# Interpolation in \Lukasiewicz logic and amalgamation of MV-algebras Daniele Mundici Dept. of Mathematics “Ulisse Dini” University of Florence, Florence,

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Interpolation in \Lukasiewicz logic and amalgamation of MV-algebras Daniele Mundici Dept. of Mathematics “Ulisse Dini” University of Florence, Florence, Italy mundici@math.unifi.it

2-simplex 0-simplex 1-simplex 3-simplex we all know what a simplex in R n is

polyhedron P= finite union of simplexes S i in R n P need not be convex, nor connected a polyhedron P = U S i is said to be rational if so are the vertices of every simplex S i

our main themes: rational polyhedra and \Lukasiewicz logic Chapter 1: Local Deduction as a main ingredient of interpolation and amalgamation

\Lukasiewicz logic L ∞ FORMULAS are exactly the same as in boolean logic any VALUATION V evaluates formulas into the real unit interval [0,1] via the inductive rules: V(¬F) = 1–V(F) V(F —> G) = min(1, 1–V(F)+V(G)) Therefore, every valuation V is uniquely determined by its values on the variables: V(X 1 ),...,V(X n ) CONSEQUENCE RELATION: F |– G means that every valuation satisfying F also satisfies G

formulas yield functions f:[0,1] n —>[0,1] as boolean formulas yield f:{0,1} n —>{0,1} every formula F(X 1,...,X n ) determines a map f F : [0,1] n —>[0,1] by f Xi = the ith coordinate map f ¬F = 1 – f F f F —> G = min(1, 1 – f F + f G )

definable functions of one variable the ONESET f F -1 (1) of f F is the set of valuations satisfying the formula F oneset(f F )=zeroset(¬f F ) for each formula F, its associated function f F is continuous, linear, and each linear piece has integer coefficients (for short, f F is a McNaughton function)

oneset of f F = Mod(F) by induction on the number of connectives in F, the oneset of f F is a rational polyhedron, and so is the oneset of f ¬F and of f F —> G EACH ZEROSET AND EACH ONESET IS A RATIONAL POLYHEDRON IN [0,1] n

(Local) Deduction Theorem Theorem. For any two formulas A and B, the following conditions are equivalent: 1. Every valuation satisfying A also satisfies B 2. For some m=1,2,... the formula A—>(A—>(A—>... —>(A—>(A—>B))... )) is a tautology 3. B is obtained from A and the tautologies via Modus Ponens PROOF. 2—>3 easy; 3—>1 induction; 1—>2 is proved geometrically

assume oneset(f A ) contained in oneset(f B ) 1 1 let T be a triangulation of [0,1] such that the functions f A and f B both formulas A and B are linear over each interval of T fAfA fBfB

f A & f A < f A 1 1 f A&A fBfB applying \Lukasiewicz conjunction to A, from the formula A&A we get obtain a minorant f A&A of f A, still with the same one set of f A Recall definition P&Q = ¬(P —> ¬Q)

f A & f A & f A < f A & f A < f A 1 1 fkAfkA fBfB by iterated application of the \Lukasiewicz conjunction we obtain a function f k A = f A &f A &...&f A with the same oneset of f A, and with the additional property that f A k ≤ f B

for large k this will hold at every simplex of T 1 1 fAfA fBfB in other words, we have the tautology A k —>B, which is the same as the desired tautology A—>(A—>(A—>... —>(A—>(A—>B))... ))

Chapter 2: Interpolation (as a main tool to amalgamation)

interpolation/amalgamation Craig interpolation theorem fails in \Lukasiewicz logic, because the tautology x  ¬x—>y  ¬y has no interpolant deductive interpolation is like Craig interpolation, with the |– symbol in place of the implication connective (more soon) over the last 25 years, several proofs have been given of deductive interpolation for \Lukasiewicz infinite-valued propositional logic deductive interpolation, together with local deduction, is a main tool to prove the amalgamation theorem for the algebras of \Lukasiewicz infinite-valued logic

amalgamation: many proofs the first proof of amalgamation used the categorical equivalence between MV-algebras and unital lattice-ordered groups (relying on Pierce's amalgamation theorem). in the early eighties I heard from Andrzej Wro\’nski during one of his visits to Florence, that the Krakow group had a proof of the amalgamation property for MV-algebras without negation (i.e., Komori’s C algebras) recent proofs, like the proof by Kihara and Ono, follow by applying to MV-algebras results in universal algebra I will present a simple geometric proof of the amalgamation theorem, using Deductive Interpolation

background literature F. Montagna, Interpolation and Beth's property in propositional many- valued logics: A semantic investigation, Annals of Pure and Applied Logic, 141: 148-179, 2006. This is based on: N.Galatos, H. Ono, Algebraization, parametrized local deduction theorem and interpolation for substructural logics over FL, Studia Logica, 83:279-308, 2006. For the proof of Theorem 5.8, the following is needed: A. Wro\'nski, On a form of equational interpolation property, In: Foundations of Logic and Linguistic, G.Dorn, P. Weingartner, (Eds.), Salzburg, June 19, 1984, Plenum, NY, 1985, 23-29. For the proof of Theorem I on page 25, the following is needed: P.D. Bacisch, Amalgamation properties and interpolation theorems for equational theories, Algebra Universalis, 5:45-55, 1975.

(Deductive) Interpolation If F |– G then there is a formula J such that F |– J, J |– G, and each variable of J is a variable of both F and G our proof will be entirely geometrical

rational polyhedra are preserved under projection the projection of a (rational) polyhedron onto a (rational) hyperplane is a (rational) polyhedron we record this fact as the PROJECTION LEMMA

rational polyhedra are preserved under perpendicular cylindrification we record this fact as the CYLINDRIFICATION LEMMA

oneset of f F = Mod(F) recall: THE ZEROSET (AND THE ONESET) OF ANY \LUKASIEWICZ FORMULA IS A RATIONAL POLYHEDRON IN [0,1] n we now prove the converse: EACH RATIONAL POLYHEDRON IN [0,1] n IS THE ZEROSET OF SOME \LUKASIEWICZ FORMULA

rational half-spaces in [0,1] n a rational line L in [0,1] 2 H is one of the half-planes bounded by L in the square [0,1] 2 PROBLEM: Does there exist a formula F such that the zeroset of f F coincides with H ? mx+ny+p=0, with m,n,p integers, m>0 H ANSWER: Yes, by induction on |m|+|n|

then every rational polyhedron is a zeroset this blue half-space is a zeroset then so is this rational triangle (formulas can express intersections) and this rational polyhedron (formulas can express unions) ANY RATIONAL POLYHEDRON IN [0,1] n IS THE ZEROSET OF SOME f F

this was known to McNaughton (1951) FOLKLORE LEMMA Rational polyhedra contained in the n-cube [0,1] n coincide with zerosets (and also coincide with onesets) of definable maps, i.e., functions of the form f F where F ranges over formulas in n variables we record the FOLKLORE LEMMA by writing: RATIONAL POLYHEDRA=ONESETS=MODELSETS

Deductive interpolation PROOF. We may write var(F) = X u Z var(G) = Y u Z, for X,Y,Z pairwise disjoint sets of variables Mod(F) = f F -1 (1) = P, which by the Folklore Lemma is a rational polyhedron in [0,1] XuZ by the Projection Lemma, the projection of P onto R Z is a rational polyhedron Q contained in [0,1] Z

Mod(G) = f G -1 (1) = R, a rational polyhedron in [0,1] YuZ Mod(F) = P X Y Z Mod(G)=R the hypothesis F |— G states that, in the space R XuYuZ Mod(F) is contained in Mod(G) Q

Q Mod(F) = P We then obtain the first half of interpolation: F |— J X Y Z by the Folklore Lemma, there is a formula J(Z) such that Q=Mod(J) regarding J as a formula in the variables X,Z, then Mod(J) is this blue rectangle! = Mod(J)

Mod(F) = P X Y Z Mod(G)=R in the space R YuZ, Mod(J) is contained in Mod(G) Q=Mod(J) regarding J as a formula in Y,Z, then Mod(J) is this blue rectangle! We then obtain the second half of interpolation: J |— G

Chapter 3: Amalgamation of the algebras of \Lukasiewicz logic, i.e., Chang MV-algebras

MV-algebras (in Wajsberg’s version) directly from \Lukasiewicz axioms A—>(B—>A) (A—>B)—>((B—>C)—>(A—>C)) ((A—>B)—>B)—> ((B—>A)—>A) (¬A—>¬B)—>(B—>A)

the amalgamation property Z A B we have

the usual setup Z A B D we want henceforth, all blue maps are one-one we have

the embedding of Z into A let us focus attention on the embedding of Z into A without loss of generality, Z is a subalgebra of A thus the set A is the disjoint union of Z and some set X, A=Z U X Z A

extending maps to homomorphisms the identity map z—>z uniquely extends to a homomorphism s Z of the free MV-algebra FREE Z onto Z similarly, the identity map a—> a uniquely extends to a homomorphism s A of FREE A onto A let ker s Z and ker s A denote the kernels of these maps Z FREE Z sZsZ

Z A FREE XUZ ker(s A ) ker(s Z ) sAsA sZsZ all blue arrows are inclusions all red arrows are surjections intuitively, this trivial Largeness Lemma states that ker(s Z ) is as large as possible in ker(s A ).

Z AB

Z AB FREE Z FREE XUZ FREE YUZ ker(s A ) ker(s B ) ker(s Z ) sAsA szsz sAsA

Z AB FREE Z FREE XUZ FREE YUZ ker(s A ) ker(s B ) ker(s Z ) sAsA szsz I = the ideal generated by ker(s A ) U ker(s B ) FREE XUYUZ sAsA

Z AB FREE Z FREE XUZ FREE YUZ ker(s A ) ker(s B ) ker(s Z ) sAsA szsz I = the ideal generated by ker(s A ) U ker(s B ) FREE XUYUZ sAsA D

Z AB FREE Z FREE XUZ FREE YUZ ker(s A ) ker(s B ) ker(s Z ) sAsA szsz i = the ideal generated by ker(s A ) U ker(s B ) FREE XUYUZ sAsA D µ µ(x/ ker(s A )) = x/i there remains to be proved that µ is one-one

e/i = 0 means that e is an element of i. In other words, (theories ~ ideals) a, b |– e for some a in ker(s A ) and b in ker(s B ) Let e be an element of FREE XUYUZ such that e/i =0. We must prove e/ker(s A ) = 0

end of the proof of amalgamation

Chapter 4: Further geometric developments on projective MV- algebras

why should we insist in giving many proofs of MV-amalgamation and interpolation? because MV-algebras provide a benchmark for other structures of interest in algebraic logic because interpolation and amalgamation are deeply related to many fundamental logical-algebraic-geometric notions: quantifier elimination, cut elimination, joint consistency, joint embedding, unification, projectives,... let us briefly review what is known about finitely generated projective MV-algebras, i.e., retracts of FREE n for some n this is joint work with Leonardo Cabrer, to appear in Communications in Contemporary Math., and based on earlier joint work on Algebra Universalis 62 (2009) 63–74.

projectives are routinely characterized by duality Every n-generated projective MV-algebra A is finitely presented (essentially, Baker) A is finitely presented iff A=M(P) for some polyhedron P lying in some n-cube [0,1] n (Baker-Beynon duality) DEFINITION P is said to be a Z-retract if the MV- algebra M(P) is projective Problem: characterize Z-retracts, among all polyhedra

this property is not easy to handle; thus, we must find equivalent conditions for a polyhedron P to be retract of [0,1] n a first property of Z-retracts: they are retracts of some cube [0,1] n

The elements of the fundamental group π 1 (P) (introduced by Poincaré) of a connected polyhedron P are the equivalence classes of the set of all paths with initial and final points at a given basepoint p, under the equivalence relation of homotopy. The fundamental groups of homeomorphic spaces are isomorphic. to check if P is a retract it suffices to check that all homotopy groups of P are trivial

equivalents for P to be a retract of [0,1] n THEOREM. For any polyhedron P in [0,1] n the following conditions are equivalent: (a) P is a retract of [0,1] n (b) P is connected and all homotopy groups π i (P) are trivial (c) P is contractible (can be continuously shrunk to a point). Proof. (a)—>(b) by the functorial properties of the homotopy groups π i. The implications (b)—>(a) and (b)— >(c) follow from Whitehead theorem in algebraic topology. (c)—>(b) is trivial. QED

M(P) is not projective P is not a Z- retract, because it is not simply connected let P be this polyhedron

M(P) is not projective a second property of Z-retracts: P must contain a vertex of [0,1] n P is not a Z-retract, because it does not contain any vertex of the unit square

PROPOSITION If P is a Z-retract, then P has a triangulation Ω such that the affine hull of every maximal simplex in Ω contains some integer point of R n a third property: strong regularity M(P) is not projective this P is not a Z- retract: for, the affine hull of the vertical red segment does not contain any integer point 0 1 1

projectiveness: 3 necessary conditions THEOREM ( L.Cabrer, D.M., 2009 ) If A is a finitely generated projective MV-algebra, then up to isomorphism, A=M(P) for some rational polyhedron lying in [0,1] n such that (i) P contains some vertex of [0,1] n, (ii) P is contractible, and (iii) P is strongly regular. are these three conditions also sufficient for an MV-algebra A to be finitely generated projective ?

yes, when the maximal spectrum is one- dimensional THEOREM ( L.Cabrer, D.M. ) Suppose the maximal spectrum of A is one-dimensional. Then A is n- generated projective if and only if A is isomorphic to M(P) for some contractible strongly regular rational polyhedron in [0,1] n containing a vertex of [0,1] n. It is not known if these three conditions are sufficient in general. They become sufficient if contractibility is strengthened to collapsibility

a sequence of collapses

a sufficient condition for P to be a Z-retract, i.e., for M(P) to be projective THEOREM ( L.Cabrer, D.M., Communications in Contemporary Mathematics ) If P has a collapsible strongly regular triangulation containing a vertex of [0,1] n then M(P) is projective.

A is finitely presented homomorphism isomorphism indecomposable A is free n-generated A is n-generated dim(maxspec(A))=d A=M(P) is projective A=M(P), P a polyhedron Z-map Z-homeomorphism P is connected P=unit cube [0,1] n P lies in [0,1] n dim(P)=d P is a Z-retract algebrageometry

\Lukasiewicz logic and MV-algebras together are a rich source of geometric inspiration thank you

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