Download Axiomatising the modal logic of affine planes

introduction
Projective and affine planes — incidence systems of points and lines
Axiomatising the modal logic of affine planes
Ian Hodkinson
joint work with Altaf Hussain
Thanks to Nick for inviting me
• axiomatic abstractions of ‘real plane geometry’
• no quantitative information (distances)
• qualitative directional information (collinearity, parallelism)
Heavily studied in mathematics as a core part of geometry.
Studied in modal logic by Stebletsova, Venema, Balbiani, Goranko,
Vakarelov, . . . Spatial logic is currently of some interest.
• Venema (1999) proposed 2-sorted treatment, and axiomatised
projective planes.
• Balbiani–Goranko (2001) axiomatised ‘weak affine planes’.
• This talk: the true affine planes are not finitely axiomatisable!
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definition — projective plane
A projective plane is a triple (P, L, E), where P, L are the sets of
points and lines (resp.), E ⊆ P × L, and
outline
• definitions, examples
• existing work (Bruck–Ryser, Venema, Balbiani–Goranko)
• proof of non-finite axiomatisability
• conclusion, problems
P1. any two distinct points lie on a unique line
∀x, y ∈ P (x 6= y → ∃ ! l ∈ L(x E l ∧ y E l))
P2. any two distinct lines meet in a unique point
∀l, m ∈ L(l 6= m → ∃ ! x ∈ P (x E l ∧ x E m))
P3. there exist four points,
^ no three of which are collinear
∃x0 x1 x2 x3 ∈ P
¬∃l ∈ L(xi E l ∧ xj E l ∧ xk E l)
i<j<k<4
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examples
definition — affine plane
An affine plane is a triple (P, L, E, ||), where P, L are the sets of
points and lines (resp.), E ⊆ P × L, || ⊆ L × L, and
A0. two lines are parallel iff they are equal or disjoint
∀l, m ∈ L(l || m ↔ l = m ∨ ¬∃x ∈ P (x E l ∧ x E m))
A1. any two distinct points lie on a unique line
∀x, y ∈ P (x 6= y → ∃ ! l ∈ L(x E l ∧ y E l))
A2. there is a unique line through any point parallel to any given line
∀x ∈ P ∀l ∈ L ∃ ! m ∈ L(x E m ∧ m || l)
A3. there exist three non-collinear points
∃x0 x1 x2 ∈ P ¬∃l ∈ L(x0 E l ∧ x1 E l ∧ x2 E l)
(equivalently: L 6= ∅, and for any l ∈ L, there is a point not on l)
The real plane R2 naturally yields an affine plane.
Replace R by a finite field — you get a finite affine plane.
It can be completed to a finite projective plane.
completion: affine plane → projective plane
In an affine plane, || is an equivalence relation. Take its equivalence
classes to be the points of a new line (the ‘line at infinity’). This gives
a projective plane — the ‘completion’ of the affine plane.
projective plane → affine plane
Rip a line and its points out of a projective plane — get affine plane.
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order
The order of a projective plane is (the number of points on a line) − 1.
This is well-defined.
modal logic of projective planes
The order of an affine plane is the number of points on a line.
This is (the number of parallel classes) − 1.
Yde Venema (1999) proposed treating projective plane (P, L, E) as
2-sorted Kripke frame (P, L, E, ∃), where ∃ is converse of E.
Modal formulas:
Bruck–Ryser theorem, 1949: If a finite projective plane has order n,
and n ≡ 1 or 2 mod 4, then n is the sum of two squares.
Exercise: For all n ≥ 0, the number 2 · 32n+1 is ≡ 2 mod 4 and is not
the sum of two squares.
So there’s no projective (or affine) plane of order 2 · 32n+1 , for any
n ≥ 0.
No affine plane has 2 · 32n+1 + 1 parallel classes, for any n ≥ 0.
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point formulas π
::= p0 | ¬π | π ∧ π | [01]λ
line formulas λ
::= p1 | ¬λ | λ ∧ λ | [10]π
The boxes [01], [10] have accessibility relations E, ∃ (resp.).
h01i, h10i are the corresponding diamonds (the usual abbreviations).
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Venema’s axioms and rules
what about affine planes?
(Sahlqvist) axioms:
all propositional tautologies
normality of [01], [10]
[01] and [10] are mutually converse
h01i⊤, h10i⊤
transitivity of h·i and h−i, where h·iπ = h01ih10iπ, h−iλ = h10ih01iλ.
(In a projective plane, these are universal modalities for their sorts.)
Regard affine planes as 2-sorted Kripke frames (P, L, E, ∃, ||).
Rules: modus ponens, generalisation for [01], [10], substitution.
This system is (strongly) sound and complete for projective planes.
Venema also proved that satisfiability for the logic of projective
planes is decidable and NE XPTIME-complete.
point formulas
π
line formulas λ
::= p0 | ¬π | π ∧ π | [01]λ
::= p1 | ¬λ | λ ∧ λ | [10]π | [||]λ
Balbiani–Goranko (2002) proposed axioms for affine planes.
Completeness was left open (though it was proved for a wider class
of structures called ‘weak affine models’).
IH–Hussain (2005): the 2-sorted modal logic of affine planes is not
finitely axiomatisable.
So B–G’s axioms are not complete.
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details — configurations (affine-like frames)
proof of non-finite axiomatisability
Idea: construct finite affine-like frames Cn (n < ω) such that:
1. each Cn is not the bounded morphic image of any affine plane,
2. their ‘limit’ C∞ (e.g., an ultraproduct) is a bounded morphic
image of an affine plane.
If the logic of affine planes were axiomatisable by a finite set Σ of
formulas, then
Let κ ≥ 1 be a cardinal.
A κ-configuration is a frame C = (P, L, E, ∃, ||) such that
C0. two lines are parallel iff they are equal or disjoint
C1. For all x, y ∈ P , there are at least κ lines l ∈ L with x, y
E
l.
C2. there is a unique line through any point parallel to any given line
C3. L 6= ∅, and for any l ∈ L, there is a point not on l.
1. each Cn would satisfy at least one of {¬σ : σ ∈ Σ},
2. C∞ would validate Σ.
With suitable construction of ‘limit’, this is impossible!
C0, C2, C3 are equivalent to affine axioms A0, A2, A3.
C1 is different from A1 (‘any two distinct points lie on a unique line’).
Any affine plane is a 1-configuration.
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making the Cn
configurations and affine planes
Lemma 1. For every finite k ≥ 1, there is a finite k-configuration Ck
with exactly c parallel classes, where c = 2 · 32n+1 + 1 for some n.
Lemma 2. Let C be a κ-configuration, A an affine plane, and
f : A → C a surjective homomorphism. Then A and C have the
same number of ||-classes.
Proof. Pick c, d, n with k ≤ 2d−2 , 4kd2 ≤ 2 · 32n+1 + 1 = c, c ≤ 2d−1 .
Take any set P with |P | = 2d. Put S := {l ⊆ P : |l| = d}.
Choose L ⊆ S with
• l ∈L⇒P \l ∈L
• ∀x, y ∈ P ∃≥k l ∈ L(x, y ∈ l)
¢
¡
d−2
≥ k.
Possible since |{l ∈ S : x, y ∈ l}| ≥ 2d−2
d−2 ≥ 2
2
Need to pick total of ≤ (2d) k sets l (plus their complements).
• |L|/2 = c.
So far, |L|/2 ≤ 4kd2 ≤ c. Just add more l ∈ S to L until |L|/2 = c.
¡ ¢
d−1
d
≥ c.
Possible since |S| = 2d
d ≥ 2 so |S|/2 ≥ 2
Check Ck = (P, L, ∈, ∋, ||) (where || is defined by C0) is as required.
Proof. Show f induces a bijection on ||-classes. Take lines l, m of A.
l || m ⇒ f (l) || f (m) — trivial.
Assume ¬(l || m) but f (l) || f (m), and get contradiction:
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By Bruck–Ryser theorem, no Ck from Lemma 1 is the bounded
morphic image of an affine plane.
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limit
coherence (soundness) conditions and defects
Let C∞ be a ‘limit’ (e.g., non-principal ultraproduct) of the Ck .
It is a |C∞ |-configuration.
S1. |Nk | < |C∞ | for each k
Lemma 3. C∞ is the bounded morphic image of an affine plane.
Proof. Build affine plane A and surjective bounded morphism
f : A → C∞ by a step-by-step construction similar to Venema (1999).
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S2. each fk is a homomorphism
S3. l || m ⇐⇒ fk (l) || fk (m), for all lines l, m of Nk
S4. any two distinct points have at most one line through them
S5. no two distinct parallel lines intersect
S6. there exist three non-collinear points
D1. ‘back-defects’ for fk for each of h01i, h10i, h||i.
D2. two points with no line joining them
D3. parallel-axiom defects
D4. non-parallel lines with no point of intersection
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D2 defects (the main case)
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danger. . .
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summary
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complexity of satisfiability problem
• Each Ck is not the bounded morphic image of an affine plane
• The limit C∞ is the bounded morphic image of an affine plane
So the modal logic of affine planes is not finitely axiomatisable.
This is a surprising contrast with projective planes.
Can view as a positive result — explicit axioms may have interesting
‘geometrical’ content.
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IH–Hussain
hybrid projective
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hybrid affine
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trivial
modal affine
:
IH–Hussain
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modal projective (NE XPTIME-complete)
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some references
1. P. Balbiani and V. Goranko, Modal logics for parallelism,
orthogonality, and affine geometries., Journal of Applied
Non-Classical Logics 12 (2002), no. 3-4, 365–398.
open questions
1. decidability, complexity of modal logic of affine planes (etc)?
2. find explicit axioms for modal logic of affine planes
3. what if we use hybrid logic?
4. what if we use strict (irreflexive) parallelism?
Balbiani–Goranko (2002) axiomatised affine planes using
non-standard ‘irreflexivity’ rules.
decidability and complexity — open
2. R H Bruck and H J Ryser, The nonexistence of certain finite
projective planes, Canad. J. Math. 1 (1949), 88–93.
3. A. Hussain, Applications of extended modal logics, Ph.D. thesis,
University of London, 2006, submitted.
4. V Stebletsova, Algebras, relations and geometries, Ph.D. thesis,
Zeno, Leiden–Utrecht Research Institute of Philosophy,
Heidelberglaan 8, 3584 CS Utretcht, Netherlands, 2000, Number
XXXII in Quæstiones Infinitæ.
5. Y. Venema, Points, lines and diamonds: a two-sorted modal logic
for projective planes, J. Logic Computat. 9 (1999), 601–621.
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