BL-Functions and Free BL-Algebra

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BL-Functions and Free BL-Algebra Simone Bova bova@unisi.it www.mat.unisi.it/ bova Department of Mathematics and Computer Science University of Siena (Italy) December 9, 008 Ph.D. Thesis Defense

Outline Motivation Basic Logic Truthfunctions Background Generic BL-Algebra Free BL-Algebra over One Generator Contribution Free Finitely Generated BL-Algebra Conclusion Summary Publications

Outline Motivation Basic Logic Truthfunctions Background Contribution Conclusion

Fuzzy Logics Fuzzy logics are propositional logics over,, st: variables X, X,... are interpreted over [0, ]; = and are interpreted over and 0; and are interpreted over binary functions on [0, ]; =. Fuzzy conjunction and implication must maintain: the behavior of Boolean counterparts over {0, } ; intuitive properties of Boolean counterparts over [0, ] ; validity (and power) of modus ponens.

Continuous Triangular Norms Definition (Continuous Triangular Norm, Residuum) A (continuous) triangular norm is a continuous binary function on [0, ] that is associative, commutative, monotone (x y implies x z y z) and has as unit (x = x). Given a continuous triangular norm, its residuum is the binary function on [0, ] defined by x y = max{z : x z y}. Triangular norms and their residua provide suitable interpretations for fuzzy conjunction and implication.

Triangular Tautologies t, s, r,... terms over,, and variables X, X,.... Definition (Triangular Tautology) t is a triangular tautology iff t evaluates identically to for every assignment of the variables in [0, ] and every interpretation of over a triangular norm,, and of over the corresponding residuum,.

Hájek s Basic Logic Hájek s Basic logic: (BL) (r s) ((s t) (r t)) (BL) (r s) r (BL) (r (r s)) (s (s r)) (BL4) (r (s t)) ((r s) t) (BL5) ((r s) t) (r (s t)) (BL6) ((r s) t) (((s r) t) t) (BL7) r (MP) r, r s BL s Theorem (Cignoli et al., 000) BL t iff t is a triangular tautology.

BL-Algebras x y = x (x y) aka divisibility, x y = ((x y) y) ((y x) x), x = x, =. Definition (BL-Algebras) A BL-algebra is an algebra (A,,, ) of type (,, 0) st: (i) (A,, ) is a commutative monoid; (ii) (A,,,, ) is a bounded lattice; (iii) residuation holds, ie x y z iff y x z ; (iv) prelinearity holds, ie (x y) (y x) =.

Free BL-Algebra and Basic Logic Truthfunctions BL-algebras form the equivalent algebraic semantics of Basic logic. The free n-generated BL-algebra is isomorphic to the Lindenbaum-Tarski algebra of the n-variate fragment of Basic logic. An explicit description of the free n-generated BL-algebra is equivalent to an explicit description of the truthfunctions of the n-variate fragment of Basic logic.

Outline Motivation Background Generic BL-Algebra Free BL-Algebra over One Generator Contribution Conclusion

Generic BL-Algebra Definition (Generic BL-Algebra) (n + )[0, ] = ([0, n + ], (n+)[0,], (n+)[0,], (n+)[0,] ) is the algebra of type (,, 0) such that (n+)[0,] = 0, and for all a, a [0, n + ], { a (n+)[0,] min{a, a } if a a a = (a j [0,] WH a j) + j if a = a = j n + if a a a (n+)[0,] a = a if a < a (a j [0,] WH a j) + j if a = a = j where a [0,] WH a = max{0, a + a }, and a [0,] WH a = min{, a + a }.

Generic BL-Algebra Case n = 0 0 a_ 0 0 a_ a_ 0 a_ 0 (a) a [0,] a. (b) a [0,] a. Figure: [0, ] = ([0, ], [0,], [0,] ), [0,] = 0).

Generic BL-Algebra Theorem (Aglianó and Montagna, 00) (n + )[0, ] generates the variety generated by all the n-generated BL-algebras. Let F n be the smallest set of n-ary functions from [0, n + ] [n] to [0, n + ] that contains the n-ary projection functions x,..., x n, the n-ary constant function 0, and is closed under pointwise application of (n+)[0,] and (n+)[0,]. Corollary The free n-generated BL-algebra is isomorphic to the set F n, equipped with the operations of (n + )[0, ] defined pointwise.

McNaughton Functions Definition (McNaughton Function) n positive integer, I [n] = {,..., n}. A continuous I -ary function f : [0, ] I [0, ] is a McNaughton function iff there are I -ary linear polynomials with integer coefficients p,..., p k : R I R st, for every a in [0, ] I, f (a) = p j (a) for some j [k]. f is said positive if f ({} I ) =, and negative ow. 5 6 x_ x_ x_ Figure: Unary McNaughton functions from [0, ] {} [0, ].

Free BL-Algebra over One Generator f : [0, ] {} [0, ] F have two forms. 5 6 x_ x_ 5 6 5 x_ (a) (b) (c) Figure: Form : (a), (b) positive McNaughton functions; (c) in F.

Free BL-Algebra over One Generator f : [0, ] {} [0, ] F have two forms. x_ x_ (a) (b) Figure: Form : (a) negative McNaughton function; (b) in F.

Free BL-Algebra over One Generator f : [0, ] {} [0, ] F have two forms. 5 6 5 x_ x_ (a) Form. (b) Form. Figure: F.

References P. Aglianó and F. Montagna. Varieties of BL-Algebras I: General Properties. Journal of Pure and Applied Algebra, 8:05 9, 00. S. Aguzzoli and B. Gerla. Normal Forms for the One-Variable Fragment of Hájek s Basic Logic. Proceedings of ISMVL 05, 84 89, 005. M. Baaz, P. Hájek, F. Montagna, and H. Veith. Complexity of t-tautologies. Annals of Pure and Applied Logic, :, 00. R. Cignoli, F. Esteva, L. Godo, and A. Torrens. Basic Fuzzy Logic is the Logic of Continuous t-norms and their Residua. Soft Computing, 4():06, 000. F. Montagna The Free BL-Algebra on One Generator. Neural Network World, 5:87 844, 000.

Outline Motivation Background Contribution Free Finitely Generated BL-Algebra Conclusion

BL n -Tree n positive integer, [n] = {,..., n}. Definition (Fubini Partition) A Fubini partition of [n] is an ordered partition of [n] into nonempty subsets. Definition (BL n -Tree) A BL n -tree is a rooted tree T = (V, E) st V is a finite multiset of Fubini partitions of [n], V contains exactly one copy of ([n]), ([n]) is the root of the tree, and (u, v) E only if the last block of u is equal to the union of the last two blocks of v.

BL n -Tree Case n =,, (a) T = (V, E). (b) T = (V, E ). Figure: BL -trees.

Labeled BL n -Tree Definition (Rational Polyhedron) A k-dimensional (rational) simplex S in [0, ] I (k I ) is the convex hull of a set of k + affinely independent (rational) vertices v,..., v k+ in [0, ] I. A (rational) simplicial complex C in [0, ] I is a collection of (rational) simplexes in [0, ] I st every face of a simplex of C is in C, and the intersection of any two simplexes of C is a face of each of them. A (rational) polyhedron in [0, ] I is the underlying space of a (rational) simplicial complex in [0, ] I, ie, the union of the (rational) simplexes in a simplicial complex in [0, ] I. No hyperplane of dimension m contains more than m + of them.

Labeled BL n -Tree Definition (Labeled BL n -Tree) A labeled BL n -tree, l(t), is a BL n -tree T = (V, E) with V E labeled as follows. Let B < B = v V with successors {u j = C < C } j [k], so B = C C. (i) l(v) McNaughton function over [0, ] B, positive if B [n]. (ii) l(v, u ),..., l(v, u k ) are st: (ii.i) l(v, u j ) is the relint of a rational polyhedron in [0, ] C. (ii.ii) {l(v, u j ) {} C j [k]} partitions the -set of l(v) in: [0, ] B minus the union of [0, ) B and {} B. l(t) is positive if l([n]) is positive (negative, ow). Display ([n]) by (, [n]). Relative interior.

Labeled BL n -Tree Case n = Sampling a labeled BL -tree, l(t). g g, f g (a) T = (V, E). (b) l(v). Figure: l sends V to McNaughton functions f, g, g, g.

Labeled BL n -Tree Case n = g g 0 0 a_ f g (a) l({, }) = f. a_ 0 (b) f : [0, ] {,} [0, ]. Figure: f is positive, thus l(t) is positive.

Labeled BL n -Tree Case n = g g f g (a) l({}, {}) = g. (b) g : [0, ] {} [0, ]. a_ Figure: g is positive.

Labeled BL n -Tree Case n = g g f g (a) l({}, {}) = g. (b) g : [0, ] {} [0, ]. a_ Figure: g is positive.

Labeled BL n -Tree Case n = g g f g (a) l({}, {}) = g. (b) g : [0, ] {} [0, ]. a_ Figure: g is positive.

Labeled BL n -Tree Case n = g g P P, f P g (a) T = (V, E). (b) l(e). Figure: l bijects btw E and a suitable set of rational polyhedra relints.

Labeled BL n -Tree Case n = a_ g g P P f P g (a) P [0, ] {}. a_ (b) P {} {}. Figure: P = {(a ) a = 0}. f P {} {} =.

Labeled BL n -Tree Case n = a_ g g P P f P g (a) P [0, ] {}. a_ (b) P {} {}. Figure: P = {(a ) 0 < a < }. f P {} {} =.

Labeled BL n -Tree Case n = a_ g g P P f P g (a) P [0, ] {}. a_ (b) P {} {}. Figure: P = {(a ) a = 0}. f P {} {} =.

Labeled BL n -Tree Case n = Suitability of {P, P, P }: (i) = {a = (a, a ) f (a) = } [0, ] {,} \ ( [0, ) {,} {} {,}) ; (ii) P {} {}, P {} {}, P {} {} form a partition of. a_ 0 0 a_ a_ 0 (a) f : [0, ] {,} [0, ]. a_ (b) Part of f s -set.

BL n -Encoding Definition (BL n -Encoding) A BL n -encoding is a pair (l(t), l (T )) of of labeled BL n -trees st l (T ) is positive if l(t) is positive.

BL n -Encoding Case n = Sampling a BL -encoding, (l(t), l (T )). g g, g g f, f g (a) T = (V, E), T = (V, E ). (b) l(v), l (V ). Figure: l(v) and l (V ) range over McNaughton functions.

BL n -Encoding Case n = g g g g f 0 0 a_ f g a_ 0 Figure: l(v) specification, f : [0, ] {,} [0, ].

BL n -Encoding Case n = g g g g f f g a_ Figure: l(v) specification, g : [0, ] {} [0, ].

BL n -Encoding Case n = g g g g f f g a_ Figure: l(v) specification, g : [0, ] {} [0, ].

BL n -Encoding Case n = g g g g f f g a_ Figure: l(v) specification, g : [0, ] {} [0, ].

BL n -Encoding Case n = g g g g f 0 0 a_ f g a_ 0 Figure: l (V ) specification, f : [0, ] {,} [0, ].

BL n -Encoding Case n = g g g g f f g a_ Figure: l (V ) specification, g : [0, ]{} [0, ].

BL n -Encoding Case n = g g g g f f g a_ Figure: l (V ) specification, g : [0, ]{} [0, ].

BL n -Encoding Case n = g P g P, g g f P P, f P g (a) T = (V, E), T = (V, E ). (b) l(e), l (E ). Figure: l(e) and l (E ) range over suitable rational polyhedra relints.

BL n -Encoding Case n = a_ g P g P g g f P P f P g a_ Figure: l(e) specification, P = {(a ) a = 0} [0, ] {}.

BL n -Encoding Case n = a_ g P g P g g f P P f P g a_ Figure: l(e) specification, P = {(a ) 0 < a < } [0, ] {}.

BL n -Encoding Case n = a_ g P g P g g f P P f P g a_ Figure: l(e) specification, P = {(a ) a = 0} [0, ] {}.

BL n -Encoding Case n = a_ 0 0 a_ a_ (a) f. 0 a_ (b) Part of f s -set. Figure: Suitability of {P, P, P }.

BL n -Encoding Case n = a_ g P g P g g f P P f P g a_ Figure: l(e) specification, P = {(a ) / < a < } [0, ] {}.

BL n -Encoding Case n = a_ g P g P g g f P P f P g a_ Figure: l(e) specification, P = {(a ) a = /} [0, ] {}.

BL n -Encoding Case n = a_ 0 0 a_ a_ (a) f. 0 a_ (b) Part of f s -set. Figure: Suitability of {P, P }.

REALIZE Algorithm REALIZE Input BL n -encoding (l(t), l (T )) and a [0, n + ] [n] Output b [0, n + ] REALIZE receives in input (l(t), l (T )) and a and:. uniquely determines a McNaughton function f in l(v) l (V ), which is responsible for a;. outputs a value b which is a function of f and a only. Proposition (Realization) For any BL n -encoding e, REALIZE(e, ) is a function, denoted by f e : [0, n + ] [n] [0, n + ].

REALIZE Pseudocode REALIZE((l(T), l (T )), a = (a,..., a n) [0, n + ] [n] ) if a i = 0 for some i [n] k(s) l(t) else 4 if l (T ) is negative 5 output 0 6 endif 7 k(s) l (T ) 8 endif 9 Q = B < < B m the Fubini partition corresponding to a (m ) 0 i c the root of S while i m U the successors of c that are copies of B < < B i < B i+ B m 4 a the point in [0, ] B i such that a j = a j a j for all j B i 5 if exists u U such that a k(c, u) 6 i i +, c u 7 else 8 a the point in [0, ] B i Bm such that a j = a j a j for all j B i B m 9 if k(c)(a ) < and a {n + } B i Bm 0 output k(c)(a ) + a j where j is any index in B i B m else output n + endif 4 endif 5 endwhile

REALIZE Case n = Sampling REALIZE on the BL -encoding ((l(t), l (T )). g g f g P g P 0 0 P P a_ a_ f P g (a) (l(t), l (T )) 0 (b) REALIZE((l(T), l (T )), ). Figure: REALIZE((l(T), l (T )), ): [0, ] {,} [0, ].

REALIZE Case n = a_ f a_ (a) f s responsibilities. 0 0 a_ (b) 0 a_ Figure: (b) REALIZE((l(T), l (T )), ) restricted to f s responsibilities.

REALIZE Case n = a_ a_ a_ g g 0 0 a_ (a) g s resp. a_ (b) g s resp. g a_ (c) g s resp. a_ 0 (d) a_ Figure: (d) REALIZE((l(T), l (T )), ) restricted to i [] g i s resp..

REALIZE Case n = a_ f a_ (a) f s responsibilities. 0 0 a_ (b) 0 a_ Figure: (b) REALIZE((l(T), l (T )), ) restricted to f s responsibilities.

REALIZE Case n = a_ a_ g g 0 0 a_ (a) g s resp. (b) g s resp. a_ a_ (c) 0 a_ Figure: (d) REALIZE((l(T), l (T )), ) restricted to i [] g i s resp..

BL n -Functions Definition (BL n -Function) A function g: [0, n + ] [n] [0, n + ] is an n-ary BL-function iff g = f e for some BL n -encoding e.

Functional Representation Theorem (Functional Representation) The set of n-ary BL-functions equipped with pointwise defined operations (n+)[0,] and (n+)[0,] is isomorphic to the free n-generated BL-algebra. t (n+)[0,] denotes the n-ary term operation of (n + )[0, ] corresponding to term t. (t, t ) iff t (n+)[0,] = t (n+)[0,] is an equivalence over terms, whose representatives are n-ary BL-functions.

Functional Representation Proof Lemma (Closure) The class of n-ary BL-functions contains all the n-ary term operations of (n + )[0, ]. Lemma (Normal Form) Let f be any n-ary BL-function. There is an algorithm T( ) that takes in input (an encoding of) f and returns in output a term T(f ) over (,, ) and {X i i [n]} st T(f ) (n+)[0,] = f. (e, e ) iff f e = f e is an equivalence over BL n -encodings, whose blocks have cardinality >, so canonicity lacks.

Outline Motivation Background Contribution Conclusion Summary Publications

Summary and Future Work We gave a concrete representation of the free BL-algebra over finitely many generators (ie, Basic logic truthfunctions) in terms of a suitable class of real functions. The problem was open, even for the case of two generators. This result encourages the development of further work on locally finite subvarieties of BL-algebras, tight countermodels to BL-quasiequations, and constructive amalgamation property.

Publications S. Aguzzoli and S. Bova. The Free n-generated BL-Algebra. Accepted to Annals of Pure and Applied Logic. S. Aguzzoli, S. Bova, and V. Marra. Applications of Finite Duality to Locally Finite Varieties of BL-Algebras. To appear in Proceedings of LFCS 09, 009. S. Bova. BL-Functions and Free BL-Algebra. Ph.D. Thesis, University of Siena, 008. S. Bova and F. Montagna. Proof Search in Hájek s Basic Logic. ACM Transactions on Computational Logic, 9(), 008. S. Bova and F. Montagna. The Consequence Relation in the Logic of Commutative GBL-Algebras is PSPACE-complete. Accepted to Theoretical Computer Science.

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