TAFA A Tool for Admissibility in Finite Algebras Christoph - - PowerPoint PPT Presentation

TAFA – A Tool for Admissibility in Finite Algebras

Christoph Röthlisberger

Mathematics Institute, University of Bern

Tableaux 2013 – Nancy, September 16–19, 2013

Christoph Röthlisberger TAFA – A Tool for Admissibility in Finite Algebras September 17, 2013 1 / 16

TAFA

Implemented in Delphi XE2 Compiled for Windows (use WINE for Mac and Linux) Most recent version of TAFA.EXE is downloadable from https://sites.google.com/site/admissibility/

Christoph Röthlisberger TAFA – A Tool for Admissibility in Finite Algebras September 17, 2013 2 / 16

Derivability vs Admissibility

Consider a system defined by two rules: Nat(0) and Nat(x) ⇒ Nat(s(x)). The following rule is derivable: Nat(x) ⇒ Nat(s(s(x))). However, this rule is only admissible: Nat(s(x)) ⇒ Nat(x). But what if we add to the system: Nat(s(−1)) ???

Christoph Röthlisberger TAFA – A Tool for Admissibility in Finite Algebras September 17, 2013 3 / 16

Motivation

Admissibility plays a fundamental role in describing properties of (classes of) algebras and logics. Checking admissibility in finite algebras with the naive approach is decidable, but not feasible. We consider a more efficient method to check admissibility in finite algebras and provide a tool to get the results.

Christoph Röthlisberger TAFA – A Tool for Admissibility in Finite Algebras September 17, 2013 4 / 16

Validity and Admissibility

If Σ ∪ {ϕ ≈ ψ} is a finite set of L-equations, then we call the

• rdered pair Σ ⇒ ϕ ≈ ψ a L-quasiequation.

For a class K of L-algebras, an L-quasiequation Σ ⇒ ϕ ≈ ψ is K-valid, Σ | =K ϕ ≈ ψ, if for every algebra A ∈ K and every homomorphism h: TmL → A: h(ϕ′) = h(ψ′) for all ϕ′ ≈ ψ′ ∈ Σ implies h(ϕ) = h(ψ). K-admissible if for every homomorphism σ: TmL → TmL: | =K σ(ϕ′) ≈ σ(ψ′) for all ϕ′ ≈ ψ′ ∈ Σ implies | =K σ(ϕ) ≈ σ(ψ). If K = {A}, we usually write A-valid and A-admissible.

Christoph Röthlisberger TAFA – A Tool for Admissibility in Finite Algebras September 17, 2013 5 / 16

Example: Kleene Lattice KL

Consider the Kleene lattice KL = {⊥, a, ⊤}, ∧, ∨, ¬:

b b b

⊥ a ⊤ Then (since no term is constantly a) {x ≈ ¬x} ⇒ x ≈ y is KL-admissible, but not KL-valid. The same holds for the following quasiequation (x y stands for x ≈ x ∧ y): {¬x x, x ∧ ¬y ¬x ∨ y} ⇒ ¬y y

Christoph Röthlisberger TAFA – A Tool for Admissibility in Finite Algebras September 17, 2013 6 / 16

Free Algebras

Let K be a class of L-algebras. Then the quotient algebra FK(X) = TmL(X) / ∼K (with ϕ ∼K ψ iff | =K ϕ ≈ ψ) with universe FK(X) = {[ϕ]∼K | ϕ ∈ TmL(X)} and operations f([ϕ1]∼K, . . . , [ϕn]∼K) = [f(ϕ1, . . . , ϕn)]∼K is called the X-generated free algebra of K. In particular, FK(n) is the free algebra on n generators of K, and if m = max{|A| : A ∈ K}, then for all ϕ, ψ ∈ TmL(m): | =K ϕ ≈ ψ iff | =FK(m) ϕ ≈ ψ.

Christoph Röthlisberger TAFA – A Tool for Admissibility in Finite Algebras September 17, 2013 7 / 16

Free Algebras and Admissibility

Theorem (Rybakov) Let K be a finite set of finite L-algebras, n = max{|A| : A ∈ K}, Σ ⇒ ϕ ≈ ψ a quasiequation. The following are equivalent:

1 Σ ⇒ ϕ ≈ ψ is K-admissible. 2 Σ ⇒ ϕ ≈ ψ is Q(K)-admissible. 3 Σ ⇒ ϕ ≈ ψ is FK(n)-valid.

Moreover, FK(n) is finite, so checking K-admissibility is

• decidable. But FK(n) usually is very big, e.g., |FC3(3)| = 43916.

We seek a set of algebras K′, called admissibility set of K, s.t. Σ ⇒ ϕ ≈ ψ is K-admissible iff Σ ⇒ ϕ ≈ ψ is K′-valid.

Christoph Röthlisberger TAFA – A Tool for Admissibility in Finite Algebras September 17, 2013 8 / 16

Checking Admissibility

Theorem Given a class of algebras K, the following are equivalent:

1 K′ ⊆ Q(FK(ω)) and K ⊆ V(K′). 2 Q(K′) = Q(FK(ω)).

Corollary Given a finite set K of finite algebras, every set K′ with K′ ⊆ S(FK(ω)) and K ⊆ H(K′) is an admissibility set of K, i.e., Σ ⇒ ϕ ≈ ψ is K-admissible iff Σ ⇒ ϕ ≈ ψ is K′-valid.

Christoph Röthlisberger TAFA – A Tool for Admissibility in Finite Algebras September 17, 2013 9 / 16

A Possible Algorithm

1: function ADMALGS(K) 2:

declare A, D : set

3:

declare B, B′ : algebra

4:

A ← ∅

5:

for all A ∈ D do

6:

B ← FREE(A, D)

7:

B′ ← SUBPREHOM(A, B)

8:

while B′ = B do

9:

B ← B′

10:

B′ ← SUBPREHOM(A, B)

11:

end while

12:

add B to A

13:

end for

14:

return A

15: end function

Christoph Röthlisberger TAFA – A Tool for Admissibility in Finite Algebras September 17, 2013 10 / 16

Example: Kleene lattice KL

Let us again look at the algebra KL.

1 KL ∈ H(FKL(1)), but KL ∈ H(FKL(2)). 2 FKL(2) has 82 elements and the smallest subalgebras

B ≤ FKL(2) with KL ∈ H(B) have 4 elements.

Christoph Röthlisberger TAFA – A Tool for Admissibility in Finite Algebras September 17, 2013 11 / 16

Minimal Generating Set

A set of finite algebras K = {A1, . . . , An} is called a minimal generating set for the quasivariety Q(K) if for every set K′ = {B1, . . . , Bk}: Q(K) = Q(K′) implies [|A1|, . . . , |An|] ≤m [|B1|, . . . , |Bk|]. Theorem If Q = Q(A1, . . . , An), A1, . . . , An are Q-subdirectly irreducible finite algebras, and Ai ∈ IS(Aj) for all i = j, then {A1, . . . , An} is a minimal generating set for Q. Moreover, this is the unique minimal generating set for Q up to isomorphism.

Christoph Röthlisberger TAFA – A Tool for Admissibility in Finite Algebras September 17, 2013 12 / 16

ADMALGS

1: function ADMALGS(K) 2:

declare A, D : set

3:

declare B, B′ : algebra

4:

D ← MINGENSET(K)

5:

A ← ∅

6:

for all A ∈ D do

7:

B ← FREE(A, D)

8:

B′ ← SUBPREHOM(A, B)

9:

while B′ = B do

10:

B ← B′

11:

B′ ← SUBPREHOM(A, B)

12:

end while

13:

add B to A

14:

end for

15:

return MINGENSET(A)

16: end function

Christoph Röthlisberger TAFA – A Tool for Admissibility in Finite Algebras September 17, 2013 13 / 16

Structural and Almost Structural Completeness

Theorem The following are equivalent for any finite set K of finite L-algebras and n = max{|C| : C ∈ K}:

1 K is structurally complete. 2 MINGENSET(K) ⊆ IS(FK(n)).

Theorem The following are equivalent for any finite set K of finite L-algebras, B ∈ S(FK(ω)) and n := max{|C| : C ∈ K}:

1 K is almost structurally complete. 2 MINGENSET({A × B : A ∈ K}) ⊆ IS(FK(n)).

Christoph Röthlisberger TAFA – A Tool for Admissibility in Finite Algebras September 17, 2013 14 / 16

Some Experiments

A |A| Language Quasivariety Q(A) n F(n) M SC Reduction BA 2 ∧, ∨, ¬, ⊥, ⊤ Boolean algebras 2 2 sc 0 % PCL2 5 ∧, ∨,∗ , ⊥, ⊤ Q(PCL2) 1 7 5 sc 29 % PCL1 3 ∧, ∨,∗ , ⊥, ⊤ Stone algebras 1 6 3 sc 50 % L3 3 →, ¬ Algebras for Ł3 1 12 6 asc 50 % P 4 ∗ Q(P) 2 6 3 sc 50 % G106 3

• Q(G106)

2 10 2,2 no 80 % M5 5 ∧, ∨ Lattices in Q(M5) 3 28 5 sc 82 % L→

3

3 → Algebras for Ł→

3

2 40 3 sc 93 % Z→

3

3 → Algebras for RM→ 2 60 3 sc 95 % KL 3 ∧, ∨, ¬ Kleene lattices 2 82 4 no 95 % N5 5 ∧, ∨ Lattices in Q(N5) 3 99 5 sc 95 % PCL3 9 ∧, ∨,∗ , ⊥, ⊤ Q(PCL3) 2 625 19 no 97 % Z→¬

3

3 →, ¬ Algebras for RM→¬ 2 264 6 asc 98 % Z3 3 ∧, ∨, →, ¬ Q(Z3) 2 1296 6 asc 100 % Lat8 5 ∧, ∨ Q(Lat8) 5 7579 2 sc 100 %

Christoph Röthlisberger TAFA – A Tool for Admissibility in Finite Algebras September 17, 2013 15 / 16

TAFA – A Tool for Admissibility in Finite Algebras

Thank you for your attention!

Christoph Röthlisberger TAFA – A Tool for Admissibility in Finite Algebras September 17, 2013 16 / 16