Bilattices with two chains of truth values
Andrew Craig1
1 Department
Miroslav Haviar2
of Pure and Applied Mathematics, University of Johannesburg
2 Matej
Bel University, Banská Bystrica, Slovakia
LATD, Phalaborwa, South Africa
29 June 2016
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
Outline
Bilattices, interlaced bilattices and their product representation
Default bilattices and the bilattices Jn
Natural duality theory
Natural duality for the quasivarieties ISP(Jn )
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
Bilattices were introduced by Ginsberg in 1986 as a generalisation of
Belnap’s four valued logic from the mid 1970’s.
>
6k
t
f
⊥
6t
The vertical axis represents the knowledge order and the horizontal axis
represents the truth order. The lattice operations ⊗ and ⊕ of the
knowledge order represent consensus and gullibility.
In addition to two sets of lattice operations, a bilattice has a unary
negation operation ¬ which preserves 6k and reverses 6t .
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
Examples of bilattices:
>
>
>
t
f
t
f
t
f
⊥
⊥
⊥
FOUR
FIVE
N IN E
A bilattice B has the algebraic signature
B = hB; ⊗, ⊕, ∧, ∨, ¬i.
This signature will often contain bounds of one (or both) of the orders.
The knowledge bounds are ⊥, > and the truth bounds are f , t. Note:
often one set of bounds will be term-definable from the other.
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
There can be varying levels of interaction between the orders. A bilattice
B is distributive if for all a, b, c ∈ B the identity
a • (b ∗ c) ≈ (a • b) ∗ (a • c)
holds for •, ∗ ∈ {⊗, ⊕, ∧, ∨}.
A bilattice is interlaced if
a 6t b
=⇒
a ⊗ c 6t b ⊗ c,
a 6t b
=⇒
a ⊕ c 6t b ⊕ c,
a 6k b
=⇒
a ∧ c 6k b ∧ c,
a 6k b
=⇒
a ∨ c 6k b ∨ c.
NB: an interlaced bilattice B has 6k as a subalgebra of B2 .
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
Product bilattices
Let L = hL; u, ti be a lattice. The operations of the product bilattice
L L = hL × L; ⊗, ⊕, ∧, ∨, ¬i
are defined for (a, b), (c, d) ∈ L × L by
(a, b) ⊗ (c, d) = (a u c, b u d)
(a, b) ⊕ (c, d) = (a t c, b t d)
(a, b) ∧ (c, d) = (a u c, b t d)
(a, b) ∨ (c, d) = (a t c, b u d)
¬(a, b) = (b, a).
Think of an element (a, b) ∈ L × L as encoding evidence about some
sentence: a is the evidence for, and b is the evidence against.
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
The following theorem was proven by various researchers at different
levels of generality: Romanowska and Trakul (1989), Fitting (1990),
Avron (1996), Rivieccio (2010) and others.
Theorem
A (bounded) bilattice B is interlaced if and only if it is isomorphic to the
bilattice product L L for some (bounded) lattice L.
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
Ginsberg proposed the bilattice SEVEN for inference in default logic.
The additional truth values dt and df represent “true by default” and
“false by default”.
>
t
f
d>
df
dt
⊥
SEVEN
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
Ginsberg proposed the bilattice SEVEN for inference in default logic.
The additional truth values dt and df represent “true by default” and
“false by default”.
>
t
dt
t
f
>
d>
⊥
d>
df
df
dt
f
⊥
6k
SEVEN
6t
Note: SEVEN is not interlaced, as d> 6k t but
d> ∧ ⊥ = df k ⊥ = t ∧ ⊥.
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
A hierarchy of default bilattices Cabrer, Craig, Priestley (2015)
>0 = >
f0
t0
t0
>i
ti
tj
fi
fj
ti
>j
tn
> = >0
>n
>j
>i
>n+1/⊥
>n
tj
fn
fj
fn
fi
tn
>n+1 = ⊥
f0
Figure: Kn in its knowledge order (left) and truth order (right); here
0<i<j<n
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
>
t0
f0
>
>
f0
t0
f1
t1
⊥
⊥
J0
J1
f0
t0
f1
t1
f2
t2
⊥
J2
Figure: The bilattices J0 , J1 and J2 , drawn with their knowledge order.
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
t0
>
t1
f0
t0
f1
t1
fn
tn
⊥
tn
>
⊥
fn
f1
f0
Figure: The bilattice Jn drawn in both its knowledge (left) and truth
(right) orders.
Note: Jn is not interlaced for n > 1. We have f0 6k > but
⊥ ∧ f0 = f0 k fn = > ∧ ⊥
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
Natural duality
Natural duality theory provides a uniform method for obtaining dual
structures for algebras in a finitely generated quasivariety.
The theory has been developed by Clark, Davey, Werner, Priestley, Haviar
and others.
It has been successfully applied to many (quasi)varieties related to logic
such as: distributive lattices (with additional operations), de Morgan
algebras, Kleene algebras, finitely-generated quasivarieties of Heyting
algebras.
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
Let M be a finite algebra and consider the quasivariety A = ISP(M).
For an algebra A ∈ A, we consider the dual, D(A), of A to be the set of
homomorphisms A(A, M).
Examples:
Let M = h{0, 1}; ∧, ∨, ¬, 0, 1i (the two-element Boolean algebra) and
B = ISP(M). Then for any Boolean algebra A, we have that the set
B(A, M) is in a one-to-one correspondence with the set of ultrafilters of
A.
For M = h{0, 1}; ∧, ∨, 0, 1i and D = ISP(M) (the variety of bounded
distributive lattices) we have that for any A ∈ D, the set D(A, M) is in
bijective correspondence with the set of prime filters of A. This is the
usual underlying set for the dual space of A when using Priestley duality.
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
Let M be a finite algebra and consider the quasivariety A = ISP(M).
Now let A ∈ A. There is a natural map
e : A → M A(A,M) given by (e(a))(f ) = f (a)
for all a ∈ A and all f ∈ A(A, M). The maps e(a) : A(A, M) → M are
maps given by evaluation.
However, there are many other maps in M A(A,M) . Natural duality
theory adds topological and relational structure to the sets A(A, M) and
M so that the only structure-preserving maps are the evaluation maps.
This gives a concrete representation of the algebra as a set of
structure-preserving maps.
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
The structure of dual spaces
Given A = ISP(M), we search for a suitable alter ego for M. We want a
structure M
∼ (the set M equipped with topology and relational structure)
such that X = ISc P+ (M
∼ ) is a category which is dually equivalent to A.
For finite k, we say that a k-ary relation R on M is an algebraic relation
if R is a subalgebra of Mk . In order for the theory to work, the relational
structure on the alter ego must consist of finitary algebraic relations.
We denote the alter ego by M
∼ . That is,
M
∼ := hM ; R, T i
where T is the discrete topology, R is a set of algebraic relations on M.
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
Observe that A(A, M) ⊆ M A . It inherits the subspace topology from
A
the product space M
∼ and as it is a closed subspace, this topology is
compact.
The set A(A, M) also inherits relational structure from M
∼ in the
following way for R ∈ R. For f, g ∈ A(A, M)
(f, g) ∈ RA
⇐⇒
∀a∈A
(f (a), g(a)) ∈ R.
Finally, the algebra A is represented as the continuous,
relation-preserving maps from D(A) to M
∼ . Formally,
D: A → X
D = A(−, M)
E: X → A
E = X (−, M
∼)
and A ' ED(A) via a 7→ e(a).
For example, in a bounded distributive lattice D is represented by the
continuous order-preserving maps from its dual space into
2 = h{0, 1}; T , 6i. This is isomorphic to the clopen up-sets of the
∼
Priestley space of prime filters (ordered by inclusion).
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
Examples
Boolean algebras:
B = ISP(M) where M = h{0, 1}; ∧, ∨, ¬, 0, 1i
+
M
∼ = h{0, 1}; T i and X = ISc P (M
∼ ) = Stone spaces
Bounded distributive lattices:
D = ISP(M) where M = h{0, 1}; ∧, ∨, 0, 1i
+
M
∼ ) = Priestley spaces
∼ = h{0, 1}; 6, T i and X = ISc P (M
Kleene algebras:
K = ISP(K) where K = h{0, a, 1}; ∧, ∨, ¬, 0, 1i
K
∼ = h{0, a, 1}; R1 , R2 , R3 , T i
a
a
0
R1
Andrew Craig, Miroslav Haviar
R2
1
R3
Bilattices with two chains of truth values
Examples
Distributive bilattices:
DB = ISP(FOUR) where FOUR = h{f , t, >, ⊥}; ∧, ∨, ¬, f , ti
FOU
∼ R = h{f , t, >, ⊥}; 6k , T i
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
We want to develop natural dualities for the quasivarieties ISP(Jn ) for
n ∈ ω.
Theorem (Clark and Davey, 1998)
(NU Duality Theorem, special case) Let M be a finite lattice-based
algebra and A = ISP(M). Let
2
M
∼ = M ; S(M ), T
where T is the discrete topology M . Then M
∼ yields a duality on A.
Therefore, to find a duality we must study the lattice S(J2n ). We have
|S(J20 )| = 4 and |S(J21 )| = 7
but
|S(J20 )| = 28 and |S(J23 )| = 200
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
We want to develop natural dualities for the quasivarieties ISP(Jn ) for
n ∈ ω.
Theorem (Clark and Davey, 1998)
(NU Duality Theorem, special case) Let M be a finite lattice-based
algebra and A = ISP(M). Let
2
M
∼ = M ; S(M ), T
where T is the discrete topology M . Then M
∼ yields a duality on A.
Therefore, to find a duality we must study the lattice S(J2n ). We have
|S(J20 )| = 4 and |S(J21 )| = 7
but
|S(J20 )| = 28 and |S(J23 )| = 200
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
We want to develop natural dualities for the quasivarieties ISP(Jn ) for
n ∈ ω.
Theorem (Clark and Davey, 1998)
(NU Duality Theorem, special case) Let M be a finite lattice-based
algebra and A = ISP(M). Let
2
M
∼ = M ; S(M ), T
where T is the discrete topology M . Then M
∼ yields a duality on A.
Therefore, to find a duality we must study the lattice S(J2n ). We have
|S(J20 )| = 4 and |S(J21 )| = 7
but
|S(J20 )| = 28 and |S(J23 )| = 200
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
17
˘
16
16
15
˘
14
14
12 13
˘ 13
˘
12
11
10
8
9
8̆
9̆
˘
10
6
4
5̆
5
7
6̆
2̆
3
2
3̆
1
Figure: The algebraic lattice S(J22 ) with its completely meet-irreducible
elements shown in bold.
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
For Jn and i < n, consider binary relation
Sn,i := {(>, >), (⊥, ⊥)} ∪ {fi+1 , . . . , fn }2 ∪ {ti+1 , . . . , tn }2
∪ {(fj , fk ) | i < j, 0 6 k 6 i} ∪ {(tj , tk ) | i < j, 0 6 k 6 i}.
>
>
>
f0
t0
f0 , f1
t0 , t1
f0 , f1 , f2
t0 , t1 , t2
f1 , f2 , f3
t1 , t2 , t3
f2 , f3
t2 , t3
f3
t3
⊥
⊥
⊥
S3,0
S3,1
S3,2
Figure: The binary relations S3,0 , S3,1 and S3,2 on J3 drawn as
quasi-orders. We draw x and y in the same block if xRy and yRx.
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
For Jn and i < n, consider binary relation
Sn,i := {(>, >), (⊥, ⊥)} ∪ {fi+1 , . . . , fn }2 ∪ {ti+1 , . . . , tn }2
∪ {(fj , fk ) | i < j, 0 6 k 6 i} ∪ {(tj , tk ) | i < j, 0 6 k 6 i}.
>
>
>
f0
t0
f0 , f1
t0 , t1
f0 , f1 , f2
t0 , t1 , t2
f1 , f2 , f3
t1 , t2 , t3
f2 , f3
t2 , t3
f3
t3
⊥
⊥
⊥
S3,0
S3,1
S3,2
Figure: The binary relations S3,0 , S3,1 and S3,2 on J3 drawn as
quasi-orders. We draw x and y in the same block if xRy and yRx.
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
Proposition
Consider the bilattice Jn . For i ∈ {0, 1, . . . , n − 1} let Sn,i ⊆ Jn2
be defined by
Sn,i := {(>, >), (⊥, ⊥)} ∪ {fi+1 , . . . , fn }2 ∪ {ti+1 , . . . , tn }2
∪ {(fj , fk ) | i < j, 0 6 k 6 i} ∪ {(tj , tk ) | i < j, 0 6 k 6 i}.
Then Sn,i is the universe of a subalgebra of J2n .
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
Proposition
Let Snn be the universe of the subalgebra of J2n generated by the
relation 6k on Jn2 . The relation Snn or its converse must always be
included in the dualising structure ∼
J n.
>
t0 , t1 , t2 , t3
f0 , f1 , f2 , f3
⊥
S3,3
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
We use a technique called “piggybacking” (developed in different
levels of generality by Davey, Werner and Priestley). The method
identifies a dualising structure M
∼ when M is a finite algebra with
a distributive lattice reduct. As observed above, Jn is always a
distributive lattice in its truth order.
Define for x 6= tn , wx : J[n → 2 by
(
1 if x 6t a
wx (a) =
0 otherwise
Now consider
Wx,y := (wx , wy )−1 (6) = { (a, b) ∈ J2n | wx (a) 6 wy (b) }
The Piggyback Theorem says that we only need the maximal
subalgebras contained in the Wx,y .
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
We use a technique called “piggybacking” (developed in different
levels of generality by Davey, Werner and Priestley). The method
identifies a dualising structure M
∼ when M is a finite algebra with
a distributive lattice reduct. As observed above, Jn is always a
distributive lattice in its truth order.
Define for x 6= tn , wx : J[n → 2 by
(
1 if x 6t a
wx (a) =
0 otherwise
Now consider
Wx,y := (wx , wy )−1 (6) = { (a, b) ∈ J2n | wx (a) 6 wy (b) }
The Piggyback Theorem says that we only need the maximal
subalgebras contained in the Wx,y .
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
Theorem
The structure ∼
J n := hJn ; Sn,n , . . . , Sn,0 , T i yields a duality on the
quasivariety ISP(Jn ).
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
A natural duality is called optimal if it is not possible to obtain a
duality if any of the relations are removed from the dualising
structure.
Theorem
The structure ∼
J n := hJn ; Sn,n , . . . , Sn,0 , T i yields an optimal
duality on the quasivariety ISP(Jn ).
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
The varieties HSP(Jn )
Con(J0 )
Con(J1 )
Con(J2 )
Con(J3 )
Proposition
For each n ∈ ω we have Con(Jn ) ∼
= (2n ) ⊕ 1.
We will have that V(Jn ) = ISP(H(Jn )) and can then use the
theory of multi-sorted natural duality to study dualities for V(Jn ).
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values
Further work
Verify that the dualities given above are full dualities.
Explore the relationship between the natural duality and
restricted Priestley duality.
Investigate adding an implication to the signature.
Andrew Craig, Miroslav Haviar
Bilattices with two chains of truth values