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 ordered 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 : Tm L → A : h ( ϕ ′ ) = h ( ψ ′ ) for all ϕ ′ ≈ ψ ′ ∈ Σ implies h ( ϕ ) = h ( ψ ) . K -admissible if for every homomorphism σ : Tm L → Tm L : = 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

b b b Example: Kleene Lattice KL Consider the Kleene lattice KL = �{⊥ , a , ⊤} , ∧ , ∨ , ¬� : ⊤ 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 ( with ϕ ∼ K ψ iff | F K ( X ) = Tm L ( X ) / ∼ K = K ϕ ≈ ψ ) with universe F K ( X ) = { [ ϕ ] ∼ K | ϕ ∈ Tm L ( X ) } and operations f ([ ϕ 1 ] ∼ K , . . . , [ ϕ n ] ∼ K ) = [ f ( ϕ 1 , . . . , ϕ n )] ∼ K is called the X -generated free algebra of K . In particular, F K ( n ) is the free algebra on n generators of K , and if m = max {| A | : A ∈ K} , then for all ϕ, ψ ∈ Tm L ( m ) : iff | = K ϕ ≈ ψ | = F K ( 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 F K ( n ) -valid. Moreover, F K ( n ) is finite, so checking K -admissibility is decidable. But F K ( n ) usually is very big, e.g., | F C 3 ( 3 ) | = 43916. We seek a set of algebras K ′ , called admissibility set of K , s.t. Σ ⇒ ϕ ≈ ψ is K ′ -valid. Σ ⇒ ϕ ≈ ψ is K -admissible iff 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 ( F K ( ω )) and K ⊆ V ( K ′ ) . 2 Q ( K ′ ) = Q ( F K ( ω )) . Corollary Given a finite set K of finite algebras, every set K ′ with K ′ ⊆ S ( F K ( ω )) and K ⊆ H ( K ′ ) is an admissibility set of K , i.e., Σ ⇒ ϕ ≈ ψ is K ′ -valid. Σ ⇒ ϕ ≈ ψ is K -admissible iff Christoph Röthlisberger TAFA – A Tool for Admissibility in Finite Algebras September 17, 2013 9 / 16

A Possible Algorithm 1: function A DM A LGS ( K ) declare A , D : set 2: declare B , B ′ : algebra 3: A ← ∅ 4: for all A ∈ D do 5: B ← F REE ( A , D ) 6: B ′ ← S UB P RE H OM ( A , B ) 7: while B ′ � = B do 8: B ← B ′ 9: B ′ ← S UB P RE H OM ( A , B ) 10: end while 11: add B to A 12: end for 13: return A 14: 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 ( F KL ( 1 )) , but KL ∈ H ( F KL ( 2 )) . 2 F KL ( 2 ) has 82 elements and the smallest subalgebras B ≤ F KL ( 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 = { A 1 , . . . , A n } is called a minimal generating set for the quasivariety Q ( K ) if for every set K ′ = { B 1 , . . . , B k } : Q ( K ) = Q ( K ′ ) implies [ | A 1 | , . . . , | A n | ] ≤ m [ | B 1 | , . . . , | B k | ] . Theorem If Q = Q ( A 1 , . . . , A n ) , A 1 , . . . , A n are Q -subdirectly irreducible finite algebras, and A i �∈ IS ( A j ) for all i � = j, then { A 1 , . . . , A n } 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

A DM A LGS 1: function A DM A LGS ( K ) declare A , D : set 2: declare B , B ′ : algebra 3: D ← M IN G EN S ET ( K ) 4: A ← ∅ 5: for all A ∈ D do 6: B ← F REE ( A , D ) 7: B ′ ← S UB P RE H OM ( A , B ) 8: while B ′ � = B do 9: B ← B ′ 10: B ′ ← S UB P RE H OM ( A , B ) 11: end while 12: add B to A 13: end for 14: return M IN G EN S ET ( A ) 15: 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 M IN G EN S ET ( K ) ⊆ IS ( F K ( n )) . Theorem The following are equivalent for any finite set K of finite L -algebras, B ∈ S ( F K ( ω )) and n := max {| C | : C ∈ K} : 1 K is almost structurally complete. 2 M IN G EN S ET ( { A × B : A ∈ K} ) ⊆ IS ( F K ( 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 0 2 2 sc 0 % ∧ , ∨ , ∗ , ⊥ , ⊤ PCL 2 5 Q ( PCL 2 ) 1 7 5 sc 29 % ∧ , ∨ , ∗ , ⊥ , ⊤ PCL 1 3 Stone algebras 1 6 3 sc 50 % L 3 3 → , ¬ Algebras for Ł 3 1 12 6 asc 50 % P 4 ∗ Q ( P ) 2 6 3 sc 50 % G 106 3 ◦ Q ( G 106 ) 2 10 2,2 no 80 % M 5 5 ∧ , ∨ Lattices in Q ( M 5 ) 3 28 5 sc 82 % L → Algebras for Ł → 3 → 2 40 3 sc 93 % 3 3 Z → Algebras for RM → 3 → 2 60 3 sc 95 % 3 KL 3 ∧ , ∨ , ¬ Kleene lattices 2 82 4 no 95 % N 5 ∧ , ∨ Lattices in Q ( N 5 ) 5 3 99 5 sc 95 % ∧ , ∨ , ∗ , ⊥ , ⊤ PCL 3 9 Q ( PCL 3 ) 2 625 19 no 97 % Z →¬ Algebras for RM →¬ 3 → , ¬ 2 264 6 asc 98 % 3 Z 3 3 ∧ , ∨ , → , ¬ Q ( Z 3 ) 2 1296 6 asc 100 % Lat 8 5 ∧ , ∨ Q ( Lat 8 ) 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