Interpolation in \Lukasiewicz logic and amalgamation of MV-algebras

1
Interpolation in \Lukasiewicz logic and
amalgamation of MV-algebras

• Daniele Mundici
• Dept. of Mathematics Ulisse Dini
• University of Florence,
• Florence, Italy
• mundici_at_math.unifi.it

2
we all know what a simplex in Rn is
0-simplex
1-simplex
2-simplex
3-simplex
3
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
4
our main themesrational polyhedra
and\Lukasiewicz logicChapter 1 Local
Deductionas a main ingredient of interpolation
and amalgamation
5
\Lukasiewicz logic L8

• 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) 1V(F)
• V(F gt G) min(1, 1V(F)V(G))
• Therefore, every valuation V is uniquely
  determined by its values on the variables
  V(X1),...,V(Xn)
• CONSEQUENCE RELATION F G means that
  every valuation satisfying F also satisfies G

6
formulas yield functions f0,1ngt0,1 as
boolean formulas yield f0,1ngt0,1

• every formula F(X1,...,Xn) determines a map fF
  0,1n gt0,1 by
• fXi the ith coordinate map
• fF 1 fF
• fF gt G min(1, 1 fF fG)

7
definable functions of one variable
for each formula F, its associated function fF is
continuous, linear, and each linear piece has
integer coefficients (for short, fF is a
McNaughton function)
the ONESET fF-1(1) of fF is the set of
valuations satisfying the formula
F oneset(fF)zeroset(fF)
8
oneset of fF Mod(F)

• by induction on the number of connectives in F,
  the oneset of fF is a rational polyhedron,
  and so is the oneset of fF and of fF gt G

EACH ZEROSET AND EACH ONESET IS A RATIONAL
POLYHEDRON IN 0,1n
9
(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 m1,2,... the formula Agt(Agt(Agt...gt
(Agt(AgtB))...)) is a tautology
3. B is obtained from A and the tautologies via
Modus Ponens
PROOF. 2gt3 easy 3gt1 induction 1gt2 is
proved geometrically
10
assume oneset(fA) contained in oneset(fB)
1
let T be a triangulation of 0,1 such that the
functions fA and fB both formulas A and B are
linear over each interval of T
fB
fA
1
11
fA fA lt fA
1
applying \Lukasiewicz conjunction to A, from the
formula AA we get obtain a minorant fAA of fA,
still with the same one set of fA Recall
definition PQ (P gt Q)
fB
fAA
1
12
fA fA fA lt fA fA lt fA
1
by iterated application of the \Lukasiewicz
conjunction we obtain a function fkA
fAfA...fA with the same oneset of fA, and
with the additional property that fAk fB
fB
fkA
1
13
for large k this will hold at every simplex of T
1
fB
in other words, we have the tautology AkgtB,
which is the same as the desired tautology
fA
1
Agt(Agt(Agt...gt(Agt(AgtB))...))
14
Chapter 2 Interpolation(as a main tool to
amalgamation)
15
interpolation/amalgamation

• Craig interpolation theorem fails in \Lukasiewicz
  logic, because the tautology xÙ xgtyÚy 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

16
(No Transcript)
17
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., Komoris 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

18
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,
  83279-308, 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, 545-55, 1975.

19
(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
20
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
21
rational polyhedra are preserved under
perpendicular cylindrification
we record this fact as the CYLINDRIFICATION LEMMA
22
oneset of fF Mod(F)
recall THE ZEROSET (AND THE ONESET) OF ANY
\LUKASIEWICZ FORMULA IS A RATIONAL POLYHEDRON IN
0,1n
we now prove the converse EACH RATIONAL
POLYHEDRON IN 0,1n IS THE ZEROSET OF SOME
\LUKASIEWICZ FORMULA
23
rational half-spaces in 0,1n
a rational line L in 0,12
PROBLEM Does there exist a formula F such that
the zeroset of fF coincides with H ?
mxnyp0, with m,n,p integers, mgt0
ANSWER Yes, by induction on mn
H
H is one of the half-planes bounded by L in the
square 0,12
24
then every rational polyhedron is a zeroset
this blue half-space is a zeroset
then so is this rational triangle (formulas can
express intersections)
ANY RATIONAL POLYHEDRON IN 0,1n IS THE ZEROSET
OF SOME fF
and this rational polyhedron (formulas can
express unions)
25
this was known to McNaughton (1951)

• FOLKLORE LEMMA
• Rational polyhedra contained in the n-cube
  0,1n 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 writing
RATIONAL POLYHEDRAONESETSMODELSETS
26
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) fF-1(1) P, which by the Folklore
Lemma is a rational polyhedron in 0,1XuZ
by the Projection Lemma, the projection of P onto
RZ is a rational polyhedron Q contained in
0,1Z
27
Mod(G) fG-1(1) R, a rational polyhedron in
0,1YuZ
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)
28
regarding J as a formula in the variables X,Z,
then Mod(J) is this blue rectangle!
by the Folklore Lemma, there is a formula
J(Z) such that QMod(J)
Z
Q
Mod(J)
Mod(F) P
Y
X
We then obtain the first half of interpolation
F J
29
regarding J as a formula in Y,Z, then Mod(J) is
this blue rectangle!
Z
QMod(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
30
Chapter 3 Amalgamationof the algebras of
\Lukasiewicz logic,i.e., Chang MV-algebras
31
MV-algebras (in Wajsbergs version)directly from
\Lukasiewicz axioms
Agt(BgtA) (AgtB)gt((BgtC)gt(AgtC)) ((AgtB)gtB)
gt ((BgtA)gtA) (AgtB)gt(BgtA)
32
the amalgamation property
Z
we have
A
B
33
the usual setup
Z
we have
A
B
we want
D
henceforth, all blue maps are one-one
34
the embedding of Z into A
let us focus attention on the embedding of Z
into A
Z
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, AZ U X
35
extending maps to homomorphisms
the identity map zgtz uniquely extends to a
homomorphism sZ of the free MV-algebra FREEZ
onto Z
FREEZ
sZ
Z
similarly, the identity map agt a uniquely
extends to a homomorphism sA of FREEA onto A
let ker sZ and ker sA denote the
kernels of these maps
36
ker(sZ)
all blue arrows are inclusions all red arrows
are surjections
FREEZ
ker(sA)
sZ
Z
FREEXUZ
sA
intuitively, this trivial Largeness Lemma states
that ker(sZ) is as large as possible in
ker(sA).
A
37
Z
A
B
38
ker(sZ)
ker(sB)
ker(sA)
FREEZ
sz
FREEXUZ
FREEYUZ
Z
sA
sA
A
B
39
ker(sZ)
ker(sB)
ker(sA)
FREEZ
sz
FREEXUZ
FREEYUZ
Z
sA
sA
A
B
FREEXUYUZ
I the ideal generated by ker(sA) U ker(sB)
40
ker(sZ)
ker(sB)
ker(sA)
FREEZ
sz
FREEXUZ
FREEYUZ
Z
sA
sA
A
B
D
FREEXUYUZ
I the ideal generated by ker(sA) U ker(sB)
41
ker(sZ)
ker(sB)
ker(sA)
FREEZ
sz
FREEXUZ
FREEYUZ
Z
sA
sA
A
B
µ
µ(x/ ker(sA)) x/i
D
there remains to be proved that µ is one-one
FREEXUYUZ
i the ideal generated by ker(sA) U ker(sB)
42
Let e be an element of FREEXUYUZ such that e/i
0. We must prove e/ker(sA) 0
e/i 0 means that e is an element of i. In
other words, (theories ideals) a, b e for
some a in ker(sA) and b in ker(sB)
43
end of the proof of amalgamation
44
Chapter 4 Further geometric developments
on projective MV-algebras
45
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-geom
  etric 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 FREEn 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) 6374.

46
projectives are routinely characterized by duality

• Every n-generated projective MV-algebra A is
  finitely presented (essentially, Baker)
• A is finitely presented iff AM(P) for some
  polyhedron P lying in some n-cube 0,1n
  (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

47
a first property of Z-retractsthey are retracts
of some cube 0,1n
this property is not easy to handle thus, we
must find equivalent conditions for a polyhedron
P to be retract of 0,1n
48
to check if P is a retract it suffices to check
that all homotopy groups of P are trivial

• The elements of the fundamental group p1(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.

       
49
equivalents for P to be a retract of 0,1n
THEOREM. For any polyhedron P in 0,1n the
following conditions are equivalent (a) P is a
retract of 0,1n (b) P is connected and all
homotopy groups pi(P) are trivial (c) P is
contractible (can be continuously shrunk to a
point).
Proof. (a)gt(b) by the functorial properties
of the homotopy groups pi . The implications
(b)gt(a) and (b)gt(c) follow from Whitehead
theorem in algebraic topology. (c)gt(b) is
trivial. QED
50
let P be this polyhedron
P is not a Z-retract, because it is not simply
connected
M(P) is not projective
51
a second property of Z-retracts P must contain a
vertex of 0,1n
P is not a Z-retract, because it does not contain
any vertex of the unit square
M(P) is not projective
52
a third property strong regularity
PROPOSITION If P is a Z-retract, then P
has a triangulation ? such that the affine hull
of every maximal simplex in ? contains some
integer point of Rn
this P is not a Z-retract for, the affine hull
of the vertical red segment does not contain any
integer point
1
M(P) is not projective
0
1
53
projectiveness 3 necessary conditions

• THEOREM (L.Cabrer, D.M., 2009) If A is a
  finitely generated projective MV-algebra, then up
  to isomorphism, AM(P) for some rational
  polyhedron lying in 0,1n such that
• P contains some vertex of 0,1n,
• P is contractible, and
• P is strongly regular.

are these three conditions also sufficient for
an MV-algebra A to be finitely generated
projective ?
54
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 0,1n containing
  a vertex of 0,1n.

It is not known if these three conditions are
sufficient in general. They become sufficient if
contractibility is strengthened to collapsibility
55
a sequence of collapses
56
a sequence of collapses
57
a sequence of collapses
58
a sequence of collapses
59
a sequence of collapses
60
a sequence of collapses
61
a sequence of collapses
62
a sequence of collapses
63
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 0,1n then
  M(P) is projective.

64
A is finitely presentedhomomorphismisomorphismi
ndecomposableA is free n-generated A is
n-generateddim(maxspec(A))d AM(P) is
projective
algebra
geometry
AM(P), P a polyhedronZ-mapZ-homeomorphismP is
connectedPunit cube 0,1n P lies in 0,1n
dim(P)d P is a Z-retract
65
\Lukasiewicz logic and MV-algebras together are
a rich source of geometric inspiration
thank you