Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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! 1 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 2 3 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. v J v Jv projective plane b v"J b" " b J " b " v v b Jv v JJ affine plane v J "b J " b " b Jv v 5 4 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. 6 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). 7 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. 8 9 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. 10 11 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: ' ' $ $ A u l f (l) ``` xu ``` ` m f f (x) f (m) ( ( ( u (((( n n′ C& & % % By Bruck–Ryser theorem, no Ck from Lemma 1 is the bounded morphic image of an affine plane. 12 13 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). ' $ yXX X XXX C∞ XXX f — surjective bounded morphism yX X & % XXXf2 . . . XXXX XXX XXX YH H f1 XX f0 6 X ' $ H t affine plane ... ⊆ N0 t AAt ⊆ N1 N2 A & % chain of networks Nk (k < |C∞ |) 14 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 15 D2 defects (the main case) ' $ C∞ need to add a line l through x, y $ ' C∞ & Nk 6fk ' x v & y % v & $ Nk % 6fk ' x v & y % v $ l? % 16 . . . but x, y already mapped by fk . . . ' $ v fk (y) v C∞ fk (x) & Nk 6fk ' x v & y . . . so need to define the fk+1 -image of l carefully ' $ C∞ % v 17 $ l? & Nk % v fk (y) v fk (x) 6fk ' x v & 18 y fk+1 (l)? % v $ l? % 19 danger. . . ' C∞ v fk (x) & v fk (m) fk (y) $ C∞ % 6fk $ ' PP v PPvP l? Nk x y P m & escape ' v fk (x) fk+1 (l) v fk (m) fk (y) & % k fk+1 Q Q 6fk Q' $ $ ' PP PP PPv l v PPvP v P ⊆ Nk x y P y Pm Nk+1 x m & % % & % 21 20 summary $ 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. 22 IH–Hussain hybrid projective Z } Z Z (ess.) Balbiani–Goranko Z Z trivial Z Z Z hybrid affine 6 trivial modal affine : IH–Hussain Z Z modal projective (NE XPTIME-complete) 23 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. 24 25