Standard Completeness I:
Proof Theoretic Approach
Agata Ciabattoni
Vienna University of Technology (TU Vienna)
Joint work with P. Baldi, N. Galatos, G. Metcalfe, L. Spendier, K. Terui
Standard Completeness
Completeness of axiomatic systems with respect to algebras
whose lattice reduct is the real unit interval [0, 1].
Standard Completeness
Completeness of axiomatic systems with respect to algebras
whose lattice reduct is the real unit interval [0, 1].
Why is the topic relevant for this workshop?
Standard Completeness
Completeness of axiomatic systems with respect to algebras
whose lattice reduct is the real unit interval [0, 1].
Why is the topic relevant for this workshop?
Why is the topic relevant?
(Hajek 1998) Formalizations of Fuzzy Logic
Uninorm (based logics)
Conjunction and implication are interpreted by a particular
uninorm/t-norm (or a class of) and its residuum.
A uninorm is a function ∗ : [0, 1]2 → [0, 1] such that for all
x, y , z ∈ [0, 1]:
x ∗ y = y ∗ x (Commutativity)
(x ∗ y ) ∗ z = x ∗ (y ∗ z) (Associativity)
x ≤ y implies x ∗ z ≤ y ∗ z (Monotonicity)
e ∈ [0, 1] e ∗ x = x (Identity)
The residuum of ∗ is a function ⇒∗ : [0, 1]2 → [0, 1] where
x ⇒∗ y = max{z | x ∗ z ≤ y }.
A t-norm is a uninorm in which e = 1.
Some standard complete logics
v : Propositions → [0, 1]
Gödel logic
v (A ∧ B) = min{v (A), v (B)}
v (A ∨ B) = max{v (A), v (B)}
v (A → B) = 1 if v (A) ≤ v (B), and v (B) otherwise
v (⊥) = 0
UL Uninorm logic (Metcalfe, Montagna 2007)
v (A B) = v (A) ∗ v (B), ∗ left continous uninorm
v (A ∨ B) = max{v (A), v (B)}
v (A → B) = v (A) ⇒∗ v (B)
v (⊥) = 0
MTL Monoidal T-norm logic (Godo, Esteva 2001)
∗ left continous t-norm
(Uninorm-based) Logics
often described by adding axioms to already known logics.
Example
UL = FLe with ((α → β) ∧ t) ∨ ((β → α) ∧ t) (linearity)
MTL = UL with weakening/integrality
Gödel logic = MTL with contraction α → α α
SUL = UL with α → α α and mingle α α → α
WMTL = MTL with ¬(α β) ∨ (α ∧ β → α β)
....
(Uninorm-based) Logics
are often described by adding axioms to already known logics.
Question Given a logic L obtained by extending UL with
α α → α (mingle)?
αn−1 → αn (n-contraction)?
¬(α β)n ∨ ((α ∧ β)n−1 → (α β)n )?
....
Is L standard complete? (is it a formalization of Fuzzy Logic?)
(Uninorm-based) Logics
are often described by adding axioms to already known logics.
Question Given a logic L obtained by extending UL with
α α → α (mingle)?
αn−1 → αn (n-contraction)?
¬(α β)n ∨ ((α ∧ β)n−1 → (α β)n )?
....
Is L standard complete? (is it a formalization of Fuzzy Logic?)
Many papers written for individual logics!
Standard Completeness: algebraic approach
Given a logic L:
1
Identify the algebraic semantics of L (L-algebras)
2
Show completeness of L w.r.t. linear, countable L-algebras
3
(Rational completeness): Find an embedding into linear,
dense countable L-algebras
4
Dedekind-Mac Neille style completion (embedding into
L-algebras with lattice reduct [0, 1])
Standard Completeness: algebraic approach
Given a logic L:
1
Identify the algebraic semantics of L (L-algebras)
2
Show completeness of L w.r.t. linear, countable L-algebras
3
(Rational completeness): Find an embedding into linear,
dense countable L-algebras
4
Dedekind-Mac Neille style completion (embedding into
L-algebras with lattice reduct [0, 1])
Step 3: problematic (mainly(∗) ad hoc solutions)
(∗) see Paolo’s talk!
Standard Completeness: proof theoretic approach
(Metcalfe, Montagna JSL 2007) Given a logic L:
Add Takeuti and Titani’s density rule (p eigenvariable)
(α → p) ∨ (p → β) ∨ γ
(density)
(α → β) ∨ γ
(= L + (density) is rational complete)
Show that density produces no new theorems (Rational
completeness)
Dedekind-Mac Neille style completion
Our result
Uniform (and automated) proofs of standard completeness for
large classes of axiomatic extensions of UL
How?
Our result
Uniform (and automated) proofs of standard completeness for
large classes of axiomatic extensions of UL
How?
(Step 1) Defining suitable calculi for axiomatic extensions of UL
(Step 2) General conditions for the elimination of the density rule
from these calculi
(Step 3) Dedekind-Mac Neille style completion
Our result
Uniform (and automated) proofs of standard completeness for
large classes of axiomatic extensions of UL
How?
(Step 1) Defining suitable calculi for axiomatic extensions of UL
(Step 2) General conditions for the elimination of the density rule
from these calculi
(Step 3) Dedekind-Mac Neille style completion
(Avron JSL ’89)
Hypersequents: Γ1 ⇒ Π1 | . . . | Γn ⇒ Πn where for all
i = 1, . . . n, Γi ⇒ Πi is an ordinary sequent
Our base calculus: FLe
α⇒α
(init)
Γ⇒>
⇒t
(>)
Γ, ⊥ ⇒ ∆
Γ⇒β
Γ⇒α∧β
α, Γ ⇒ Π
(∧r )
β, Γ ⇒ Π
α ∨ β, Γ ⇒ Π
Γ⇒α ∆⇒β
Γ, ∆ ⇒ α β
f ⇒
(∨l)
Γ⇒α
( r )
αi , Γ ⇒ Π
(∧l)
β, ∆ ⇒ Π
Γ, α → β, ∆ ⇒ Π
α, β, Γ ⇒ Π
α β, Γ ⇒ Π
α, ∆ ⇒ Π
Γ, ∆ ⇒ Π
α 1 ∧ α2 , Γ ⇒ Π
Γ⇒α
(fl)
(⊥)
Γ⇒
(fr )
Γ⇒f
Γ⇒Π
(tl)
t, Γ ⇒ Π
Γ⇒α
(tr )
(→ l)
( l)
Γ ⇒ αi
Γ ⇒ α1 ∨ α2
α, Γ ⇒ β
Γ⇒α→β
(cut)
(∨r )
(→ r )
Calculi for axiomatic extensions of FLe
E.g. UL = FLe + ((α → β) ∧ t) ∨ ((β → α) ∧ t) (linearity)
Cut elimination is not preserved when axioms are added
(Idea) Axioms are transformed into
‘good’ structural rules
in the ‘appropriate’ formalism
Hypersequent Calculus for UL
(UL = FLe + ((α → β) ∧ t) ∨ ((β → α) ∧ t))
Hypersequent: Γ1 ⇒ Π1 | . . . | Γn ⇒ Πn
This calculus is obtained
embedding sequents into hypersequents in FLe
α, Γ ⇒ β
(→, r )
Γ⇒α→β
i.e.
G|α, Γ ⇒ β
(→, r )
G|Γ ⇒ α → β
Hypersequent Calculus for UL
(UL = FLe + ((α → β) ∧ t) ∨ ((β → α) ∧ t))
Hypersequent: Γ1 ⇒ Π1 | . . . | Γn ⇒ Πn
This calculus is obtained
embedding sequents into hypersequents in FLe
adding suitable rules to manipulate the additional layer of
structure.
G|Γ ⇒ α|Γ ⇒ α
G
(ew)
(ec)
G|Γ ⇒ α
G|Γ ⇒ α
Hypersequent Calculus for UL
(UL = FLe + ((α → β) ∧ t) ∨ ((β → α) ∧ t))
Hypersequent: Γ1 ⇒ Π1 | . . . | Γn ⇒ Πn
This calculus is obtained
embedding sequents into hypersequents in FLe
adding suitable rules to manipulate the additional layer of
structure.
G|Γ ⇒ α|Γ ⇒ α
G
(ew)
(ec)
G|Γ ⇒ α
G|Γ ⇒ α
G | Γ, Γ0 ⇒ α G | Γ1 , Γ01 ⇒ α0
(com)
G | Γ, Γ1 ⇒ α | Γ0 , Γ01 ⇒ α0
(Avron 1991)
An example
β ⇒β
α⇒α
(com)
α ⇒ β|β ⇒ α
(→,r)
α ⇒ β| ⇒ β → α
(→,r)
⇒ t ⇒ α → β| ⇒ β → α
⇒t
2x(∧,r)
⇒ (α → β) ∧ t | ⇒ (β → α) ∧ t
(∨i ,r)
⇒ (α → β) ∧ t | ⇒ ((α → β) ∧ t) ∨ ((β → α) ∧ t)
(∨i ,r)
⇒ ((α → β) ∧ t) ∨ ((β → α) ∧ t) | ⇒ ((α → β) ∧ t) ∨ ((β → α) ∧ t)
(EC)
⇒ ((α → β) ∧ t) ∨ ((β → α) ∧ t)
Our result
Uniform (and automated) proofs of standard completeness for
large classes of axiomatic extensions of UL
(Step 1) Defining suitable calculi for axiomatic extensions of UL
(Step 2) General conditions for the elimination of the density rule
from these calculi
(Step 3) Dedekind-Mac Neille style completion
Algorithmic introduction of analytic calculi I
Definition (Classification; -, Galatos and Terui, LICS 2008)
The classes Pn , Nn of positive and negative axioms/equations
are:
P0 ::= N0 ::= atomic formulas
Pn+1 ::= Nn | Pn+1 ∨ Pn+1 | Pn+1 Pn+1 | t | ⊥
Nn+1 ::= Pn | Pn+1 → Nn+1 | Nn+1 ∧ Nn+1 | f | >
Examples
Class
N2
P2
P3
N3
Axiom
α → t, ⊥ → α
α→αα
αα→α
αn → αm
¬(α ∧ ¬α)
α ∨ ¬α
(α → β) ∨ (β → α)
¬α ∨ ¬¬α
¬(α β) ∨ (α ∧ β → α β)
((α → β) → β) → ((β → α) → α)
(α ∧ β) → α (α → β)
Name
weakening
contraction
expansion
knotted axioms
weak contraction
excluded middle
prelinearity
weak excluded middle
(wnm)
Lukasiewicz axiom
divisibility
Algorithmic introduction of analytic calculi II
Theorem (AC, Galatos, Terui 2008)
Algorithm to transform (almost all)
axioms α up to the class N2 into good structural rules in
sequent calculus
axioms α up to the class P3 into good structural rules in
hypersequent calculus
Algorithmic introduction of analytic calculi II
Theorem (AC, Galatos, Terui 2008)
Algorithm to transform (almost all)
axioms α up to the class N2 into good structural rules in
sequent calculus
axioms α up to the class P3 into good structural rules in
hypersequent calculus
(AC, Galatos, Terui 2011,2012,Submitted) and (almost all)
algebraic equations 1 ≤ α up to the class N2 are
preserved under DM-completion
algebraic equations 1 ≤ α up to the class P3 are preserved
under DM-completion when applied to s.i. algebras
Our result
Uniform (and automated) proofs of standard completeness for
large classes of axiomatic extensions of UL
(Step 1) Defining suitable calculi for axiomatic extensions of UL
(Step 2) General conditions for the elimination of the density rule
from these calculi
(Step 3) Dedekind-Mac Neille style completion
Density vs Cut
Takeuti and Titani’s rule (p eigenvariable)
(α → p) ∨ (p → β) ∨ γ
(α → β) ∨ γ
Density vs Cut
Takeuti and Titani’s rule (p eigenvariable)
(α → p) ∨ (p → β) ∨ γ
(α → β) ∨ γ
G | Γ ⇒ p | Σ, p ⇒ ∆
(density )
G | Γ, Σ ⇒ ∆
where p is does not occur in the conclusion.
G | Γ ⇒ A G | Σ, A ⇒ ∆
(cut)
G | Γ, Σ ⇒ ∆
Density elimination
Similar to cut-elimination
Proof by induction on the length of derivations
Density elimination
Similar to cut-elimination
Proof by induction on the length of derivations
(AC, Metcalfe 2008) Given a density-free derivation, ending in
· 0
·d
·
G|Γ ⇒ p|p ⇒ ∆
(density)
G|Γ ⇒ ∆
Density elimination
(AC, Metcalfe 2008) Given a density-free derivation, ending in
· 0
·d
·
G|Γ ⇒ p|p ⇒ ∆
(density)
G | Γ, Σ ⇒ ∆
Asymmetric substitution: p is replaced
With ∆ when occuring on the right
With Γ when occuring on the left
· 0
·d
·
G|Γ ⇒ ∆|Γ ⇒ ∆
(EC)
G|Γ ⇒ ∆
Problem with (com)
p⇒p
·
·
·
Π⇒Ψ
Γ⇒∆
·
·
·
Π⇒Ψ
(com)
Π ⇒ p|p ⇒ Ψ
·
·
·d
·
·
G|Γ ⇒ p|p ⇒ ∆
(com)
Π ⇒ ∆|Γ ⇒ Ψ
·
· ∗
·d
·
·
G|Γ ⇒ ∆|Γ ⇒ ∆
(D)
G|Γ ⇒ ∆
p ⇒ p axiom
Γ ⇒ ∆ not an axiom
(EC)
G|Γ ⇒ ∆
Solution (with weakening)
(AC, Metcalfe 2008)
p⇒p
·
·
·
G|Γ ⇒ p|p ⇒ ∆
·
·
·
Π⇒Ψ
·
·
·
Π⇒Ψ
(com)
Π ⇒ p|p ⇒ Ψ
·
·
·d
·
·
G|Γ ⇒ p|p ⇒ ∆
(cut)
Π ⇒ ∆|Γ ⇒ Ψ
·
· ∗
·d
·
·
G|Γ ⇒ ∆|Γ ⇒ ∆
(D)
G|Γ ⇒ ∆
(EC)
G|Γ ⇒ ∆
Axiomatic extensions of MTL
MTL = UL + weakening/integrality
Theorem (Baldi, A.C. ,Spendier 2013, 2014)
MTL + all P3 axioms leading to semi-anchored rules admits
density elimination
G | Γ2 , Γ1 , ∆1 ⇒ Π1 G | Γ1 , Γ3 , ∆1 ⇒ Π1
G | Γ1 , Γ1 , ∆1 ⇒ Π1 G | Γ2 , Γ3 , ∆1 ⇒ Π1
(wnm)
Ex.
G | Γ2 , Γ3 ⇒ | Γ1 , ∆1 ⇒ Π1
Axiomatic extensions of MTL
MTL = UL + weakening/integrality
Theorem (Baldi, A.C. ,Spendier 2013, 2014)
MTL + all P3 axioms leading to semi-anchored rules admits
density elimination
G | Γ2 , Γ1 , ∆1 ⇒ Π1 G | Γ1 , Γ3 , ∆1 ⇒ Π1
G | Γ1 , Γ1 , ∆1 ⇒ Π1 G | Γ2 , Γ3 , ∆1 ⇒ Π1
(wnm)
Ex.
G | Γ2 , Γ3 ⇒ | Γ1 , ∆1 ⇒ Π1
Automated transformation of axioms into rules and check
whether the latter are semi-anchored:
http://www.logic.at/people/lara/axiomcalc.html
Solution (without weakening)
(AC, Metcalfe 2008)
Π, p ⇒ p
·
·
·d
·
·
G|Γ ⇒ p|p ⇒ ∆
Π⇒t
·
· ∗
·d
·
·
G|Γ ⇒ ∆|Γ ⇒ ∆
(EC)
(D)
G|Γ ⇒ ∆
G|Γ ⇒ ∆
We substitute:
p ⇒ p with ⇒ t.
p with ∆ when occurring on the right.
p with Γ when occurring on the left.
We introduce suitable cuts
Works for UL. How about extensions?
Axiomatic extensions of UL?
Standard completeness has been shown for:
UL + contraction and mingle (Metcalfe and Montagna, JSL
2007)
UL with n-contraction αn−1 → αn and n-mingle αn → αn−1
(n > 2) (Wang, FSS 2012)
UL + with knotted axioms αk → αj (j, k > 1) (Baldi, Soft
Computing 2014)
... many open problems and no uniform method ...
A case study: UL with A → A A
(P. Baldi, AC 2014)
p⇒p
·
·
·
Π, p, p ⇒ p
(c)
Π, p ⇒ p
·
·
·
G|Γ ⇒ p|p ⇒ ∆
(D)
G|Γ ⇒ ∆
⇒t
·
·
·
Π, Γ ⇒ t
(?)
Π⇒t
·
·
·
G|Γ ⇒ ∆|Γ ⇒ ∆
(ec)
G|Γ ⇒ ∆
We substitute:
p ⇒ p with ⇒ t.
p with ∆ when occurring on the right.
p with Γ when occurring on the left.
We introduce suitable cuts
A case study: UL with A → A A
(P. Baldi, AC 2014)
p⇒p
·
·
·
Π, p, p ⇒ p
(c)
Π, p ⇒ p
·
·
·
G|Γ ⇒ p|p ⇒ ∆
(D)
G|Γ ⇒ ∆
⇒t
·
·
·
Π, Γ ⇒ t
(?)
Π⇒t
·
·
·
G|Γ ⇒ ∆|Γ ⇒ ∆
(ec)
G|Γ ⇒ ∆
(?) is replaced by a subderivation obtained by substituting:
p ⇒ p (axiom) with Π, Γ ⇒ t (derivable).
p with ∆ when occurring on the right.
p with Γ when occurring on the left.
We introduce suitable cuts
A case study: UL with A → A A
p⇒p
·
·
·
Π, p, p ⇒ p
⇒t
·
·
·
Π, Γ ⇒ t
(c)
Π, p ⇒ p
·
·
·
G|Γ ⇒ p|p ⇒ ∆
(D)
G|Γ ⇒ ∆
(?)
Π⇒t
·
·
·
G|Γ ⇒ ∆|Γ ⇒ ∆
(ec)
G|Γ ⇒ ∆
The same idea works for (mingle) and for all sequent structural
rules (= N2 axioms)
G|S1
. . . G|Sm
(r )
G|Π, Γ1 , . . . , Γn ⇒ Ψ
s.t. if R(Si ) = Ψ then none of Γi appears only once in L(Si ).
Axiomatic extensions of UL
Theorem (AC, Baldi Submitted 2014)
UL + nonlinear N2 axioms (and/or mingle) admits density
elimination
N2 axioms
Nonlinear N2
axioms
Axiomatic extensions of UL
Theorem (AC, Baldi Submitted 2014)
UL + nonlinear N2 axioms (and/or mingle) admits density
elimination
N2 axioms
Nonlinear N2
axioms
Conjecture: UL + N2 axioms admits density elimination
Closing the cycle
Uniform (and automated) proofs of standard completeness for
large classes of axiomatic extensions of UL
(Step 1) Defining suitable calculi for axiomatic extensions of UL
(Step 2) General conditions for the elimination of the density rule
from these calculi
(Step 3) Dedekind-Mac Neille style completion
Our methods applies to
Known Logics
UL + contraction and mingle (Metcalfe, Montagna 2007)
MTL + αn−1 → αn (n ≥ 3) (Baldi, 2014)
MTL + ¬(α β) ∨ ((α ∧ β) → (α β)) (Noguera et al.08)
MTL + αn−1 → αn (AC, Esteva, Godo 2002)
...
New Fuzzy Logics
UL + contraction or mingle
UL + f αk → αn (n > 1)
MTL + ¬(α β)n ∨ ((α ∧ β)n−1 → (α β)n ), for all n > 1
...
The big picture
Theory and tools for the investigation of non-classical logics
Analytic calculi (sequent, hypersequent, nested, display
calculi ...)
Exploitation:
standard completeness
new semantic foundations (e.g. paraconsistent logics)
interpolation
properties of algebraic structures
....
”Non-classical Proofs: Theory, Applications and Tools”, research project
2012-2017 (START prize – Austrian Research Fund)