n-dimensional Boolean algebras

Transcription

1 n-dimensional Boolean algebras Antonio Bucciarelli IRIF, Univ Paris Diderot joint work with Antonio Ledda, Francesco Paoli and Antonino Salibra

2 Outline 1 Church algebras and the λ-calculus 2 n-dimensional Boolean algebras 3 n-dimensional propositional calculus 4 Rewriting n-pc formulas

3 Semantic completeness The problem Given a class of models of the λ-calculus, is it the case that any λ-theory is representable in it? Some (ad hoc) incompleteness results : Sott semantics (Honsell and Ronchi della Rocca 1992), Stable semantics (Gouy 1995), Strongly stable semantics (Bastonero 1998). A general incompleteness result, based on Universal Algebra (Manzonetto and Salibra 2006) : The λ-models arising from cpo-based semantics are indecomposable, and there exist λ-theories admitting only decomposable models.

4 Church Algebras An algebra A is a Church algebra if it admits a ternary operator q(x, y, z) and two constants 0, 1 such that the following identities are satisfied : q(1, x, y) = x q(0, x, y) = y Examples : Rings with unit : q(x, y, z) = xy + (1 x)z. Heyting algebras : q(x, y, z) = (x z) ((x 0) y). Combinatory algebras : q(x, y, z) = (xy)z; 1 = k; 0 = sk. Lambda abstraction algebras : q(x, y, z) = (xy)z 1 = λxy.x; 0 = λxy.y.

5 Factor congruences and decomposition Decomposition A = B C θ, θ such that B = A/θ and C = A/θ θ, θ as above, and moreover θ θ = and θ θ = (θ, θ) as in the third line above is called a pair of complementary factor congruences. Fact If (θ, θ) is a CFC-pair, then the map is a homomorphism from A A into A (x, y) The unique z such that xθzθy An algebra A is directly indecomposable if it does not admit any non trivial CFC-pair. simple if Con(A) = {, }.

6 Central elements Definition An element e of a double-pointed algebra is central if (θ(1, e), θ(0, e)) is a CFC-pair. Within Church Algebras The BA of central elements The BA of CFC-pairs e Ce(A) (θ(1, e), θ(0, e)) (θ, θ) The unique e such that 1θeθ0 BA-operations on central elements x y = q(x, y, 0); x y = q(x, 1, y); x = q(x, 0, 1) Equational Characterization of the central element e 1. q(e, x, x) = x 2. q(e, q(e, x, y), z) = q(e, x, z) = q(e, x, q(e, y, z)) 3. q(e, f (x), f (y)) = f (q(e, x 1, y 1 ),..., q(e, x n, y n)) (for every f in the type) 4. q(e, 1, 0) = e

7 Non-trivial central elements in λ-models The λ-term Ω = (λx.xx)(λx.xx) is an easy term : it can be consistently equated to any other closed λ-term. Fact In the term model of the λ-theory θ(1, Ω) θ(0, Ω), where 1 = λxy.x, 0 = λxy.y, Ω is a non-trivial central element (hence the term model is decomposable). Lemma Indecomposable lambda-algebras are closed by subalgebras. Corollary Any model of the λ-theory θ(1, Ω) θ(0, Ω) is decomposable (since if M is a model of the λ-calculus then the term model of Th(M) is a subalgebra of M).

8 The Scott semantics is incomplete Scott λ-algebras are simple, hence indecomposable ( Scott is always simple ) Sketch of the proof : Let φ be a congruence on a Scott λ-algebra A. Suppose aφb with a b. { c if x b Given c A, g c(x) = is continuous. otherwise Let d A representing g c (for all x, dx = g c(x)). Hence c = da φ db =, and φ = Theorem The Scott semantics of the λ-calculus is incomplete. (no Scott model may have θ(1, Ω) θ(0, Ω) as theory)

9 Outline 1 Church algebras and the λ-calculus 2 n-dimensional Boolean algebras 3 n-dimensional propositional calculus 4 Rewriting n-pc formulas

10 Algebras of dimension n An algebra A is an algebra of dimension n (n-a) if it admits a (n + 1)-ary operator q(x, y 1,..., y n) and n constants e 1,..., e n such that for all 1 i n the following identity is satisfied : q(e i, y 1,... y n) = y i Examples : Lambda abstraction algebras : e i = λx 1... x n.x i q(x, y 1,..., y n) = xy 1... y n. n-sets on a set U : {(X 1,..., X n) : X i U} ; e i = (,...,, U,,..., ), }{{} (i 1) q(x, Y 1,..., Y n ) = ( n j=1 X j Y j 1,..., n j=1 X j Y j n).

11 n-central elements If A is an n-da, then c A is called n-central if the sequence (θ(c, e 1 ),..., θ(c, e n)) is an n-tuple of complementary factor congruences of A, meaning that : i n θ(c, e i ) = ;. for all a 1,..., a n A, q(c, a 1,..., a n) is the unique element such that a i θ(c, e i ) q(c, a 1,..., a n) for all 1 i n ; If c is n-central, then A = A/θ(c, e 1 )... A/θ(c, e n). Equational characterisation c A is n-central iff : B1 q(c, e 1,..., e n) = c. B2 q(c, x, x,..., x) = x for every x A. B3 If σ ν has arity k, then q(c, σ(x 11,..., x 1k ),..., σ(x n1,..., x nk )) = σ(q(c, x 11,..., x n1 ),..., q(c, x 1k,..., x nk )).

12 n-bas Proposition The set of n-central elements of an n-da A is a subalgebra of the pure reduct of A (i.e. : if x, y 1,..., y n are n-central, then q(x, y 1,..., y n) is n-central.) A Boolean algebra of dimension n (n-ba) is an n-da all of whose elements are n-central. Examples : The pure n-ba ({e 1,..., e n}, q, e 1,..., e n) of generalized truth values. n-partitions of set U : {(X 1,..., X n) : X i U, i X i = U, X i X j = } ;

13 BAs and n-bas A first representation theorem Any pure n-ba is isomorphic to a subalgebra of the n-ba of the n-partitions of a set X. The 2-elements Boolean algebra is primal : all finite Boolean functions are definable by AND, OR, NOT. This property is inherited by n-bas. Primality in n-bas Let A be a finite τ-algebra of cardinality n. Then A is primal if and only if it is an n-ba.

14 A second representation theorem The inner Boolean algebra of an n-ba Let A be an n-ba. The set B A = {x A : q(x, y, y, e 3,..., e n) = y} with the operations : x y = q(x, e 1, y, e 3,..., e n); x y = q(x, y, e 2, e 3,..., e n); is a Boolean algebra, written B A. x = q(x, e 2, e 1, e 3..., e n); 0 = e 1 ; 1 = e 2, Intuition : the elements of B A correspond to the n-partitions of the form (X, U \ X,,..., ), for X U. Theorem Any n-ba A is isomorphic to the n-ba of n-central elements of the Boolean vector space B A n.

15 Outline 1 Church algebras and the λ-calculus 2 n-dimensional Boolean algebras 3 n-dimensional propositional calculus 4 Rewriting n-pc formulas

16 n-dimensional propositional calculus Formulas : Φ, Ψ := x Var e i n = {e 1,..., e n} q(φ, Ψ 1,..., Ψ n) Interpretation : given ρ : Var n [x] ρ = ρ(x) [e i ] ρ = e i [q(φ, Ψ 1,..., Ψ n)] ρ = [Ψ j ] ρ where [Φ ρ] = e j (i-)validity : Γ = (n,ei ) if for all ρ : Var n : if for all Φ Γ [Phi] ρ = e i then there exists Ψ such that [Ψ] ρ = e i Simmetry of truth values Γ = (n,ei ) iff Γ (i,j) = (n,ej ) (i,j) e n is chosen as designated value.

17 Encoding of tabular logics into n-pc Theorem All tabular logic = (V,1) may be faithfully encoded into n-pc. Example : the 3-valued Post logic V = {0, 1, 1}, C = {,, } 2 x y = min(x, y), x y = max(x, y) 0 = 1, 1 2 = 0, = 1, 1 = 1 2 Encoding into 3-PC : 0 = e 1, 1 2 = e2, 1 = e 3. (x y) = q(x, e 1, q(y, e 1, e 2, e 2 ), y ) (x y) = q(x, y, q(y, e 2, e 2, e 3 ), e 3 ) ( x) = q(x, e 3, e 1, e 2 ). Fact = (V,1) Φ = ({e1,e 2,e 3 },e 3 ) Φ

18 Outline 1 Church algebras and the λ-calculus 2 n-dimensional Boolean algebras 3 n-dimensional propositional calculus 4 Rewriting n-pc formulas

19 Rewriting n-pc formulas (h 0 ) q(e i, x 1,..., x n) hnf x i (h 1 ) q(q(x, y 1..., y n), z 1,..., z n) hnf q(x, q(y 1, z 1,..., z n),..., q(y n, z 1,..., z n)). (r 1 ) q(x, y,..., y) y ; (r 2 ) q(x, e 1,..., e n) x ; (r i 3 ) q(x, y 1,..., y i 1, x, y i+1,..., y n) q(x, y 1,..., y i 1, e i, y i+1,..., y n) ; (r i 4 ) q(x, y 1,..., y i 1, q(x, z 1,..., z n), y i+1,..., y n) q(x, y 1,..., y i 1, z i, y i+1,..., y n) ; (r i 5 ) If x < x then q(x, y 1,..., y i 1, q(x, z 1,..., z n), y i+1,..., y n) q(x, q(x, y 1,..., y i 1, z 1, y i+1,..., y n),..., q(x, y 1,..., y i 1, z n, y i+1,..., y n)) (r i 6 ) If x < x then q(x, y 1,..., y i 1, x, y i+1,..., y n) q(x, q(x, y 1,..., y i 1, e 1, y i+1,..., y n),..., q(x, y 1,..., y i 1, e n, y i+1,..., y n))

20 Properties of the rewriting system The rewriting system is terminating and confluent. For any formula Φ of any given tabular logic = (V,1) : = (V,1) Φ = (n,en) Φ Φ e n

21 Rewriting binary decision diagrams PC 2-PC BDD x 1 x x 3 φ = x 1 (x 2 x 3 ) φ = q(x 1, q(x 2, 0, x 3 ), 1) 0 1 In the binary case, the rewriting system of head-normal 2-PC formulas is a reformulation of Zantema and van de Pol rewriting system for binary decision diagrams (2001).

22 Rewriting multi-valued decision diagrams 3-PC MDD x 1 x 2 x 2 x 2 φ = q(x 1, q(x 2, 0, 1, 1), q(x 2, 1, 0, 1), q(x 2, 1, 1, 0))

23 Ongoing and further work The algebraic theory of n-bas : Skew n-bas, multi-ideals,... Proof theory of n-pc. Rewriting of multi-valued decision diagrams.

24 Thanks

25 Axioms and rules for the n-pc Γ (i,j) i (i,j) (Sym) Γ j (Const) i e i π 1 (i) = ρ 1 (i) X π i X ρ Γ (i,j) i F, (i,j) i k Γ, F (j,k) (Neg1) j (Id) Γ (i,j), F i (i,j) i j Γ j F, (Neg2) n j=1 Γ (j,i), F, G (j,i) j j (j,i) Γ, q(f, G 1,..., G n) i (ql) Γ, F i Γ i F, (Cut) Γ i n j=1 Γ (j,i), F j G (j,i), (j,i) j Γ i q(f, G 1,..., G n), (qr) Γ Γ Γ i Γ i (Weak)