Falsity, measures, and repelling assignments Giovanni Panti - - PowerPoint PPT Presentation

Falsity, measures, and repelling assignments

Giovanni Panti

Department of Mathematics and Computer Science University of Udine

Falsum-free product logic

Consider the propositional language ·, →, 1 and the set of axioms: commutative monoid identities for ·, 1, x → x = 1, z → (y → x) = (z · y) → x, x · (x → y) = y · (y → x), y → (x · y) = x.

Falsum-free product logic

Consider the propositional language ·, →, 1 and the set of axioms: commutative monoid identities for ·, 1, x → x = 1, z → (y → x) = (z · y) → x, x · (x → y) = y · (y → x), y → (x · y) = x. They define a rather basic propositional logic, which has as canonical algebraic semantics the half-open real unit interval (0, 1] with conjunction · interpreted as ordinary product and → as the corresponding residuum (explicitely, x → y = min(1, y/x)).

Truncated subtraction

It is remarkable that the above axioms capture precisely the equational theory of truncated subtraction in the nonnegative reals.

Truncated subtraction

It is remarkable that the above axioms capture precisely the equational theory of truncated subtraction in the nonnegative reals. Indeed, fixing and arbitrary real number 0 < c < 1, the exponential function in base c provides an order-reversing isomorphism ([0, +∞), +, · −, 0) → ((0, 1], ·, →, 1),

Truncated subtraction

It is remarkable that the above axioms capture precisely the equational theory of truncated subtraction in the nonnegative reals. Indeed, fixing and arbitrary real number 0 < c < 1, the exponential function in base c provides an order-reversing isomorphism ([0, +∞), +, · −, 0) → ((0, 1], ·, →, 1), and it is a —not really trivial— fact that an identity is true in (R≥0, +, · −, 0) iff it can be deduced from the axioms, which now read commutative monoid identities for +, 0, x · − x = 0, (x · − y) · − z = x · − (y + z), x + (y · − x) = y + (x · − y), (x + y) · − y = x.

Cancellative hoops

Let us stick to the +, · −, 0 language. The models of the axioms are called [cancellative] hoops. All hoops we consider will be separating subhoops of C(X)≥0, for X some compact Hausdorff space; they are just positive cones of lattice-ordered abelian groups.

Cancellative hoops

Let us stick to the +, · −, 0 language. The models of the axioms are called [cancellative] hoops. All hoops we consider will be separating subhoops of C(X)≥0, for X some compact Hausdorff space; they are just positive cones of lattice-ordered abelian groups. (P., LNAI 2007). The free n-generated hoop is the set of all functions from [0, 1]n−1 to R≥0 which are generated by x1, . . . , xn−1,1 l − (x1 ∨ · · · ∨ xn−1) under pointwise operations. These functions are continuous and piecewise-linear with integer coefficients, and all such functions are thus obtainable.

Measuring things

The prototypical cancellative hoop is R≥0. Nonnegative real numbers are excellent for measuring

Measuring things

The prototypical cancellative hoop is R≥0. Nonnegative real numbers are excellent for measuring but

Measuring things

The prototypical cancellative hoop is R≥0. Nonnegative real numbers are excellent for measuring but we cannot measure until we fix a gauge scale.

Measuring things

The prototypical cancellative hoop is R≥0. Nonnegative real numbers are excellent for measuring but we cannot measure until we fix a gauge scale. We usually fix 1, but in the real numbers the choice is irrelevant: since the automorphism group of R≥0 acts transitively on positive reals, all gauge scales are created equal.

Belying our politician

Gauge scales must have the archimedean property. Therefore, we define a unit u in the cancellative hoop H to be an archimedean element: ∀h ∈ H ∃n ≥ 1 h ≤ nu.

Belying our politician

Gauge scales must have the archimedean property. Therefore, we define a unit u in the cancellative hoop H to be an archimedean element: ∀h ∈ H ∃n ≥ 1 h ≤ nu. For the class of hoops we are considering (the subhoops of some C(X)≥0), the units are precisely the functions which are never zero.

Belying our politician

Gauge scales must have the archimedean property. Therefore, we define a unit u in the cancellative hoop H to be an archimedean element: ∀h ∈ H ∃n ≥ 1 h ≤ nu. For the class of hoops we are considering (the subhoops of some C(X)≥0), the units are precisely the functions which are never zero. Upon defining x ⊕ y = (x + y) ∧ u and ¬x = u · − x, the interval [0, u] = {x ∈ H : x ≤ u} becomes an MV-algebra (the algebraic conterparts of Lukasiewicz many-valued logic), with u denoting “absolute falsity”.

Belying our politician

Gauge scales must have the archimedean property. Therefore, we define a unit u in the cancellative hoop H to be an archimedean element: ∀h ∈ H ∃n ≥ 1 h ≤ nu. For the class of hoops we are considering (the subhoops of some C(X)≥0), the units are precisely the functions which are never zero. Upon defining x ⊕ y = (x + y) ∧ u and ¬x = u · − x, the interval [0, u] = {x ∈ H : x ≤ u} becomes an MV-algebra (the algebraic conterparts of Lukasiewicz many-valued logic), with u denoting “absolute falsity”. Note that in free hoops not all units are created equal: the automorphism group of Freen CH does not act transitively. Different units will usually give rise to nonisomorphic MV-algebras.

Average truth-values

Fix your favorite finitely presented cancellative hoop H, and your favorite unit u in it.

Average truth-values

Fix your favorite finitely presented cancellative hoop H, and your favorite unit u in it. I’m interested in the following question:

Average truth-values

Fix your favorite finitely presented cancellative hoop H, and your favorite unit u in it. I’m interested in the following question:

• can we assign an “average truth value” to all elements of H

in a consistent way?

Average truth-values

Fix your favorite finitely presented cancellative hoop H, and your favorite unit u in it. I’m interested in the following question:

• can we assign an “average truth value” to all elements of H

in a consistent way? More formally, is there a monoid homomorphism (which usually is not a hoop homomorphism) s : H → R≥0 that maps u to 1 and is automorphism-invariant (i.e., s(σh) = s(h) for every h and every automorphism σ of H that fixes u)?

Average truth-values

Fix your favorite finitely presented cancellative hoop H, and your favorite unit u in it. I’m interested in the following question:

• can we assign an “average truth value” to all elements of H

in a consistent way? More formally, is there a monoid homomorphism (which usually is not a hoop homomorphism) s : H → R≥0 that maps u to 1 and is automorphism-invariant (i.e., s(σh) = s(h) for every h and every automorphism σ of H that fixes u)? s(h) must be thought of as the average truth-value of h. The set of all such s’s (usually called the state space) is a compact convex subset of RH

≥0, whose boundary is precisely the

set of hoop homomorphism.

Basic stuff

• Kroupa, P. (independently, 2005). States correspond 1-1

to finite measures on the dual space of H (which is the —unique up to homeo— space X such that H is embeddable in C(X)≥0).

Basic stuff

• Kroupa, P. (independently, 2005). States correspond 1-1

to finite measures on the dual space of H (which is the —unique up to homeo— space X such that H is embeddable in C(X)≥0).

• P. (2007) The duals of automorphisms of Freen CH are

precisely the piecewise-fractional GLn Z-homeomorphisms

• f [0, 1]n−1,

p1 p2 p3 p4 p5 p6

σ∗

q1 q2 q3 q4 q5 q6

and a state is σ-invariant iff the corresponding measure is σ∗-invariant.

Basic stuff

• Kroupa, P. (independently, 2005). States correspond 1-1

to finite measures on the dual space of H (which is the —unique up to homeo— space X such that H is embeddable in C(X)≥0).

• P. (2007) The duals of automorphisms of Freen CH are

precisely the piecewise-fractional GLn Z-homeomorphisms

• f [0, 1]n−1,

p1 p2 p3 p4 p5 p6

σ∗

q1 q2 q3 q4 q5 q6

and a state is σ-invariant iff the corresponding measure is σ∗-invariant.

• Mundici (1995) The Lebesgue measure on [0, 1]n−1 is

invariant under all the [duals of the] elements in Stab(1 l) < Aut Freen CH.

Actually, the latter is a characterization

• P. (Comm. in Alg. 2008), Marra (J. Group Th. 2009). The

Lebesgue measure on [0, 1]n−1 is the only nonatomic measure that is invariant under Stab(1 l).

Actually, the latter is a characterization

• P. (Comm. in Alg. 2008), Marra (J. Group Th. 2009). The

Lebesgue measure on [0, 1]n−1 is the only nonatomic measure that is invariant under Stab(1 l). This really means that once you fixed the constant 1 l as your gauge scale for falsity, then your policician’s statements have precisely one nonambiguous average truth-value: their integrals w.r.t. Lebesgue measure.

Actually, the latter is a characterization

• P. (Comm. in Alg. 2008), Marra (J. Group Th. 2009). The

Lebesgue measure on [0, 1]n−1 is the only nonatomic measure that is invariant under Stab(1 l). This really means that once you fixed the constant 1 l as your gauge scale for falsity, then your policician’s statements have precisely one nonambiguous average truth-value: their integrals w.r.t. Lebesgue measure. My proof exploits a rigidity phenomenon common in ergodic theory: a single homeo of a space X tipically determines a plethora of invariant measure, but if you ask for invariance w.r.t. two independent homeos, then this plethora often reduces to the obvious ones.

Actually, the latter is a characterization

• P. (Comm. in Alg. 2008), Marra (J. Group Th. 2009). The

Lebesgue measure on [0, 1]n−1 is the only nonatomic measure that is invariant under Stab(1 l). This really means that once you fixed the constant 1 l as your gauge scale for falsity, then your policician’s statements have precisely one nonambiguous average truth-value: their integrals w.r.t. Lebesgue measure. My proof exploits a rigidity phenomenon common in ergodic theory: a single homeo of a space X tipically determines a plethora of invariant measure, but if you ask for invariance w.r.t. two independent homeos, then this plethora often reduces to the obvious ones. This implies that Stab(1 l) is a “higly chaotic” subgroup of Aut Freen CH. Note that, beyond being chaotic in the large, Stab(1 l) also contains single elements which are chaotic by themselves —namely, have the Bernoulli property— (P., JPAA 2007).

More recent work

We’ve seen that the choice of the unit 1 l in the free hoop on n generators determines uniquely an automorphism-invariant measure (namely, Lebesgue) on the dual space [0, 1]n−1.

More recent work

We’ve seen that the choice of the unit 1 l in the free hoop on n generators determines uniquely an automorphism-invariant measure (namely, Lebesgue) on the dual space [0, 1]n−1. Obviously, this calls for generalization in two directions:

More recent work

We’ve seen that the choice of the unit 1 l in the free hoop on n generators determines uniquely an automorphism-invariant measure (namely, Lebesgue) on the dual space [0, 1]n−1. Obviously, this calls for generalization in two directions:

1 replace Freen CH with an arbitrary finitely presented hoop;

More recent work

We’ve seen that the choice of the unit 1 l in the free hoop on n generators determines uniquely an automorphism-invariant measure (namely, Lebesgue) on the dual space [0, 1]n−1. Obviously, this calls for generalization in two directions:

1 replace Freen CH with an arbitrary finitely presented hoop; 2 replace 1

l with an arbitrary unit.

More recent work

We’ve seen that the choice of the unit 1 l in the free hoop on n generators determines uniquely an automorphism-invariant measure (namely, Lebesgue) on the dual space [0, 1]n−1. Obviously, this calls for generalization in two directions:

1 replace Freen CH with an arbitrary finitely presented hoop; 2 replace 1

l with an arbitrary unit. The first generalization was accomplished by Mundici (DCDS 2008): the distinguished measure exists, and is a certain weighted average of d-dimensional Lebesgue measures, for 0 ≤ d ≤ (number of generators) − 1.

Denominators

Both generalizations are accomplished simultaneously by P.-Ravotti (JSL 2013); moreover, the construction is representation-independent.

Denominators

Both generalizations are accomplished simultaneously by P.-Ravotti (JSL 2013); moreover, the construction is representation-independent. It works as follows:

Denominators

Both generalizations are accomplished simultaneously by P.-Ravotti (JSL 2013); moreover, the construction is representation-independent. It works as follows:

• For each d ≥ 1 the set of all extreme states s from H onto

d−1Z≥0 that map u to 1 is finite; these are called discrete states of denominator d.

Denominators

Both generalizations are accomplished simultaneously by P.-Ravotti (JSL 2013); moreover, the construction is representation-independent. It works as follows:

• For each d ≥ 1 the set of all extreme states s from H onto

d−1Z≥0 that map u to 1 is finite; these are called discrete states of denominator d.

• The dual space of d−1Z≥0 is a singleton, so there’s only one

measure on that singleton.

Denominators

Both generalizations are accomplished simultaneously by P.-Ravotti (JSL 2013); moreover, the construction is representation-independent. It works as follows:

• For each d ≥ 1 the set of all extreme states s from H onto

d−1Z≥0 that map u to 1 is finite; these are called discrete states of denominator d.

• The dual space of d−1Z≥0 is a singleton, so there’s only one

measure on that singleton.

• Pulling back that measure gives a Dirac mass δx(s) on

X = MaxSpec H.

Denominators

Both generalizations are accomplished simultaneously by P.-Ravotti (JSL 2013); moreover, the construction is representation-independent. It works as follows:

• For each d ≥ 1 the set of all extreme states s from H onto

d−1Z≥0 that map u to 1 is finite; these are called discrete states of denominator d.

• The dual space of d−1Z≥0 is a singleton, so there’s only one

measure on that singleton.

• Pulling back that measure gives a Dirac mass δx(s) on

X = MaxSpec H.

• We list arbitrarily all discrete states s1, s2, s3, . . . of H,

under the only restriction of nondecreasing denominators.

Geometry

All of this can be seen geometrically: X is a rational polytopal set in [0, 1]n−1, discrete states correspond to primitive points in Cone(X) ∩ Zn, and a discrete state has denominator d iff the corresponding point lies in the level set {u = d}, where u is the projectivization of u.

X {u = d}

The main results

As d increases the level set {u = d} lifts, and more and more primitive integer points are radially projected to X.

The main results

As d increases the level set {u = d} lifts, and more and more primitive integer points are radially projected to X. The key fact is that this rain of points equidistributes according to a specific probability on X.

The main results

As d increases the level set {u = d} lifts, and more and more primitive integer points are radially projected to X. The key fact is that this rain of points equidistributes according to a specific probability on X.

• Theorem. The Ces`

aro limit lim

k→∞

1 k

• t≤k

δx(st) exists in the weak-∗ sense, does not depend on the enumeration, and the limit measure µ is automorphism-invariant.

The main results

As d increases the level set {u = d} lifts, and more and more primitive integer points are radially projected to X. The key fact is that this rain of points equidistributes according to a specific probability on X.

• Theorem. The Ces`

aro limit lim

k→∞

1 k

• t≤k

δx(st) exists in the weak-∗ sense, does not depend on the enumeration, and the limit measure µ is automorphism-invariant.

• Theorem. Say that µ1 and µ2 correspond to the units u1 and

u2 as above. Then each of µ1 and µ2 is absolutely continuous w.r.t. the other, with dµ2 = C u1 u2 d+2 dµ1, where C is a constant and d is the topological dimension of X.

Many open problems

Many open problems

1 We’ve seen that Stab(1

l) < Aut Freen CH acts chaotically. Is this true for every Stab(u), where u is an arbitrary unit?

Many open problems

1 We’ve seen that Stab(1

l) < Aut Freen CH acts chaotically. Is this true for every Stab(u), where u is an arbitrary unit?

2 If (as I suspect) the answer is “no”, can we classify units

according to the complexity of their stabilizers?

Many open problems

1 We’ve seen that Stab(1

l) < Aut Freen CH acts chaotically. Is this true for every Stab(u), where u is an arbitrary unit?

2 If (as I suspect) the answer is “no”, can we classify units

according to the complexity of their stabilizers?

3 Which is the “right” topology on these stabilizers? For

example, Stab(1 l) is not amenable w.r.t. the discrete topology, but maybe it might become so under some weaker topology.

Many open problems

1 We’ve seen that Stab(1

l) < Aut Freen CH acts chaotically. Is this true for every Stab(u), where u is an arbitrary unit?

2 If (as I suspect) the answer is “no”, can we classify units

according to the complexity of their stabilizers?

3 Which is the “right” topology on these stabilizers? For

example, Stab(1 l) is not amenable w.r.t. the discrete topology, but maybe it might become so under some weaker topology.

4 Is any of these stabilizers finitely generated? Is their

conjugacy problem decidable?

Many open problems

1 We’ve seen that Stab(1

l) < Aut Freen CH acts chaotically. Is this true for every Stab(u), where u is an arbitrary unit?

2 If (as I suspect) the answer is “no”, can we classify units

according to the complexity of their stabilizers?

3 Which is the “right” topology on these stabilizers? For

example, Stab(1 l) is not amenable w.r.t. the discrete topology, but maybe it might become so under some weaker topology.

4 Is any of these stabilizers finitely generated? Is their

conjugacy problem decidable?

5 Is it true that every automorphism of Aut Freen CH that

fixes a continuous density must necessarily fix a unit? (the answer is negative if the density is not assumed continuous).

Many open problems

1 We’ve seen that Stab(1

l) < Aut Freen CH acts chaotically. Is this true for every Stab(u), where u is an arbitrary unit?

2 If (as I suspect) the answer is “no”, can we classify units

according to the complexity of their stabilizers?

3 Which is the “right” topology on these stabilizers? For

example, Stab(1 l) is not amenable w.r.t. the discrete topology, but maybe it might become so under some weaker topology.

4 Is any of these stabilizers finitely generated? Is their

conjugacy problem decidable?

5 Is it true that every automorphism of Aut Freen CH that

fixes a continuous density must necessarily fix a unit? (the answer is negative if the density is not assumed continuous).

6 Replace finitely many generators with countably many. Is

it true that Stab(1 l) acting on the Hilbert cube [0, 1]ω fixes the product Lebesgue measure?