Deductive interpolation in Lukasiewicz logic and amalgamation of MV-algebras Daniele Mundici Dept. of Mathematics “Ulisse Dini” University of Florence, Florence, Italy
[email protected]
many proofs •
over the last 25 years, several proofs have been given of deductive interpolation for Lukasiewicz infinite-valued propositional logic, and amalgamation for MV-algebras
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the first proof of amalgamation used the categorical equivalence between MV-algebras and unital l-groups (relying on Pierce's amalgamation theorem for l-groups).
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next proof from Panti's analysis of prime l-group ideals
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among others, also Czelakowski, Galatos, Montagna, Pigozzi, Tsinakis have universal algebraic proofs...
other proofs, like the proof by Kihara and Ono, follow by applying to MV-algebras results in universal algebra
my main motivation •
F. Montagna, Interpolation and Beth's property in propositional manyvalued logics: A semantic investigation, Annals of Pure and Applied Logic, 141: 148-179, 2006. BASED ON:
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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 PAPER IS NEEDED:
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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 PAPER IS NEEDED:
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P.D. Bacisch, Amalgamation properties and interpolation theorems for equational theories, Algebra Universalis, 5:45-55, 1975.
the aim of this talk • • • • • • •
to give a proof of (deductive) interpolation for Lukasiewicz logic and get the amalgamation of MV-algebras as a corollary using no literature using no deep result of universal algebra without Chang completeness and McNaughton theorem hopefully, the shortest possible proof geared towards the nonspecialist in MV-algebras...
introducing rational polyhedra
we all know what a simplex in Rn is 0-simplex
1-simplex 2-simplex
3-simplex
polyhedron P= finite union of simplexes Si in Rn
P need not be convex, nor connected a polyhedron P = USi is said to be rational if so are the vertices of every simplex Si
rational polyhedra are preserved under projection F
C
D
E A
the projection of a (rational) polyhedron onto a (rational) hyperplane is a (rational) polyhedron
B
for instance, the projection of this cube onto our retina is the union of the four triangles ABD, BEC, DCF and BCD
we record this fact as the PROJECTION LEMMA
rational polyhedra are preserved under perpendicular cylindrification along I=[0,1]
I
P
rational polyhedra are preserved under perpendicular cylindrification along [0,1]
I
P x I
we record this fact as the CYLINDRIFICATION LEMMA
Lukasiewicz infinitevalued propositional logic in one slide
Lukasiewicz logic L∞ (original definition) • • • • •
FORMULAS: exactly the same as in boolean logic
•
CONSEQUENCE RELATION: F |– G means that the oneset fF-1(1) of fF is contained in the oneset of fG
VALUATIONS: V evaluates formulas into [0,1] via the rule V(¬F) = 1–V(F) and V(F —> G) = min(1, 1–V(F)+V(G)) V is uniquely determined by V(X1),...,V(Xn) any formula F(X1,...,Xn) gives a map fF : [0,1]n —>[0,1] by fXi=ith coordinate map, f¬F=1–fF and fF —> G=min(1, 1–fF+fG)
Repeated: the most important and peculiar properties of infinite-valued Lukasiewicz logic each valuation V can be identified with its restriction to the variables (V(X1),...,V(Xn))=(V1,...,Vn)=a point xV in [0,1]n
V
each formula F determines a unique function fF:[0,1]n—>[0,1] such that V(F) = fF(xV). Variables go to identity functions The remaining functions fF are obtained from the coordinate functions by pointwise negation and truncated sum
properties of the definable maps fF • •
fix n=1,2,..., and the variables X1,...,Xn
• • •
e.g., DEF1 consists of the following maps f:[0,1]–>[0,1]
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we are not interested in characterizing DEFn
a map f is definable if it has the form fF for some formula F(X1,...,Xn). Let DEFn denote the definables the identity x (corresponding to the variable X1) and any other map obtainable from x using the pointwise operations of negation and truncated sum
definable functions of one variable the oneset fF-1(1) of fF is the set of valuations satisfying the formula F once we know ONESET(fF) and ONESET(fG) we can check if F |– G
the ONESET of fF (one variable) By induction on the number of connectives in F, the ONESET, like the ZEROSET of fF , is a rational polyhedron contained in [0,1]
N.B. This is not McNaughton theorem—it is a trivial exercise
definable functions of two variables •
the zeroset of x is the segment x=1 in [0,1]2, a rational polyhedron
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the zeroset of y is the segment y=1 in [0,1]2, a rational polyhedron
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inductively, the zeroset of f¬F is a rational polyhedron, and so is we record this fact as: EACH ZEROSET AND EACH the zeroset of fF —> G
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oneset(f)=zeroset(¬f)
ONESET IS A RATIONAL POLYHEDRON IN [0,1]n
a short proof of the converse: Each rational polyhedron in [0,1]n is a zeroset (again, without McNaughton theorem)
half-spaces in a rational line in [0,1]2
mx+ny+p=0, with m,n,p integers, m>0
H H
H is one of its half-planes
n [0,1] QUESTION: Does there exist a formula F such that the zeroset of fF coincides with H ?
half-spaces are zerosets Let us define g(x,y)=mx+ny+p and gclip = max(0,min(1,g)). Observe that the zeroset of gclip equals H. It suffices to show that gclip is definable by some formula
mx+nz+p=0
H
one of its half-planes, H
gclip
half-spaces are zerosets By induction on |m|+|n|, there are definable functions d=(g-x)clip and e=(g-x-1)clip. Then, by direct inspection, gclip = max[0,d+min(e+x,1)-1] = (e or x) and d
mx+nz+p=0
whence this map is definable
using the connectives “not”, “and”, “or” we conclude
this half-space is definable
and this rational polyhedron
then so is this rational triangle
RATIONAL POLYHEDRA CONTAINED IN [0,1]n ARE ONESETS
summing up: FOLKLORE LEMMA (Since 1950 at least) 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 fF where F ranges over formulas in n variables
we record the FOLKLORE LEMMA by the slogan: RATIONAL POLYHEDRA=ONESETS=MODELSETS
COROLLARY OF THE FOLKLORE LEMMA The deductive interpolation theorem for Lukasiewicz logic
Deductive interpolation THEOREM If F |– G then there is a formula J such that F |– J, J |– G, and var(J) = var(F)« var(G) PROOF. We may write var(F)=XuZ var(G)=YuZ, for X,Y,Z pairwise disjoint sets of variables Mod(F) = fF-1(1) = P, which by the Folklore Lemma is a rational polyhedron in [0,1]XuZ Mod(G) = fG-1(1) = R, another rational polyhedron in [0,1]YuZ by the Projection Lemma, the projection of P onto RZ is a rational polyhedron Q contained in [0,1]Z
Q is the projection of Mod(F) onto the Z-axis Z Mod(G)=R
Q Mod(F) = P Y X
the hypothesis F |— G states that, in the space RXuYuZ Mod(F) is contained in Mod(G)
regarding J as a formula in the variables X,Z, then Mod(J) is this blue rectangle!
Z Q = Mod(J) Mod(F) = P Y
by the Folklore Lemma, there is a formula J(Z) such that Q=Mod(J)
X
We then obtain the first half of interpolation: F |— J
regarding J as a formula in Y,Z, then Mod(J) is this blue rectangle!
Z Q=Mod(J) Mod(F) = P
Mod(G)=R Y X
in the space RYuZ , Mod(J) is contained in Mod(G) We then obtain the second half of interpolation: J |— G
a retrospective • we have given an elementary geometric proof of deductive interpolation in Lukasiewicz logic
• only using the Folklore Lemma stating that rational polyhedra = onesets of definable maps in Lukasiewicz logic
• no use has been made of McNaughton theorem, and of Chang completeness theorem
• because deductive interpolation is a very elementary fact of Lukasiewicz logic
a disclaimer • now we will prove the amalgamation property as a corollary of deductive interpolation
• we will get it as an elementary consequence of the special geometric features of MV-algebras
• I expect that an even shorter proof springs out from this outstanding audience of experts
• however, have a look at my proof:
MV-algebras have the amalgamation property
the usual setup
Z A
we have B
the usual setup
Z A
we have B we want
D henceforth, all blue maps are one-one
A, B and Z have an enveloping structure, arising from general equational folklore Z A let us focus attention on the embedding of Z into A
without loss of generality , Z is a subalgebra of A and the set A is the disjoint union of Z and some set X, A=Z U X
the scaffolding of Z and A the identity map z—>z uniquely extends to a homomorphism sZ of FREEZ onto Z the identity map a—> a uniquely extends to a homomorphism sA of FREEA onto A let
ker sZ and ker sA denote their kernels
ker(sZ) all blue arrows are inclusions all red arrows are surjections FREEZ ker(sA)
sZ Z
FREEXUZ sA A
LEMMA
ker(sZ) = ker(sA) « FREEZ
intuitively, this trivial Largeness Lemma states that ker(sZ) is as large as possible in ker(sA).
Z A
B
ker(sZ) ker(sA)
FREEZ
ker(sB)
sz
FREEXUZ sA A
Z
FREEYUZ sA B
ker(sZ) ker(sA)
FREEZ
ker(sB)
sz
FREEXUZ sA
Z
A
FREEYUZ sA B
FREEXUYUZ I = the ideal generated by ker(sA) U ker(sB)
ker(sZ) ker(sA)
FREEZ
ker(sB)
sz
FREEXUZ sA
Z
A
FREEYUZ sA B
D FREEXUYUZ I = the ideal generated by ker(sA) U ker(sB)
ker(sZ) ker(sA)
FREEZ
ker(sB)
sz
FREEXUZ
Z
sA A µ(x/ ker(sA)) = x/i
FREEYUZ sA B
µ D FREEXUYUZ
i = the ideal generated by ker(sA) U ker(sB)
courtesy of Leo Cabrer
clearly, this diagram commutes there remains to be proved that µ is one-one if this is to be the shortest possible proof of amalgamation for MV-algebras, patiently allow me to show you that µ is indeed one-one, in 2 slides, stressing the key role of deductive interpolation and of the deduction theorem
Let e be an element of FREEXUYUZ such that e/i =0, with the intent of proving e/ker(sA) = 0
e/i = 0 means that e belongs to i, which means, up to a trivial rearrangement, (theories ~ ideals) that a, b |– e for some element a in ker(sA) and b in ker(sB)
7 lines to end the proof From a, b |– e, by Deduction Theorem b |– anÆe. Deductive Interpolation yields j Œ FREEZ with n b |– j and j |– a Æ e. Now b |– j means j Œ ker(sB). Since j Œ FREEZ, by the Largeness Lemma j Œ ker(sZ)Õ ker(sA). From j |– an –> e, by the Deduction Theorem we get { j, a } |– e. Since j, a Œ ker(sA) then e Œ ker(sA). QED
Concluding remarks, in view of generalizations to other logics
why do MV-algebras have the amalgamation property? because the Lukasiewicz consequence relation has the following two basic properties:
deduction theorem
deductive interpolation theorem
why does Lukasiewicz logic satisfy the deductive interpolation theorem? because the sets Mod(F), as F ranges over formulas F(X1,...,Xn), are all possible rational polyhedra in [0,1]n by the Folklore Lemma
and rational polyhedra are preserved under projection and cylindrification
why should we insist in proving and reproving amalgamation and interpolation? •
because interpolation and amalgamation are deeply related to all fundamental logical-algebraic-geometric keywords:
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quantifier elimination, cut elimination, joint consistency, free products, joint embedding, limit constructions, pullback and pushouts,...
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mathematicians always want to simplify proofs of fundamental theorems, in the hope of making them crystal clear
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I have given a short proof of interpolation and amalgamation, stressing what seemed to me the main ideas
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I hope somebody finds a shorter proof or else, generalizing the key ideas of this proof, finds nice generalizations to other algebras and logics
thank you