Standard Completeness II: a novel
algebraic approach
Paolo Baldi
(Joint work with A. Ciabattoni, K.Terui)
Workshop on Admissible Rules II - Les Diablerets
2.02.2015
A novel algebraic approach.
• Reformulation of the proof theoretic methods, with no
reference in principle to the proof theory.
• Showing more concretely how to build an embedding into
dense algebras
• Extension to the noncommutative cases.
Recall: The usual way to Standard Completeness
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
(L-chains)
3. (Rational completeness): Find an embedding of countable
L-chains into dense countable L-chains
4. Dedekind-Mac Neille style completion (embedding into
L-algebras with lattice reduct [0, 1])
Algebraic semantics (Step 1.)
• FL-algebra A = (A, ∧, ∨, ·, /, \, t, f )
• (A, ∧, ∨) lattice
• (A, ·, t) monoid
• f ∈ A.
• x · y ≤ z ⇔ x ≤ z/y ⇔ y ≤ x\z for any x, y , z ∈ A
Algebraic semantics (Step 1.)
• FL-algebra A = (A, ∧, ∨, ·, /, \, t, f )
• (A, ∧, ∨) lattice
• (A, ·, t) monoid
• f ∈ A.
• x · y ≤ z ⇔ x ≤ z/y ⇔ y ≤ x\z for any x, y , z ∈ A
• An FLe -algebra is an FL-algebra where the operation · is
commutative (thus \ = / and both are denoted with →)
• An FLw -algebra is an FL-algebra where f and t are minimum
and maximum of the lattice ordering.
Algebraic semantics (Step 1.)
An FL-algebra A is
• A Chain if the lattice ordering is total.
• Dense if for any a, b ∈ A such that a < b there is a c ∈ A
such that a < c < b.
• Bounded If the lattice ordering has a least element ⊥ and a
maximum element ⊤
• Complete if, for any X ⊆ A, we have ∨X , ∧X ∈ A .
Semilinear logics (Step 2.)
Logics complete w.r.t. chains:
• UL
⇐⇒
bounded FLe -chains
• MTL
⇐⇒
bounded FLew -chains
•
psULr
⇐⇒
bounded FL-chains
•
psMTLr
⇐⇒
bounded FLw -chains
Semilinear logics (Step 2.)
Logics complete w.r.t. chains:
• UL + α
⇐⇒
bounded FLe -chains sat. t ≤ α
• MTL + α
⇐⇒
bounded FLew -chains sat. t ≤ α
•
psULr
•
psMTLr
+ α
+ α
⇐⇒
bounded FL-chains sat. t ≤ α
⇐⇒
bounded FLw -chains sat. t ≤ α
Densifiability (Step 3.)
Definition. A subvariety V of FL algebras is densifiable, if for any
chain A in V and a, b ∈ A such that a < b and for no c ∈ A we
have a < c < b (a, b form a “gap”, a ≺ b)
Densifiability (Step 3.)
Definition. A subvariety V of FL algebras is densifiable, if for any
chain A in V and a, b ∈ A such that a < b and for no c ∈ A we
have a < c < b (a, b form a “gap”, a ≺ b)
b
a
A
Densifiability (Step 3.)
Definition. A subvariety V of FL algebras is densifiable, if for any
chain A in V and a, b ∈ A such that a < b and for no c ∈ A we
have a < c < b (a, b form a “gap”, a ≺ b), there is a chain B in
V , p ∈ B and an embedding e : A → B such that e(a) < p < e(b)
b
e(b)
a
e(a)
A
p
B
Densifiability
Theorem. (Baldi, Terui 2015) Let V be a densifiable variety.
Then every (nontrivial) finite or countable chain in V is
embeddable into a countable dense chain in V .
Preframe - Residuated Frames
• A preframe is a structure (W , W ′ , N, ◦, ε, ǫ) such that
• (W , ◦, ε) is a monoid
• N ⊆ W × W′
• ǫ ∈ W′
Preframe - Residuated Frames
• A preframe is a structure (W , W ′ , N, ◦, ε, ǫ) such that
• (W , ◦, ε) is a monoid
• N ⊆ W × W′
• ǫ ∈ W′
• A residuated frame is a preframe with additional operations
\\ and // satisfying
x ◦ yNz ⇔ yNx\\z ⇔ xNz//y
for any x, y ∈ W , z ∈ W ′ .
From preframe to residuated frames
We can always extend a preframe W = (W , W ′ , N, ◦, ε, ǫ) to a
residuated frame W̃ = (W , W̃ ′ , Ñ, ◦, ε, (ε, ǫ, ε)) , letting
• W̃′ := W × W ′ × W
• x Ñ (v1 , z, v2 )
⇐⇒
v1 xv2 N z.
From preframe to residuated frames
We can always extend a preframe W = (W , W ′ , N, ◦, ε, ǫ) to a
residuated frame W̃ = (W , W̃ ′ , Ñ, ◦, ε, (ε, ǫ, ε)) , letting
• W̃′ := W × W ′ × W
• x Ñ (v1 , z, v2 )
⇐⇒
v1 xv2 N z.
• (v1 , z, v2 )//y = (v1 , z, yv2 )
(v1 , z, v2 )\\x = (v1 x, z, v2 )
From preframe to residuated frames
We can always extend a preframe W = (W , W ′ , N, ◦, ε, ǫ) to a
residuated frame W̃ = (W , W̃ ′ , Ñ, ◦, ε, (ε, ǫ, ε)) , letting
• W̃′ := W × W ′ × W
• x Ñ (v1 , z, v2 )
⇐⇒
v1 xv2 N z.
• (v1 , z, v2 )//y = (v1 , z, yv2 )
(v1 , z, v2 )\\x = (v1 x, z, v2 )
Notice that:
• xy Ñ (v1 , z, v2 ) ⇔ x Ñ (v1 , z, yv2 ) ⇔ y Ñ (v1 x, z, v2 )
Examples of Residuated frames
• Let A = (A, ∧, ∨, ·, /, \, t, f ) be an FL-algebra.
WA = (A, A, N, ◦, f , t)
where N is the lattice ordering ≤ of A and ◦ = ·, is a
residuated frame.
Letting \\ = \ and // = / we have that
x ◦ yNz ⇔ yNx\\z ⇔ xNz//y
is just the residuation property.
Examples of Residuated frames
• Let Fm be the set of formulas in our language, ⊢FL the
derivability relation defined by the sequent calculus FL.
WFL = (Fm∗ , Fm, N, ◦, ε, ε)
where ◦ is the comma and N is defined as
α1 ◦ · · · ◦ αn N β
⇔
⊢FL α1 , . . . , αn ⇒ β
is a preframe. It can be extended in the canonical way to a
residuated frame.
Nuclei on residuated frames
(W , W ′ , N, ◦, ε, ǫ) residuated frame, X ⊆ W , Y ⊆ W ′
• X ⊲ = {y ∈ W ′ : XNy }
• Y ⊳ = {w ∈ W : wNY }
• γN (X ) = X ⊲⊳
Nuclei on residuated frames
(W , W ′ , N, ◦, ε, ǫ) residuated frame, X ⊆ W , Y ⊆ W ′
• X ⊲ = {y ∈ W ′ : XNy }
• Y ⊳ = {w ∈ W : wNY }
• γN (X ) = X ⊲⊳
γN (X ) is a closure operator (1-3), in addition it is a nucleus (4).
1. X ⊆ γN (X )
2. X ⊆ Y ⇒ γN (X ) ⊆ γN (Y )
3. γN (γN (X )) = γN (X )
4. γN (X ) ◦ γN (Y ) ⊆ γN (X ◦ Y )
The dual algebra
From a Residuated frame W = (W , W ′ , N, ◦, ε, ǫ) we can build a
complete FL-algebra, the dual algebra of W.
W+ = (γN [P(W )], ∩, ∪γN , ◦γN , \, /, γN (ε), ǫ⊳ )
Where
• γN [P(W )] = {X ⊆ W such that γN (X ) = X }
• X ◦γN Y := γN (X ◦ Y )
• X ∪γN Y := γN (X ∪ Y )
• X \Y := {y : X ◦ {y } ⊆ Y }
• Y /X := {y : {y } ◦ X ⊆ Y }
Densifiability for FLw chains
Let A = (A, ∧, ∨, ·, /, \, t, f ) be a FLw -chain which is not dense.
Assume a, b ∈ A form a “gap”, a ≺ b.
Densifiability for FLw chains
Let A = (A, ∧, ∨, ·, /, \, t, f ) be a FLw -chain which is not dense.
Assume a, b ∈ A form a “gap”, a ≺ b. We want to add an element
p in between a and b, hence we require:
xNp ⇔ x ≤ a
pNy ⇔ b ≤ y
pNp
Densifiability for FLw chains
Let A = (A, ∧, ∨, ·, /, \, t, f ) be a FLw -chain which is not dense.
Assume a, b ∈ A form a “gap”, a ≺ b. We want to add an element
p in between a and b, hence we require:
xNp ⇔ x ≤ a
pNy ⇔ b ≤ y
pNp
• Preframe: ((A ∪ {p})∗ , A ∪ {p}, N, ◦, ε, f ), with ◦ string
concatenation, N defined as:
• x[p]Nc
• xNp
• x[p]Np
⇔
⇔
x[b] ≤ c .
x ≤a.
always holds.
p
• We call W̃A
the corresponding residuated frame.
Recall: Density elimination for MTL
Given a density-free derivation, ending in
·· d ′
·
G |Γ ⇒ p|p ⇒ ∆
G |Γ ⇒ ∆
(density)
• Asymmetric substitution: p is replaced
• With ∆ when occurring on the right
• With Γ when occurring on the left
Recall: Density elimination for MTL
·· d ′
·
G |Γ ⇒ ∆|Γ ⇒ ∆
G |Γ ⇒ ∆
(EC)
• Asymmetric substitution: p is replaced
• With ∆ when occurring on the right
• With Γ when occurring on the left
Densifiability for FLw chains
Let A = (A, ∧, ∨, ·, /, \, t, f ) be an FLw -chain, a, b ∈ A such that
a ≺ b. We want to add an element p in between a and b, hence
we require:
xNp ⇔ x ≤ a
pNy ⇔ b ≤ y
pNp
• Preframe: ((A ∪ {p})∗ , A ∪ {p}, N, ◦, ε, f ), with ◦ string
concatenation, N defined as:
• x[p]Nc
• xNp
• x[p]Np
⇔
⇔
x[b] ≤ c .
x ≤a.
always holds.
p
• We call W̃A
the corresponding residuated frame.
Densifiability for FLw chains
Theorem. (Baldi, Terui 2015) FLw -chains are densifiable.
Proof Idea. Let A be an FLw chain with a, b ∈ A such that
p
a ≺ b, W̃A
residuated frame. We show the following
p+
1. A is an FLw chain −→ W̃A
is a complete FLw -chain.
2. There is an embedding
p+
e : x ∈ A → {x}⊲⊳ ∈ W̃A
p+
3. W̃A
“fills the gap” between a and b, i.e.
{a}⊲⊳ ⊂ {p}⊲⊳ ⊂ {b}⊲⊳
Densifiability for FLw chains
1.
A is an FLw chain
p+
W̃A
is an FLw chain
Densifiability for FLw chains
1.
p+
W̃A
is an FLw chain
A is an FLw chain
p
W̃A
satisfies (com),(w )
xNz yNw
(com)
xNw or yNz
uv N z (w )
uxv N z
Densifiability for FLw chains
Theorem. (Baldi, Terui 2015) FLw -chains are densifiable.
Proof Idea. Let A be an FLw chain with a, b ∈ A such that
p
a ≺ b, W̃A
residuated frame. We show the following
p+
1. A is an FLw chain −→ W̃A
is a complete FLw -chain.
2. There is an embedding
p+
e : x ∈ A → {x}⊲⊳ ∈ W̃A
p+
3. W̃A
“fills the gap” between a and b, i.e.
{a}⊲⊳ ⊂ {p}⊲⊳ ⊂ {b}⊲⊳
From Densifiability to Standard Completeness
• FLw -chains are densifiable. We can embedd any FLw chain A
p+
in the complete FLw chain W̃A
.
From Densifiability to Standard Completeness
• FLw -chains are densifiable. We can embedd any FLw chain A
p+
in the complete FLw chain W̃A
.
• We can embedd any FLw -chain into a dense, complete
FLw -chain.
From Densifiability to Standard Completeness
• FLw -chains are densifiable. We can embedd any FLw chain A
p+
in the complete FLw chain W̃A
.
• We can embedd any FLw -chain into a dense, complete
FLw -chain.
• The logic PsMTLr is Standard Complete.
From Densifiability to Standard Completeness
• FLw -chains are densifiable. We can embedd any FLw chain A
p+
in the complete FLw chain W̃A
.
• We can embedd any FLw -chain into a dense, complete
FLw -chain.
• The logic PsMTLr is Standard Complete.
What about subvarieties / Axiomatic extensions of PsMTLr ?
Densifiability for FLw chains
Problem. For which equations t ≤ α the FLw -chains satisfying
t ≤ α are densifiable?
Proof Idea. Let A be an FLw chain satisfying t ≤ α with a, b ∈ A
p
such that a ≺ b, W̃A
residuated frame. We need to show
p+
1. A is an FLw chain satisfying t ≤ α −→ W̃A
is a complete
FLw -chain satisfying t ≤ α
2. There is an embedding
p+
e : x ∈ A → {x}⊲⊳ ∈ W̃A
p+
“fills the gap” between a and b, i.e.
3. W̃A
{a}⊲⊳ ⊂ {p}⊲⊳ ⊂ {b}⊲⊳
Densifiability for FLw chains
1.
A is an FLw chain sat. t ≤ α
p+
W̃A
is a FLw chain sat. t ≤ α
Densifiability for FLw chains
1.
A is an FLw chain sat. t ≤ α
p+
W̃A
is a FLw chain sat. t ≤ α
p
W̃A
satisfies (com),(w ), (q)
The substructural hierarchy for FL
♣♣
✻
♣♣
♣♣
♣♣
♣
P3
♣♣
✻
♣♣
♣♣
♣♣
♣
N3
(Ciabattoni, Galatos, Terui 2012).
✻❅
■
❅ ✒ ✻
Sets Pn , Nn of terms defined by:
❅
❅
P2
N2 P0 , N0 := Variables
✻❅
■
❅ ✒ ✻ Pn+1 := Nn | Pn+1 · Pn+1 | Pn+1 ∨ Pn+1 | t | ⊥
❅
Nn+1 := Pn | Pn+1 \Nn+1 | Nn+1 /Pn+1
❅
:= Nn+1 ∧ Nn+1 | f | ⊤
P1
N1
✻❅
■
❅ ✒ ✻
❅
❅
P0
N0
The substructural hierarchy
♣♣
✻
♣♣
♣♣
♣♣
♣
P3
♣♣
✻
♣♣
♣♣
♣♣
♣
N3
✻❅
■
❅ ✒ ✻
❅
❅
P2
N2
✻❅
■
❅ ✒ ✻
❅
❅
P1
N1
✻❅
■
❅ ✒ ✻
❅
❅
P0
N0
• If a residuated frame W̃ satisfies an analytic quasi
equation
t1 Nu1 and . . . and tm Num
(q)
t0 Nu0
W + satisfies the corresponding N2 equation
t ≤ α.
The substructural hierarchy
♣♣
✻
♣♣
♣♣
♣♣
♣
P3
♣♣
✻
♣♣
♣♣
♣♣
♣
N3
✻❅
■
❅ ✒ ✻
❅
❅
P2
N2
✻❅
■
❅ ✒ ✻
❅
❅
P1
N1
✻❅
■
❅ ✒ ✻
❅
❅
P0
N0
• If a residuated frame W̃ satisfies an analytic quasi
equation
t1 Nu1 and . . . and tm Num
(q)
t0 Nu0
W + satisfies the corresponding N2 equation
t ≤ α.
• If a residuated frame W̃ satisfies an analytic
clause
t1 Nu1 and . . . and tm Num
(q)
tm+1 Num+1 or . . . or tn Nun
W + satisfies the corresponding P3 equation t ≤ α
Examples
N2 includes:
t
t
t
t
≤ xy \yx
≤ x\xx
≤ x k \x n
≤∼ (x∧ ∼ x)
(e)
(c)
(knotted axioms, n, k ≥ 0)
(no-contradiction)
P3 includes:
t ≤ x∨ ∼ x
t ≤∼ x∨ ∼∼ x
t ≤∼ (x · y ) ∨ (x ∧ y \x · y )
t ≤∼ (x · y )n ∨ ((x ∧ y )n−1 \(x · y )n )
t ≤ p0 ∨ (p0 \p1 ) ∨ · · · ∨ (p0 ∧ · · · ∧ pk−1 \pk )
t ≤ (x n−1 \x · y ) ∨ (y \x · y )
(excluded middle)
(weak excluded middle)
(wnm)
(wnmn )
(bounded size k)
(Ωn )
Densifiability for FLw chains
1.
A is an FLw chain sat. t ≤ α
p+
W̃A
is a FLw chain sat. t ≤ α
N2 , P3
p
W̃A
satisfies (com),(w ), (q)
Densifiability for FLw chains
1.
A is an FLw chain sat. t ≤ α
semi-anchored
p+
W̃A
is a FLw chain sat. t ≤ α
N2 , P3
p
W̃A
satisfies (com),(w ), (q)
Example
The equation t ≤∼ (x · y ) ∨ (x ∧ y \x · y ) is equivalent to the clause
xu N z and xy N z and uy N z and uu N z
(wnm)
xy N ǫ or u N z
• (x, z ) unanchored
Example
The equation t ≤∼ (x · y ) ∨ (x ∧ y \x · y ) is equivalent to the clause
xu N z and xy N z and uy N z and uu N z
(wnm)
xy N ǫ or u N z
• (x, z ) unanchored
• (u, z ) anchored
Example
The equation t ≤∼ (x · y ) ∨ (x ∧ y \x · y ) is equivalent to the clause
xu N z and xy N z and uy N z and uu N z
(wnm)
xy N ǫ or u N z
• xu N z contains (x, z ) unanchored =⇒ uu N z contains (u, z)
anchored .
Example
The equation t ≤ (x 2 \x · y ) ∨ (y \x · y ) is equivalent to the
analytic clause
yx N z1 and wx N z1 and yx N z2 and wx N z2
(Ω3 )
w y N z2 or x N z1
• (y , z1 ) unanchored
Example
The equation t ≤ (x 2 \x · y ) ∨ (y \x · y ) is equivalent to the
analytic clause
yx N z1 and wx N z1 and yx N z2 and wx N z2
(Ω3 )
w y N z2 or x N z1
• (y , z1 ) unanchored
• (y , z2 ) anchored
Example
The equation t ≤ (x 2 \x · y ) ∨ (y \x · y ) is equivalent to the
analytic clause
y x N z1 and wx N z1 and yx N z2 and wx N z2
(Ω3 )
wy N z2 or x N z1
• y x N z1 contains (y , z1 ) unanchored =⇒ y x N z2 contains
(y , z2 ) anchored
Densifiability for FLw chains
1.
A is an FLw chain sat. t ≤ α
semi-anchored
p+
W̃A
is a FLw chain sat. t ≤ α
N2 , P3
p
W̃A
satisfies (com),(w ), (q)
Densifiability for FLw chains
Theorem. (Baldi, Terui 2015) FLw chains satisfying a
semi-anchored equation t ≤ α are densifiable.
Proof Idea. Let A be an FLw chain satisfying t ≤ α with a, b ∈ A
p
such that a ≺ b, W̃A
residuated frame. We show that the
following hold
p+
1. A is an FLw chain satisfying t ≤ α −→ W̃A
is a complete
FLw -chain satisfying t ≤ α
2. There is an embedding
p+
e : x ∈ A → {x}⊲⊳ ∈ W̃A
p+
3. W̃A
“fills the gap” between a and b, i.e.
{a}⊲⊳ ⊂ {p}⊲⊳ ⊂ {b}⊲⊳
From Densifiability to Standard Completeness
• FLw -chains satisfying a semi-anchored equation t ≤ α are
densifiable.
• We can embedd any FLw -chain satisfying t ≤ α into a dense,
complete FLw -chain satisfying t ≤ α.
• The logic PsMTLr + α is standard complete.
Densifiability for FLe chains
Let A = (A, ∧, ∨, ·, /, \, t, f ) be an FLe -chain and a, b ∈ A such
that a ≺ b .
• Preframe ((A ∪ {p})∗ , A ∪ {p}, N, t, f ), with N defined as
follows:
• xp n Nc
⇔ xb n ≤ c .
• xNp
⇔ x ≤a.
• xp n−1 pNp ⇔ xb n−1 ≤ t.
p
• Residuated frame W̃A
defined in the standard way.
Recall: Density Elimination for UL
(Ciabattoni, Metcalfe 2008)
Π, p ⇒ p
··
·· d
·
G |Γ ⇒ p|p ⇒ ∆
G |Γ ⇒ ∆
(D)
• We substitute:
• p ⇒ p with ⇒ t (axiom)
• p with ∆ when occurring on the right.
• p with Γ when occurring on the left.
Recall: Density Elimination
(Ciabattoni, Metcalfe 2008)
Π⇒ t
··
·· d ∗
·
G |Γ ⇒ ∆|Γ ⇒ ∆
G |Γ ⇒ ∆
(EC)
• We substitute:
• p ⇒ p with ⇒ t.
• p with ∆ when occurring on the right.
• p with Γ when occurring on the left.
Conclusions
• Standard completeness for axiomatic extensions of UL with
nonlinear N2 axioms.
• Standard completeness for extensions of MTL, psMTLr with
semi-anchored axioms/equations
• An algebraic version of the proof theoretical approach via
residuated frames.
Some open problems
• Find a necessary condition for standard completeness for MTL
extended with axioms within the class P3 in the substructural
hierarchy.
• Prove that any axiomatic extension of UL with axioms within
the class N2 in the substructural hierarchy is standard
complete.
• Logics with involutive negation. Long standing open problem:
standard completeness of IUL.