ x → z
4. 0 ∈ P .
Many-valued logics= FLe + axioms
Cut elimination is not preserved when axioms are added
(Idea) Transform axioms into ‘good’ structural rules
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On the structural rules
Example
Contraction: α → α ⊗ α
Weakening l: α → 1
Weakening r: 0 → α
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On the structural rules
Example
Contraction: α → α ⊗ αA,A, Γ ⇒ ΠA, Γ ⇒ Π
(c)
Weakening l: α → 1Γ ⇒ Π
Γ, A ⇒ Π(w, l)
Weakening r: 0 → αΓ ⇒
Γ ⇒ A(w, r)
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On the structural rules
Example
Contraction: α → α ⊗ αA,A, Γ ⇒ ΠA, Γ ⇒ Π
(c)
Weakening l: α → 1Γ ⇒ Π
Γ, A ⇒ Π(w, l)
Weakening r: 0 → αΓ ⇒
Γ ⇒ A(w, r)
Equivalence between rules and axioms
⊢FLe+(axiom) = ⊢FLe+(rule)
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Our preliminary results
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Our preliminary results
The sets Pn,Nn of formulas defined by:
P0, N0 := Atomic formulas
Pn+1 := Nn | Pn+1 ⊗ Pn+1 | Pn+1 ∨ Pn+1 | 1 | ⊥
Nn+1 := Pn | Pn+1 → Nn+1 | Nn+1 ∧Nn+1 | 0 | ⊤
P and N
Positive connectives 1,⊥,⊗,∨ haveinvertible left rules:
Negative connectives ⊤,0,∧,→ have in-vertible right rules:
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Our preliminary results
P3 N3
P2 N2
P1 N1
P0 N0
p
p
p
p
p
p
p
p
p
6
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The sets Pn,Nn of formulas defined by:
P0, N0 := Atomic formulas
Pn+1 := Nn | Pn+1 ⊗ Pn+1 | Pn+1 ∨ Pn+1 | 1 | ⊥
Nn+1 := Pn | Pn+1 → Nn+1 | Nn+1 ∧Nn+1 | 0 | ⊤
P and N
Positive connectives 1,⊥,⊗,∨ have invertible
left rules:
Negative connectives ⊤,0,∧,→ have invertible
right rules:
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Examples
Class Axiom Name
N2 α → 1, ⊥ → α weakening
α → α ⊗ α contraction
α ⊗ α → α expansion
⊗αn → ⊗αm knotted axioms (n, m ≥ 0)
¬(α & ¬α) weak contraction
P2 α ∨ ¬α excluded middle
(α → β) ∨ (β → α) prelinearity
P3 ¬α ∨ ¬¬α weak excluded middle
¬(α ⊗ β) ∨ (α ∧ β → α ⊗ β) (wnm)
N3 ((α → β) → β) → ((β → α) → α) Lukasiewicz axiom
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Our preliminary results
P3 N3
P2 N2
P1 N1
P0 N0
p
p
p
p
p
p
p
p
p
6
p
p
p
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Algorithm to transform:
axioms up to the class N2 into "good" structural
rules in sequent calculus
axioms up to the class P3 into "good" structural
rules in hypersequent calculus
(-, N. Galatos and K. Terui. LICS 2008) and
( -, L. Strassburger and K. Terui. CSL 2009)
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Our preliminary results
P3 N3
P2 N2
P1 N1
P0 N0
p
p
p
p
p
p
p
p
p
6
p
p
p
p
p
p
p
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p
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6
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@@@I
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Algorithm to transform:
axioms up to the class N2 into "good" structural
rules in sequent calculus
axioms up to the class P3 into "good" structural
rules in hypersequent calculus
(-, N. Galatos and K. Terui. LICS 2008) and
( -, L. Strassburger and K. Terui. CSL 2009)
Prolog program: AxiomCalc
http://www.logic.at/people/lara/axiomcalc.html
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Hypersequent calculus
It is obtained embedding sequents into hypersequents
Γ1 ⇒ Π1 | . . . |Γn ⇒ Πn
where for all i = 1, . . . n, Γi ⇒ Πi is an ordinary sequent.
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Hypersequent calculus
Γ ⇒ A A, ∆ ⇒ ΠΓ, ∆ ⇒ Π
CutA ⇒ A
Identity
Γ ⇒ A B, ∆ ⇒ ΠΓ, A → B, ∆ ⇒ Π
→ lA, Γ ⇒ B
Γ ⇒ A → B→ r
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Hypersequent calculus
G|Γ ⇒ A G|A, ∆ ⇒ Π
G|Γ, ∆ ⇒ ΠCut
G|A ⇒ AIdentity
G|Γ ⇒ A G|B, ∆ ⇒ Π
G|Γ, A → B, ∆ ⇒ Π→ l
G|A, Γ ⇒ B
G|Γ ⇒ A → B→ r
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Hypersequent calculus
G|Γ ⇒ A G|A, ∆ ⇒ Π
G|Γ, ∆ ⇒ ΠCut
G|A ⇒ AIdentity
G|Γ ⇒ A G|B, ∆ ⇒ Π
G|Γ, A → B, ∆ ⇒ Π→ l
G|A, Γ ⇒ B
G|Γ ⇒ A → B→ r
and adding suitable rules to manipulate the additional layerof structure.
GG |Γ ⇒ A
(ew)G |Γ ⇒ A |Γ ⇒ A
G |Γ ⇒ A(ec)
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From axioms to analytic rules
Step 1
Step 2
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From axioms to analytic rules
Step 1Transformation of any N2 (P3) axiom into an equivalent(set of) structural rule(s).
Step 2Analytic completion of the generated rules
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From axioms to analytic rules
Step 1Transformation of any N2 (P3) axiom into an equivalent(set of) structural rule(s).
Step 2Analytic completion of the generated rules
How? Using
the invertibility of the rules (∨, l), (&, r), (⊗, l), (→, r).
the Lemma: Any axiom A ⇒ B is equivalent to
α ⇒ Aα ⇒ B and also to
B ⇒ β
A ⇒ β
for α, β fresh variables.
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An example
(α → β) ∨ (β → α)
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An example
(α → β) ∨ (β → α)
is equivalent to
G | ⇒ α → β | ⇒ β → α
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An example
(α → β) ∨ (β → α)
and toG |α ⇒ β | β ⇒ α
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An example
(α → β) ∨ (β → α)
G |α ⇒ β | β ⇒ α
and by the Lemma: Any sequent α′ ⇒ β′ is equivalent to
Γ ⇒ α′
Γ ⇒ β′ and also toβ′, Γ ⇒ ∆
α′, Γ ⇒ ∆
(for Γ, ∆ fresh meta-variables)
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An example
(α → β) ∨ (β → α)
G |α ⇒ β | β ⇒ α
by the Lemma: Any sequent α′ ⇒ β′ is equivalent to
Γ ⇒ α′
Γ ⇒ β′ and also toβ′, Γ ⇒ ∆
α′, Γ ⇒ ∆
(for Γ, ∆ fresh meta-variables) is equivalent to
G |Γ ⇒ α
G |Γ ⇒ β | β ⇒ α
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An example
(α → β) ∨ (β → α)
G |α ⇒ β | β ⇒ α
is equivalent to
G |Γ ⇒ α G |Γ′ ⇒ β G |Σ, β ⇒ ∆ G |Σ′, α ⇒ ∆′
G |Γ, Σ ⇒ ∆ |Γ′, Σ′ ⇒ ∆′
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An example
(α → β) ∨ (β → α)
G |α ⇒ β | β ⇒ α
G |Γ ⇒ α G |Γ′ ⇒ β G |Σ, β ⇒ ∆ G |Σ′, α ⇒ ∆′
G |Γ, Σ ⇒ ∆ |Γ′, Σ′ ⇒ ∆′
is equivalent to
G |Γ, Σ′ ⇒ ∆′ G |Γ′, Σ ⇒ ∆
G |Γ, Σ ⇒ ∆ |Γ′, Σ′ ⇒ ∆′(com)
(Avron, Annals of Math and art. Intell. 1991)
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Some examples
⊗αn → ⊗αm
{∆i1 , . . . , ∆im , Γ ⇒ Π}i1,...,im∈{1,...,n}
∆1, . . . , ∆n, Γ ⇒ Π(knotnm)
¬α ∨ ¬¬αG |Γ1, Γ2
G |Γ1 ⇒ |Γ2 ⇒(lq)
¬(α ⊗ β) ∨ (α ∧ β → α ⊗ β)
G |Γ2, Γ1, ∆1 ⇒ Π1
G |Γ1, Γ1, ∆1 ⇒ Π1
G |Γ1, Γ3, ∆1 ⇒ Π1
G |Γ2, Γ3, ∆1 ⇒ Π1
G |Γ2, Γ3 ⇒ |Γ1, ∆1 ⇒ Π1
(wnm)
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Uniform cut-elimination
TheoremThe cut rule is admissible in (the hypersequent version of)FLe extended with any completed rule.
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Uniform cut-elimination
TheoremThe cut rule is admissible in (the hypersequent version of)FLe extended with any completed rule.
Syntactic argument:elimination procedure
Cut-ful Proofs =⇒ Cut-free Proofs
Semantic argument:Quasi-DM completion
CRL ⇐= ‘Intransitive’ CRL
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Expressive powers of (hyper)sequents
P3 N3
P2 N2
P1 N1
P0 N0
p
p
p
p
p
p
p
p
p
6
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Hilbert axioms = equations over CRL
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Expressive powers of (hyper)sequents
P3 N3
P2 N2
P1 N1
P0 N0
p
p
p
p
p
p
p
p
p
6
p
p
p
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Sequent structural rules: only equations
that hold in Heyting algebras (IL)
closed under DM completion
(-, N. Galatos and K. Terui. APAL 2012)
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Expressive powers of (hyper)sequents
P3 N3
P2 N2
P1 N1
P0 N0
p
p
p
p
p
p
p
p
p
6
p
p
p
p
p
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Sequent structural rules: only equations
that hold in Heyting algebras (IL)
closed under DM completion
Hypersequent structural rules: only equations
closed under regular completions
(-, N. Galatos and K. Terui. Draft 2012)
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An application
A logic is standard complete when it is complete w.r.t.evaluations on [0, 1]
Algebraically:
A logic is standard complete ⇔ it is complete w.r.talgebras order-isomorphic to [0, 1]
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SC: algebraic approach
Given a logic L:
1. Identify the algebraic semantics of L (L-algebras)
2. Show completeness of L w.r.t. linear, countableL-algebras
3. (Rational completeness): Find an embedding ofcountable L-algebras into dense countable L-algebras
4. Dedekind-Mac Neille style completion (embedding intoL-algebras with lattice reduct [0, 1])
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SC: algebraic approach
Given a logic L:
1. Identify the algebraic semantics of L (L-algebras)
2. Show completeness of L w.r.t. linear, countableL-algebras
3. (Rational completeness): Find an embedding ofcountable L-algebras into dense countable L-algebras
4. Dedekind-Mac Neille style completion (embedding intoL-algebras with lattice reduct [0, 1])
Step 1, 2, 4: routine
Step 3: problematic (only ad hoc solutions)
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SC: proof-theoretic approach
(Metcalfe, Montagna JSL 2007) Given a logic L:
Define a suitable hypersequent calculus
Add the density rule
G |Γ ⇒ p |Σ, p ⇒ ∆
G |Γ, Σ ⇒ ∆(density)
(= L + (density) is rational complete)
Show that the addition of density produces no newtheorems
Dedekind-Mac Neille style completion
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SC: proof-theoretic approach
(Metcalfe, Montagna JSL 2007) Given a logic L:
Define a suitable hypersequent calculus
Add the density rule
G |Γ ⇒ p |Σ, p ⇒ ∆
G |Γ, Σ ⇒ ∆(density)
(= L + (density) is rational complete)
Show that the addition of density produces no newtheorems
Dedekind-Mac Neille style completion
A ⇒ p | p ⇒ B
A ⇒ B(density)
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Density elimination
Similar to cut-elimination
Proof by induction on the length of derivations
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Density elimination
Similar to cut-elimination
Proof by induction on the length of derivations
(-, Metcalfe TCS 2008)Given a density-free derivation, ending in
···
G |Σ, p ⇒ ∆ |Γ ⇒ p(D)
G |Γ, Σ ⇒ ∆
Asymmetric substitution: p is replacedWith Σ ⇒ ∆ when occuring on the rightWith Γ when occuring on the left
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Method for density elimination
···
G |Σ, p ⇒ ∆ |Γ ⇒ p(D)
G |Γ, Σ ⇒ ∆
becomes:
···
G |Γ, Σ ⇒ ∆ |Γ, Σ ⇒ ∆(EC)
G |Γ, Σ ⇒ ∆
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A problem
Asymmmetric substitutions do not preserve derivability, as :
A sequent Π, p ⇒ p is derivable from p ⇒ p +(weakening)
Π, Γ, Σ ⇒ ∆ is not derivable
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A problem
Asymmmetric substitutions do not preserve derivability, as :
A sequent Π, p ⇒ p is derivable from p ⇒ p +(weakening)
Π, Γ, Σ ⇒ ∆ is not derivable
We can solve the problem with a suitable restructuringof the derivation...
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Our Results
(Baldi, - ,Spendier: Wollic 2012)
Theorem: Hypersequent calculus for MTL +convergent rules admits density elimination
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Our Results
(Baldi, - ,Spendier: Wollic 2012)
Theorem: Hypersequent calculus for MTL +convergent rules admits density elimination
i.e. rules whose premises do not mix "too much" theconclusion
Example :
G |Γ2, Γ1, ∆1 ⇒ Π1
G |Γ1, Γ1, ∆1 ⇒ Π1
G |Γ1, Γ3, ∆1 ⇒ Π1
G |Γ2, Γ3, ∆1 ⇒ Π1
G |Γ2, Γ3 ⇒ |Γ1, ∆1 ⇒ Π1(wnm)
Axiom: ¬(α ⊗ β) ∨ (α ∧ β → α ⊗ β)
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Our Results
(Baldi, - ,Spendier: Wollic 2012)
Theorem: Hypersequent calculus for MTL +convergent rules admits density elimination
(Baldi, - ,Spendier: In Preparation)
Theorem: Hypersequent calculus for UL + sequentrules admits density elimination.
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SC: Automated Proofs
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SC: Automated Proofs
Let L be a suitable axiomatic extension of MTL (UL )
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SC: Automated Proofs
Let L be a suitable axiomatic extension of MTL (UL )
define a hypersequent calculus for L
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SC: Automated Proofs
Let L be a suitable axiomatic extension of MTL (UL )
define a hypersequent calculus for L
check whether the calculus satisfies the condition fordensity elimination (rational completeness)
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SC: Automated Proofs
Let L be a suitable axiomatic extension of MTL (UL )
define a hypersequent calculus for L
check whether the calculus satisfies the condition fordensity elimination (rational completeness)
standard completeness follows by (-, Galatos, TeruiAlgebra Universalis 2011)
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SC: Automated Proofs
Let L be a suitable axiomatic extension of MTL (UL )
define a hypersequent calculus for L
check whether the calculus satisfies the condition fordensity elimination (rational completeness)
standard completeness follows by (-, Galatos, TeruiAlgebra Universalis 2011)
http://www.logic.at/people/lara/axiomcalc.html
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Example
Known Logics
MTL + ¬(α ⊗ β) ∨ ((α ∧ β) → (α ⊗ β))
MTL + ¬α ∨ ¬¬α
MTL + αn−1 → αn
UL + αn−1 → αn
...
New Fuzzy Logics
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Example
Known Logics
MTL + ¬(α ⊗ β) ∨ ((α ∧ β) → (α ⊗ β))
MTL + ¬α ∨ ¬¬α
MTL + αn−1 → αn
UL + αn−1 → αn
...
New Fuzzy Logics
MTL + ¬(α⊗ β)n ∨ ((α∧ β)n−1 → (α⊗ β)n), for all n > 1
UL + ¬α ∨ ¬¬α
UL + αm → αn
...
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Open problems
P3 N3
P2 N2
P1 N1
P0 N0
p
p
p
p
p
p
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p
6
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Uniform treatment of axioms behond P3
Systematic introduction of analytic calculi
first-order logic
modal and temporal logic
logics with different connectives
more on standard completeness
Proving useful properties for classes
logics in a uniform and systematic way
... many more ...
"Non-classical Proofs: Theory, Applications and
Tools", research project 2012-2017
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How far can we go?
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How far can we go?
Analytic Calculi
sequent, hypersequent calculi...
display calculi
nested sequents, deep inference, calculus of structures
labelled systems
......
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How far can we go?
Analytic Calculi
sequent, hypersequent calculi...
display calculi
nested sequents, deep inference, calculus of structures
labelled systems
......
(Strongly) Analytic calculus = weaker for of Herbrandtheorem: If ∃xB(x) is provable, where B isquantifier-free, so is
∨ni=1 B(ti), for some n.
Proof theory for many valued logics – some applications – p.31/33
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A negative result
Let L be a first-order logic satisfying:
1. ⊢LA → A
2. ⊢L∀xA(x) → B → ∃x(A(x) → B)
3. ⊢L(B → ∀xA(x)) → ∀x(B → A(x))
4. ⊢L∀xA(x) → A(t), for any term t
5. there is an atomic formula A in L such that for no n
⊢L
n∨
i=1
A(xi) → A(xi+1)
then L does not admit any strongly analytic calculus.
(Baaz, -, Work in progress)
Proof theory for many valued logics – some applications – p.32/33
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Corollary
The following logics do not admit any strongly analyticcalculus:
witnessed logics (∀= min and ∃= max), e.g. Gödel logicwith truth values in [0, 1], ∧= min, ∨= max,v(A) → v(B) = 1 iff v(A) ≤ v(B), v(B) otherwise. ∀=min and ∃= max.
(fragments of) first-order Lukasiewicz logic
Gödel logic with set of truth values{1 − 1/n : n ≥ 1} ∪ {1}
first-order nilpotent minimum logic NM with set of truthvalues {1/n : n ≥ 1} ∪ {1 − 1/n : n ≥ 1}
....
Proof theory for many valued logics – some applications – p.33/33