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Inductive Formalization of PLCA

Inductive Formalization of PLCA. Graduate School of Science and Technology, Kwansei Gakuin University Mizuki G oto. Motivation. We proposed PLCA as qualitative spatial expression in lab and proved property satisfied PLCA. But, it was pen-and- peper proof

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Inductive Formalization of PLCA

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  1. Inductive Formalization of PLCA Graduate School of Science and Technology, KwanseiGakuin University MizukiGoto

  2. Motivation • We proposed PLCA as qualitative spatial expression in lab and proved property satisfied PLCA. • But, it was pen-and-peper proof • The mistakes may exist in that proof • We check PLCA in theorem proverCoq • Many theorem provers match inductive model • Inductive PLCA 9th theorem proving and Provers meeting

  3. Agenda • Definition of PLCA • Inductive formalization • Proof of validity • Future work 9th theorem proving and Provers meeting

  4. PLCA • PLCA expression is a kind of qualitative spatial reasoning • PLCA provides a symbolic representation for spatial data using the simple four objects (point, line, circuit and area) • For a given figure in a two-dimensional plane, there exists a PLCA expression, that is unique in its pattern of connections between regions. intersect e.g. External connect non tagentialconnect tagential connect Disconnect 9th theorem proving and Provers meeting

  5. PLCA • Merit • small data size • Compared to quantitative measurement • Intuitive thinking • application • Artificial Intelligence • Image analysis • Geometrical analysis • Another framework • RCC (Region Connection Calculus)(Randell et al., 1992) 9th theorem proving and Provers meeting

  6. PLCA • Point • Represents dot • Line • Represents line connecting two points • Adding + or -, represents directed line • example： ll • l.pointsmeans the set of point included line p1 l p1 p2 l = (p1, p2) - + 9th theorem proving and Provers meeting

  7. PLCA • Circuit • Represents loop lines • Circuithas direction • clock-wise, counter clock-wise ⇔ inner circuit, outer circuit l6 c1 l1 c1 = [l1,l2,l3,l4,l5,l6 ] + + + + + + l1= (p1, p2), l2 = (p2,p3), ...... l6 = (p6, p1) l5 l2 c1 = [(p1,p2),(p2,p3),........(p6,p1) ] l4 l3 9th theorem proving and Provers meeting

  8. PLCA • Area • Represents space divided by circuit • the outside space is not area c1 a1 c2 a1 = {c2,c3} a2 = {c4} c3 c4 a2 9th theorem proving and Provers meeting

  9. PLCA • outermost • The outside circuit is called “outermost” • The instance of PLCA has only one outermost outermost 9th theorem proving and Provers meeting

  10. PLCA outermost outermost outermost 9th theorem proving and Provers meeting

  11. PLCA P = {p1, p2, p3, p4, p5} l1 = (p1,p2) l2 = (p2,p3) l3 = (p3,p4) l4 = (p4,p1) l5 = (p5,p5) L = {l1, l2, l3, l4, l5} c1 = outermost l1 p1 p2 c2 c3 c4 a2 l4 l2 - - - - c1 = [l1,l4,l3,l2 ] c2 = [l1,l2,l3,l4 ] c3 = [l5 ] c4 = [l5 ] C = {c1, c2, c3, c4} p5 l5 a1 + + + + + p3 p4 - l3 a1 = {c2,c3} a2 = {c4} A = {a1, a2} 9th theorem proving and Provers meeting

  12. PLCA P = {p1, p2, p3, p4, p5} L = {l1, l2, l3, l4, l5} c1 = outermost l1 p1 p2 C = {c1, c2, c3, c4} c2 A = {a1, a2} c3 c4 a2 l4 l2 p5 l5 a1 p3 p4 E = <P, L, C, A, outermost> l3 9th theorem proving and Provers meeting

  13. PLCA • Notification • The lengths of lists circuit, area are more than one • Circuits in the same area don’t connect by point or line p l c = [l] a = [c] c a c1? c2? a2 a3 a1 9th theorem proving and Provers meeting

  14. PLCA • PLCAconnect • This predicate means relationship of PLCA objects • two arguments • PLCAconnect A B ∧ PLCAconnect B C iffPLCAconnect A C • PLCAconnect (point p) (Line l)iff p ∈ l.points • PLCAconnect (Line l)(Circuit c)iff l ∈ c.points • PLCAconnect (Circuit c)(Area a)iffc∈ a.points • PLCAconnect A B iffPLCAconnect B A 9th theorem proving and Provers meeting

  15. PLCA • Consistent PLCA • Consistency of Point-Line • ∀p ∈ P, ∃l ∈ L, p ∈ l.points • ∀ l ∈ L, p ∈ l.points→ p ∈ P • Consistency of Line-Circuit • ∀l ∈ L, ∃c ∈ C , l ∈ c.lines • ∀c ∈ C, l ∈ c.lines→ l∈ L • ∀l ∈ L, l ∈ c1.lines,l ∈ c2.lines → c1=c2 • Consistency of Circuit-Area • ∀c ∈ C, ∃a ∈ A , c∈ a.circuits • ∀a ∈ A, c ∈ a.circuits→ c∈ C • ∀c ∈ C, c ∈ a1.circuits, c ∈ a2.circuits → a1=a2 If these properties are satisfied, then PLCA is consistent 9th theorem proving and Provers meeting

  16. PLCA • Consistent PLCAconnect PLCA and |P| - |L| -|C| + 2 * |A| = 0 ⇔PLCA is planar (Takahashi et al,2008) • pen-and-paper proof • RxPLCAtool is implemented in java and prolog(Sumitomo et al., 2005) 9th theorem proving and Provers meeting

  17. Agenda • Definition of PLCA • Inductive formalization • inductive PLCA • inductive PLCAconnect • Proof of validity • Future work 9th theorem proving and Provers meeting

  18. Inductive PLCA Definition Point := nat. Definition Line := Point * Point. Definition Circuit := list Line. Definition Area := list Circuit. Definition reverse(l : Line) := (snd l, fst l). Definition Point_In_Line(p : Point)(l : Line) : Prop :=fst l = p ∨ snd l = p. Definition InL (l : Line)(L : set Line) : Prop :=set_In l L∨ set_In (reverse l) L. ← conditions + - l ⇔ l p ∈ l.points l ∈ L 16 9th theorem proving and Provers meeting

  19. Inductive PLCA all points all lines Inductive PATH (P : set Point)(L : set Line) : Point → Point → list Point → list Line → Prop := | nil_path : ………trace lists = nil………… | step_path : ………trace lists = y → trace lists = x::y…………… start point end point trace lists nil_path step_path 9th theorem proving and Provers meeting

  20. Inductive PLCA Inductive PATH (P : set Point)(L : set Line) : Point → Point → list Point → list Line → Prop := | nil_path : forall(p : Point),set_In p P→ PATH P L p p (p::nil) nil | step_path : forall(p1 p2 p3 : Point)(pl : list Point)(ll : list Line), PATH P L p2 p1 pl ll →InL (p3, p2) L →set_In p3 P → ~In p3 pl → PATH P L p3 p1 (p3::pl) ((p3, p2)::ll). Circuit (list Line) = [l6::l5::l4::…l1] InL l6 L ∧ PATH P L p6 p1 (p6::p5….p1) (l5::l4::…l1) p1 l6 l1 p6 p2 l5 l2 p5 l3 l4 p3 p4 9th theorem proving and Provers meeting

  21. Inductive PLCA Inductive E : set Point → set Line → set Circuit → set Area → Type := | e_outermost : ………… | connect_circuit : ………. | isolate_circuit : ………… all points all lines all circuits all areas Three constructors 9th theorem proving and Provers meeting

  22. e_outermost basecase a1 c1 outermost 9th theorem proving and Provers meeting

  23. Inductive PLCA Inductive E : set Point → set Line → set Circuit → set Area → Type := | e_outermost :forall(P : set Point)(L : set Line)(pl : list Point)(ll : list Line)(x y : Point), PATH P L x y pl ll →InL (x, y) L → length ll≧ 2 → Epl((y,x)::ll)(((y,x)::ll)::((x,y)::(reverse_circuitll))::nil)((((y,x)::ll)::nil)::nil) circuit conditions 9th theorem proving and Provers meeting

  24. connect_circuit basecase→ connect_circuit basecase c2 a1 c1 c1’ outermost 9th theorem proving and Provers meeting

  25. Inductive PLCA variables | connect_circuit :forall(P : set Point)(L : set Line)(C : set Circuit)(A : set Area)(c : Circuit)(a : Area)(l1 l2 : Line)(x y z s e : Point)(AP : set Point)(AL : set Line)(ap pl pxypyzpzx : list Point)(al lxylyzlzx : list Line),E P L C A→set_In a A→set_In c a → (last C nil) <> c →PATH pl c x y pxylxy→ PATH pl c y z pyzlyz→ PATH pl c z x pzxlzx→c = lxy ++ lyz ++ lzx→ PATH AP AL s e ap al → length al >= 2 →(forall(p : Point), In p ap -> ~In p P) → E (ap++P)((y,s)::(e,z)::(al++L))((((s,y)::lyz)++((z,e)::(reverse_circuit al)))::(lxy++((y,s)::al)++((e,z)::lzx))::(set_removeeq_listline_dec c C))(((lxy++((y,s)::al)++((e,z)::lzx))::(set_removeeq_listline_dec c a))::((((s,y)::lyz)++((z,e)::(reverse_circuit al)))::nil)::(set_removeeq_listcircuit_dec a A)) conditions all points all lines all circuits all areas 9th theorem proving and Provers meeting

  26. isolate_circuit basecase→ isolate_circuit→ connect_circuit Basecase→ isolate_circuit basecase c4 c4 c2 a3 c3 a2 a1 c1 outermost 19 9th theorem proving and Provers meeting

  27. Inductive PLCA variables | isolate_circuit :forall(P : set Point)(L : set Line)(C : set Circuit)(A : set Area)(a : Area)(x y : Point)(AP : set Point)(AL : set Line)(ap : list Point)(al : list Line),E P L C A →set_In a A → PATH AP AL x y ap al → length al >= 2 → (forall(p : Point), In p ap -> ~In p P) → E(ap++P)(((y,x)::al)++L)(((y,x)::al)::((x,y)::(reverse_circuit al))::C)((((x,y)::(reverse_circuit al))::nil)::(((y,x)::al)::a)::(set_removeeq_listcircuit_dec a A)). conditions all points all lines all circuits all areas 9th theorem proving and Provers meeting

  28. Inductive PLCAconnect • PLCAconnect • PLCAconnect A B ∧ PLCAconnect B C iffPLCAconnect A C • PLCAconnect (point p) (Line l)iff p ∈ l.points • PLCAconnect (Line l)(Circuit c)iff l ∈ c.points • PLCAconnect (Circuit c)(Area a)iffc∈ a.points • PLCAconnect A B iffPLCAconnect B A Inductive object := | o_point : Point → object | o_line : Line → object | o_circuit : Circuit → object | o_area : Area → object. e.g.PLCAconnect (o_point p) (o_line l) PLCAconnect : object → object → Prop 9th theorem proving and Provers meeting

  29. Inductive PLCAconnect Inductive PLCAconnect(P : set Point)(L : set Line)(C : set Circuit)(A : set Area) (e : E P L C A) : object → object → Prop := | PLcon : | LPcon : | LCcon : | CLcon : | CAcon : | ACcon : | OPcon: | OLcon : | OCcon : | OAcon : 9th theorem proving and Provers meeting

  30. Inductive PLCAconnect Inductive PLCAconnect(P : set Point)(L : set Line)(C : set Circuit)(A : set Area) (e : E P L C A) : object → object → Prop := | PLcon : forall(p : Point)(l : Line), In p P→InL l L→Point_In_Line p l → PLCAconnect P L C A e (o_point p) (o_line l) 9th theorem proving and Provers meeting

  31. Inductive PLCAconnect Inductive PLCAconnect(P : set Point)(L : set Line)(C : set Circuit)(A : set Area) (e : E P L C A) : object → object → Prop := | PLcon : forall(p : Point)(l : Line), In p P→InL l L→Point_In_Line p l →PLCAconnect P L C A e (o_point p) (o_line l) | LPcon : forall(p : Point)(l : Line), In p P→InL l L→Point_In_Line p l → PLCAconnect P L C A e (o_line l) (o_point p) | LCcon : forall(l : Line)(c : Circuit), InL l L→ In c C→InL l →PLCAconnect P L C A e (o_line l) (o_circuit c) | CLcon : forall(l : Line)(c : Circuit), InL l L→ In c C→InL l c →PLCAconnect P L C A e (o_circuit c) (o_line l) | CAcon : forall(c : Circuit)(a : Area), In c C→ In a A→ In c a →PLCAconnect P L C A e (o_circuit c) (o_area a) | ACcon : forall(c : Circuit)(a : Area), In c C→ In a A→ In c a →PLCAconnect P L C A e (o_area a) (o_circuit c) 9th theorem proving and Provers meeting

  32. Inductive PLCAconnect Transitivity of PLCAconnect | OPcon : forall(p : Point)(o1 o2 : object), In p P→PLCAconnect P L C A e o1 (o_point p) →PLCAconnect P L C A e (o_point p) o2 → PLCAconnect P L C A e o1 o2 | OLcon : forall(l : Line)(o1 o2 : object), InL l L→PLCAconnect P L C A e o1 (o_line l) →PLCAconnect P L C A e (o_line l) o2→ PLCAconnect P L C A e o1 o2 | OCcon : forall(c : Circuit)(o1 o2 : object), In c C→PLCAconnect P L C A e o1 (o_circuit c) → PLCAconnect P L C A e (o_circuit c) o2 → PLCAconnect P L C A e o1 o2 | OAcon : forall(a : Area)(o1 o2 : object), In a A→PLCAconnect P L C A e o1 (o_area a) →PLCAconnect P L C A e (o_area a) o2→ PLCAconnect P L C A e o1 o2. 9th theorem proving and Provers meeting

  33. Inductive PLCAconnect Transitivity of PLCAconnect | OPcon : forall(p : Point)(o1 o2 : object), In p P→PLCAconnect P L C A e o1 (o_point p) →PLCAconnect P L C A e (o_point p) o2 → PLCAconnect P L C A e o1 o2 | OLcon : forall(l : Line)(o1 o2 : object), InL l L→PLCAconnect P L C A e o1 (o_line l) →PLCAconnect P L C A e (o_line l) o2→ PLCAconnect P L C A e o1 o2 | OCcon : forall(c : Circuit)(o1 o2 : object), In c C→PLCAconnect P L C A e o1 (o_circuit c) → PLCAconnect P L C A e (o_circuit c) o2 → PLCAconnect P L C A e o1 o2 | OAcon : forall(a : Area)(o1 o2 : object), In a A→PLCAconnect P L C A e o1 (o_area a) →PLCAconnect P L C A e (o_area a) o2→ PLCAconnect P L C A e o1 o2. 9th theorem proving and Provers meeting

  34. Agenda • Definition of PLCA • Inductive formalization • Proof of validity • Future work 9th theorem proving and Provers meeting

  35. Proof of validity • Planarity • Consistent PLCAconnected PLCA and |P| - |L| -|C| + 2 * |A| = 0 ⇔PLCA is planar (Takahashi et al,2008) • Inductive PLCA • Consistency • |P| - |L| -|C| + 2 * |A| = 0 • PLCAconnect Validity of PLCA 9th theorem proving and Provers meeting

  36. Consistency • Consistent PLCA • Consistency of Point-Line • ∀p ∈ P, ∃l ∈ L, p ∈ l.points • ∀ l ∈ L, p ∈ l.points→ p ∈ P • Consistency of Line-Circuit • ∀l ∈ L, ∃c ∈ C , l ∈ c.lines • ∀c ∈ C, l ∈ c.lines→ l∈ L • ∀l ∈ L, l ∈ c1.lines,l ∈ c2.lines → c1=c2 • Consistency of Circuit-Area • ∀c ∈ C, ∃a ∈ A , c∈ a.circuits • ∀a ∈ A, c ∈ a.circuits→ c∈ C • ∀c ∈ C, c ∈ a1.circuits, c ∈ a2.circuits → a1=a2 150 lines 100 lines 100 lines 1400 lines 9th theorem proving and Provers meeting

  37. Consistency Lemma six_consistent : forall{P : set Point}{L : set Line}{C : set Circuit}{A : set Area}(e : E P L C A), (forall(a : Area)(c : Circuit), set_In a A→ In c a →set_In c C) ∧ (forall(c : Circuit)(l : Line), set_In c C→ In l c →InL l L) ∧ (forall(p : Point)(l : Line), set_In l L→Point_In_Line p l →set_In p P) ∧ (forall(a : Area), ~set_In a (set_removeeq_listcircuit_dec a A)) ∧ (forall(c : Circuit)(a1 a2 : Area), c <> (outermost e) →set_In a1 A →set_In a2 A → In c a1 → In c a2 → a1 = a2) ∧ (forall(a : Area)(c : Circuit), set_In a A→ ~In c (set_removeeq_listline_dec c a)). Proved About 1000 lines 9th theorem proving and Provers meeting

  38. Proof of validity • Planarity • Consistent PLCAconnected PLCA and |P| - |L| -|C| + 2 * |A| = 0 ⇔PLCA is planar (Takahashi et al,2008) • Inductive PLCA • Consistency • PLCAconnect • |P| - |L| -|C| + 2 * |A| = 0 Validity of PLCA 9th theorem proving and Provers meeting

  39. PLCAconnect Theorem I_PLCA_connect : forall(P : set Point)(L : set Line)(C : set Circuit)(A : set Area)(e : E P L C A), (forall(p : Point)(l : Line), In p P→InL l L→PLCAconnect P L C A e (o_point p) (o_line l)) ∧(forall(l : Line)(c : Circuit), InL l L→ In c C→PLCAconnect P L C A e (o_line l) (o_circuit c)) ∧(forall(c : Circuit)(a : Area), In c C→ In a A→PLCAconnect P L C A e (o_circuit c) (o_area a)). Proved About 1500 lines without lemmas 9th theorem proving and Provers meeting

  40. Proof of validity • Planarity • Consistent PLCAconnected PLCA and |P| - |L| -|C| + 2 * |A| = 0 ⇔PLCA is planar (Takahashi et al,2008) • Inductive PLCA • Consistency • PLCAconnect • |P| - |L| -|C| + 2 * |A| = 0 Validity of PLCA 9th theorem proving and Provers meeting

  41. The number of objects • |P| - |L| - |C| + 2 * |A| = 0 • point– line– Circuit+ 2 * Area = 0 • easy to prove by induction • Because the set is represented by list in order to count the number of objects c1 l1 p1 p2 c2 |P| = 5 |L| = 5 |C| = 4 |A| = 2 5 – 5 - 4 + 2 * 2 = 0 c3 c4 a2 l4 l2 p5 l5 a1 p3 p4 l3 9th theorem proving and Provers meeting

  42. The number of objects Theorem PLCArelation : forall(P : set Point)(L : set Line)(C : set Circuit)(A : set Area)(e : E P L C A),length P + 2 * length A - length L - length C = 0. This theorem permutes position of “2 * legnth A” because nat doesn’t include negative numbers Proved About 100 lines 9th theorem proving and Provers meeting

  43. Proof of validity • Planarity • Consistent PLCAconnected PLCA and |P| - |L| -|C| + 2 * |A| = 0 ⇔PLCA is planar (Takahashi et al,2008) • Inductive PLCA • Consistency • PLCAconnect • |P| - |L| -|C| + 2 * |A| = 0 Validity of PLCA Inductive PLCA is planar? 9th theorem proving and Provers meeting

  44. Controversial definition • Sets of P, L, C, A are represented by lists • The reason is that proof of validity needs to count the number of objects • The arguments of inductive PLCA can’t permute elements of list • Original PLCA can permute elements because of set 9th theorem proving and Provers meeting

  45. Future work • Inductive PLCA has three validity, but don’t prove planarity • Consisntent , PLCAconnect and |P| - |L| -|C| + 2 * |A| = 0 → planarity • validity → Inductive PLCA • If proved, then validity and inductive PLCA are equivalent. • construct better inductive or another definition 9th theorem proving and Provers meeting

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