Venn Diagrams for Boolean Algebra

Transcription

1 Venn Diagrams for oolean lgebra Renata de Freitas IME UFF Petrucio Viana II Workshop on Logic and Semantics Ilha Grande, ugust 26-30, 2013 This is a post-workshop version. We thank the participants of the II WLS for the fruitful discussions.

2 Outline Euler diagrams Venn diagrams Shin s system Venn II Venn diagrams for oolean algebra

3 Reasoning on categorical propositions The four different standard forms of categorical propositions are: () Every is. (O) Not every is. (I) Some is. (E) No is. where and are classes of objects.

4 ategorical propositions are naturally related to statements written in a fragment of the language of Set Theory: () Every is : (O) Not every is : (I) Some is : (E) No is : = We know some ways to reason about sets.

5 sk a mathematician: this is probably correct, but maybe incomplete. Translate to first order logic: this is provably correct and, depending of the target theory, complete. pply the algebra of sets: this can be made correct, complete, and mechanized but I don t like it! It does have a real road for inferences on categorical propositions?

6 Euler to Princess

7 Euler diagrams () Every is :

8 Euler diagrams (O) Not every is :

9 Euler diagrams (I) Some is :

10 Euler diagrams (E) No is : =

11 Example 1 ll men are mammals. ll mammals are mortal. ll men are mortal. =

12 Diagrammatic proof Unify superposing curves with the same label =

13 Diagrammatic proof Erase curve =

14 Example 2 Some men are wise. ll men are philosophers. Some philosophers are wise. =

15 Diagrammatic proof Unify superposing curves with the same label =?????????????????????????????????????????? Which choice we should make?

16 General diagrams General diagrams are diagrams representing every way an element of an universe can-or-cannot be an element of a certain number of subsets of this universe. In a general diagram, sets are represented by simple, closed and connected curves on the plane. In the simpler cases, these curves are represented by circles or elipses. not connected not simple not closed Let us call minimal region each region on the plane determined by the curves representing the sets that cannot be described as the union of other regions.

17 General diagram to 1 set: U The minimal regions in this diagram are and.

18 General diagram to 2 sets: U The minimal regions in this diagram are ( ), ( ), ( ) and ( ).

19 General diagram to 3 sets: U The minimal regions in this diagram are ( ), ( ), ( ), ( ), ( ), ( ), ( ) and ( ).

20 Proposition (Venn, 1880) Given any n N, there exists a general diagram to n sets. The general diagram to n sets has 2 n minimal regions. These diagrams can be difficult to draw, when n 4.

21 Venn diagrams ll men are mammals. ll mammals are mortal. ll men are mortal. =

22 Diagrammatic proof dd curve = =

23 Diagrammatic proof Unify superposing curves with the same labels =

24 Diagrammatic proof Erase shading =

25 Diagrammatic proof Erase curve =

26 Peirce s diagrams Some men are wise. ll men are philosophers. Some philosophers are wise. =

27 Diagrammatic proof dd curve = =

28 Diagrammatic proof Unify =

29 Diagrammatic proof Delete shaded =

30 Diagrammatic proof Erase shading =

31 Diagrammatic proof Erase curve =

32 Non-valid syllogism ll men are philosophers. Some philosophers are wise. ll men are wise. =

33 dd curve = =

34 Unify =

35 ??? = /

36 Possible conclusion =

37 Split -sequence =

38 ompound diagram Some men are wise, or some wise people are not men, or both.

39 Syntax Let Var be a set of variables for sets. unitary diagram is a triple d = (L, S, X ), where L Var is a finite set of labels, S P(L) is the set of shaded regions of d, and X P(P(L)) \ { } is the set of -sequences of d. d = (L, S, X ) L = {,, } S = {{}, {, }} X = {{{,, }, {, }}}

40 Syntax compound diagram is a finite set D of unitary diagrams. D = {d 1, d 2 } d 1 = (L 1, S 1, X 1 ) L 1 = {, } S 1 = X 1 = {{{, }}} d 2 = (L 2, S 2, X 2 ) L 2 = {, } S 2 = X 2 = {{{}}}

41 Syntax multi-diagram is a finite set of compound diagrams. = { {d}, {d 1, d 2 } }

42 Semantics model is a pair M = (M, I ), where M is a set and I : Var P(M). Let d = (L, S, X ) be a unitary diagram and M = (M, I ) be a model. We say that M satisfies d, denoted by M = d, when for each r S, we have {IY : Y r} {IZ : Z r} =, for each s X, we have { {IY : Y r} {IZ : Z r} : r s }.

43 Semantics Let D be a compound diagram and M = (M, I ) be a model. We say that M satisfies D, denoted by M = D, when M = d, for some d D. Let be a multi-diagram and M = (M, I ) be a model. We say that M satisfies, denoted by M =, when M = D, for every D.

44 Semantics Let Σ be a set of multi-diagrams and be a multi-diagram. We say that is a consequence of Σ, denoted by Σ =, when for every model M, if M = σ, for each σ Σ, then M =. Let be a multi-diagram. We say that is valid, denoted by =, when =.

45 Inference rules Rule 0 xiom {{(,, )}}

46 Inference rules Rule 1 Erase shading {D {(L, S S, X )}} = {D {(L, S, X )}} =

47 Inference rules Rule 1 Erase -sequence {D {(L, S, X X )}} = {D {(L, S, X )}} =

48 Inference rules Rule 1 Erase curve {D {(L, S, X )}} = {D {(L, S, X )}}, with L = L {c} S = {r P(L ) : r S and r = r {c}, for some r P(L) st r S and r r } X = {s P(L ) : s X } =

49 Inference rules Rule 2 Erase a shaded part of an -sequence {D {(L, S, X )}} = {D {(L, S, (X {T }) {T })}}, with T = T {s}, for some T X, s S st s T =

50 Inference rules Rule 3 Spread s {D {(L, S, X )}} = {D {(L, S, (X {T }) {T T })}} =

51 Inference rules Rule 4 dd a curve {D {(L, S, X )}} = {D {(L {c}, S +c,{t +c :T X })}}, with T +c = {s {c} : s T } =

52 Inference rules Rule 5 ontradiction {D {(L, S T, X {T })}} = \ =

53 Inference rules Rule 6 Unify {D {(L,S 1,X 1 ), (L,S 2,X 2 )}} = {D {(L, S 1 S 2, X 1 X 2 )}} =

54 Inference rules Rule 7 Split an -sequence {D {(L, S, X )}} = {D {(L, S, (X {T }) {T 1 }), (L, S, (X {T }) {T 2 })}}, if T X and {T 1, T 2 } is a partition of T. =

55 Inference rules Rule 8 Excluded middle {D {(L, S, X )}} = {D {(L, S T, X ), (L, S, X {T })}} =

56 Inference rules Rule 9 onnect a diagram {D {d}} = {D {d, d }} =

57 Inference rules Rule 10 Proof by ases {{d 1, d 2,..., d n }} = {{d}}, if {{d i }} = {{d}}, for each i = 1,..., n. d 1 = d d 2 = d d 1 d 2... d n = d if. d n = d

58 Let Σ be a set of multi-diagrams and be a multi-diagram. We say that is derivable from Σ, denoted by Σ, when there is a sequence ( 1,..., n ) such that 1 Σ { {{(,, )}} }, i+1 is a consequence of i by one of the inference rules, for all i = 1,..., n, n =.

59 Soundness and completeness Theorem (Shin / Hammer-Danner) Σ and multi-diagram, For all set of multi-diagrams Σ Σ =.

60 Problem How one can use this system to prove that a oolean equality, like is a valid? ( ) =,

61 Not so good solution Introduce new variables D, E, F, G, H, I, J, K and consider ( ) = iff = D, D = E, E = F, = G, G = H, H = I, I F = J, J = K = K = E

62 Example 3 Let s see how it works in a simpler example. = iff =, = = =

63 Example 3 =, = = = =

64 Diagrammatic proof dd curve = =

65 Diagrammatic proof Unify superposing curves with the same labels =

66 Diagrammatic proof Erase shading =

67 Diagrammatic proof Erase curve =

68 better ideia We propose an alternative diagrammatic system in which, to test the validity of an inclusion X Y, we can do the following: build the diagram of X, put the diagram of X in normal form, build the diagram of Y, put the diagram of Y in normal form, compare the normal forms. In our system, diagrams do not represent relations between sets. In our system, diagrams represent sets.

69 The normal form of the diagram of is the diagram of

70 Example 4 Let s go back to the not so simple example and use our system to prove that the following oolean equality is a valid. ( ) =

71 Diagram of X ( )

72 Putting the diagram of X in normal form ( ) = ( )

73 Putting the diagram of X in normal form ( ) = ( )

74 Putting the diagram of X in normal form ( ) = ( )

75 Putting the diagram of X in normal form ( ) =

76 Putting the diagram of X in normal form =

77 Putting the diagram of X in normal form =

78 Putting the diagram of X in normal form =

79 Putting the diagram of X in normal form =

80 Putting the diagram of X in normal form =

81 Putting the diagram of X in normal form =

82 Putting the diagram of X in normal form =

83 Putting the diagram of X in normal form =

84 Putting the diagram of X in normal form =

85 Putting the diagram of X in normal form =

86 Putting the diagram of X in normal form =

87 The normal form of the diagram of X

88 Putting the diagram of Y in normal form =

89 Putting the diagram of Y in normal form =

90 Putting the diagram of Y in normal form =

91 Putting the diagram of Y in normal form =

92 Putting the diagram of Y in normal form =

93 The normal form of the diagram of Y

94 omparing the normal forms =

95 Syntax Let Var be a set of variables for sets and Trm be the set of terms given by X := U O X X X X X unitary diagram is a pair d = (L, ), where L Trm is a finite set of labels and P(L). ompound diagrams are sets of unitary diagrams and multi-diagrams are sets of compound diagrams.

96 Example O d = (L, ) L = {, O, } = {{}, {, O}, {O}, {, O, }, }

97 Semantics model is a pair M = (M, I ), where M is a set and I : Var P(M). Let d = (L, ) be a unitary diagram and M = (M, I ) be a model. The meaning of d in M is [[d]] M = {Ib : b }, where Ib = {It : t b} {It : t b}. Define also [[D]] M = {[[d]] M : d D} [[ ]] M = {[[D]] M : d }

98 Diagram of a oolean term Let t Trm. The diagram of t is t = { {({t}, {{t}})} }. t Proposition For all term t Trm and model M, [[t]] M = [[ t ]] M.

99 Normal form Let be a multi-diagram and L Trm. We say that is in normal form w.r.t. L when = { {(L, )} } and L Var.

100 Rules to put a diagram in normal form U in {D {({U}, {{U}})}} {D {(, { })}} U

101 Rules to put a diagram in normal form U out {D {({U}, { })}} {D {(, )}} U

102 Rules to put a diagram in normal form O in {D {({O}, {{O}})}} {D {(, )}} O

103 Rules to put a diagram in normal form O out {D {({O}, { })}} {D {(, { })}} O

104 Rules to put a diagram in normal form in {D {({ }, {{ }})}} {D {({}, {{}})}, {({}, {{}})}}

105 Rules to put a diagram in normal form out {D {({ }, { })}} {D {({}, {{}}), ({}, {{}})}}

106 Rules to put a diagram in normal form in {D {({ }, { })}} {D {({}, {{}}), ({}, {{}})}}

107 Rules to put a diagram in normal form out {D {({ }, {{ }})}} {D {({}, {{}})}, {({}, {{}})}}

108 Rules to put a diagram in normal form in {D {({}, {{}})}} {D {({}, { })}}

109 Rules to put a diagram in normal form out {D {({}, { })}} {D {({}, {{}})}}

110 Rules to put a diagram in normal form dd curve {D {(L, )}} {D {(L {c}, {r {c} : r })}}

111 Rules to put a diagram in normal form Unify {D {(L, 2 ), (L, 2 )}} {D {(L, 1 2 })}} {D {(L, 2 )}, {(L, 2 )}} {D {(L, 1 2 })}}

112 Equivalence between diagrams Let 1 and 2 be multi-diagrams. We say that 1 and 2 are equivalent when, for every model M, [[ 1 ]] M = [[ 2 ]] M. Lemma For all diagram = (L, ) and L Var(L), there is a diagram st and are equivalent, is in normal form w.r.t. L.

113 Theorems We say that an inclusion 1 2 is a theorem, denoted by 1 2, when NF L 1 can be derived from NF L 2 by rule dd bullet (next slide). NF L 1 (respect. NF L 2 ) is a diagram in normal form w.r.t. L that is equivalent to 1 (respect. 2 ) and L = {Var(t) : t is a label in 1 or 2 }.

114 Rule to compare diagrams in normal form dd bullet {D {(L, )}} = {D {(L, })}} =

115 (Weak) Soundness and completeness Theorem For every inclusion 1 2, 1 2 [[ 1 ]] M [[ 2 ]] M, for every model M. Proof outline (weak completeness): Suppose 1 2. Hence, given NF L 1 = (L, 1 ) and NF L 2 = (L, 2 ), we have that there is some a L st a 1 and a 2. Take a model M = (M, I ) st M = {a} and { {a}, if X L and a X IX =, otherwise We have that [[ 1 ]] M = {a} [[ 2 ]] M.

116 Perpectives onsequence from hypotheses. Extension to non-inclusions X Y.