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1 Discrete Mathematics Yi Li Software School Fudan University June 12, 2017 Yi Li (Fudan University) Discrete Mathematics June 12, / 16
2 Review Soundness and Completeness Theorem Compactness Theorem Size of model Compactness theorem Yi Li (Fudan University) Discrete Mathematics June 12, / 16
3 Outline Application of Logic Limitation of First Order Logic Yi Li (Fudan University) Discrete Mathematics June 12, / 16
4 Application Example (linear order) A structure A =< A, <> is called an ordering if it is a model of the following sentences: Yi Li (Fudan University) Discrete Mathematics June 12, / 16
5 Application Example (linear order) A structure A =< A, <> is called an ordering if it is a model of the following sentences: Solution. ( x)( x < x), Φ ord = ( x)( y)( z)((x < y y < z) x < z), ( x)( y)(x < y x = y y < x). Yi Li (Fudan University) Discrete Mathematics June 12, / 16
6 Application(Cont.) Example (dense order) In order to describe dense linear orders, we could add into linear order the following sentence x y(x < y z(x < z z < y)) Yi Li (Fudan University) Discrete Mathematics June 12, / 16
7 Application(Cont.) Example (Graphs) Let L = {R} where R is a binary relation. We can characterize undirected irreflexive graphs with Yi Li (Fudan University) Discrete Mathematics June 12, / 16
8 Application(Cont.) Example (Graphs) Let L = {R} where R is a binary relation. We can characterize undirected irreflexive graphs with 1 x R(x, x), Yi Li (Fudan University) Discrete Mathematics June 12, / 16
9 Application(Cont.) Example (Graphs) Let L = {R} where R is a binary relation. We can characterize undirected irreflexive graphs with 1 x R(x, x), 2 x y(r(x, y) R(y, x)). Yi Li (Fudan University) Discrete Mathematics June 12, / 16
10 Application(Cont.) Example (Groups) Let L = {, e} where is a binary relation and e is a constant symbol. The class of group is described as Yi Li (Fudan University) Discrete Mathematics June 12, / 16
11 Application(Cont.) Example (Groups) Let L = {, e} where is a binary relation and e is a constant symbol. The class of group is described as 1 xe x = x e = x, Yi Li (Fudan University) Discrete Mathematics June 12, / 16
12 Application(Cont.) Example (Groups) Let L = {, e} where is a binary relation and e is a constant symbol. The class of group is described as 1 xe x = x e = x, 2 x y zx (y z) = (x y) z, Yi Li (Fudan University) Discrete Mathematics June 12, / 16
13 Application(Cont.) Example (Groups) Let L = {, e} where is a binary relation and e is a constant symbol. The class of group is described as 1 xe x = x e = x, 2 x y zx (y z) = (x y) z, 3 x yx y = y x = e. Yi Li (Fudan University) Discrete Mathematics June 12, / 16
14 Application(Cont.) Example (Equivalence relation) The equivalence relation can be formalized with a single binary relation symbols as follows: Yi Li (Fudan University) Discrete Mathematics June 12, / 16
15 Application(Cont.) Example (Equivalence relation) The equivalence relation can be formalized with a single binary relation symbols as follows: Solution. ( x)r(x, x), Φ equ = ( x)( y)(r(x, y) R(y, x), ( x)( y)( z)((r(x, y) R(y, z)) R(x, z)). Yi Li (Fudan University) Discrete Mathematics June 12, / 16
16 Application(Cont.) Example Suppose R is a binary relation. If it is non-trival, which means no isolated element, transitive and symmetric, then it must be reflexive. Yi Li (Fudan University) Discrete Mathematics June 12, / 16
17 Application(Cont.) Example Suppose R is a binary relation. If it is non-trival, which means no isolated element, transitive and symmetric, then it must be reflexive. Solution. We can represent these properties as: Yi Li (Fudan University) Discrete Mathematics June 12, / 16
18 Application(Cont.) Example Suppose R is a binary relation. If it is non-trival, which means no isolated element, transitive and symmetric, then it must be reflexive. Solution. We can represent these properties as: 1 nontriv = ( x)( y)r(x, y). Yi Li (Fudan University) Discrete Mathematics June 12, / 16
19 Application(Cont.) Example Suppose R is a binary relation. If it is non-trival, which means no isolated element, transitive and symmetric, then it must be reflexive. Solution. We can represent these properties as: 1 nontriv = ( x)( y)r(x, y). 2 sym = ( x)( y)(r(x, y) R(y, x)). Yi Li (Fudan University) Discrete Mathematics June 12, / 16
20 Application(Cont.) Example Suppose R is a binary relation. If it is non-trival, which means no isolated element, transitive and symmetric, then it must be reflexive. Solution. We can represent these properties as: 1 nontriv = ( x)( y)r(x, y). 2 sym = ( x)( y)(r(x, y) R(y, x)). 3 ref = ( x)r(x, x). Yi Li (Fudan University) Discrete Mathematics June 12, / 16
21 Application(Cont.) Example Suppose R is a binary relation. If it is non-trival, which means no isolated element, transitive and symmetric, then it must be reflexive. Solution. We can represent these properties as: 1 nontriv = ( x)( y)r(x, y). 2 sym = ( x)( y)(r(x, y) R(y, x)). 3 ref = ( x)r(x, x). 4 trans = ( x)( y)( z)((r(x, y) R(y, z)) R(x, z)). Yi Li (Fudan University) Discrete Mathematics June 12, / 16
22 Application(Cont.) Example Suppose R is a binary relation. If it is non-trival, which means no isolated element, transitive and symmetric, then it must be reflexive. Solution. We can represent these properties as: 1 nontriv = ( x)( y)r(x, y). 2 sym = ( x)( y)(r(x, y) R(y, x)). 3 ref = ( x)r(x, x). 4 trans = ( x)( y)( z)((r(x, y) R(y, z)) R(x, z)). Then {trans, sym, nontriv} = ref. Yi Li (Fudan University) Discrete Mathematics June 12, / 16
23 Expressibility Example Let L = {, +, <, 0, 1} and Th(N ) be the full theory of N. There is M = Th(N ) and a M such that a is larger than every member. Yi Li (Fudan University) Discrete Mathematics June 12, / 16
24 Expressibility Example Let L = {, +, <, 0, 1} and Th(N ) be the full theory of N. There is M = Th(N ) and a M such that a is larger than every member. Proof. Let L = L {c}, where c is a new constant symbol. We can construct a set of sentence S = {φ n = } {{ + 1 } < c, n 1}. n Then apply compactness theorem. Yi Li (Fudan University) Discrete Mathematics June 12, / 16
25 Limitation Example We can construct a L which can only hold in an infinite model. Let α = ( x)( y)(r(x, y) x y ( z)(r(x, z) R(z, y) z x z y)). Remark The notion of being finite can not be captured using the machinery of classical first-order logic according to the last Example and Theorem. Yi Li (Fudan University) Discrete Mathematics June 12, / 16
26 Limitation Example The property of being strongly-connected is not a first-order property of directed graphs. Yi Li (Fudan University) Discrete Mathematics June 12, / 16
27 Limitation Example The property of being strongly-connected is not a first-order property of directed graphs. Proof. Assume that sentence Φ SC represents the property of being strongly-connected. Define sentences Φ SL, Φ IN and Φ out as follows. Yi Li (Fudan University) Discrete Mathematics June 12, / 16
28 Limitation Example The property of being strongly-connected is not a first-order property of directed graphs. Proof. Assume that sentence Φ SC represents the property of being strongly-connected. Define sentences Φ SL, Φ IN and Φ out as follows. Let Φ SL = ( x)( E(x, x)). Yi Li (Fudan University) Discrete Mathematics June 12, / 16
29 Limitation Example The property of being strongly-connected is not a first-order property of directed graphs. Proof. Assume that sentence Φ SC represents the property of being strongly-connected. Define sentences Φ SL, Φ IN and Φ out as follows. Let Φ SL = ( x)( E(x, x)). Let Φ OUT = ( x)( y)( z)(e(x, y) E(x, z) y = z). Yi Li (Fudan University) Discrete Mathematics June 12, / 16
30 Limitation Example The property of being strongly-connected is not a first-order property of directed graphs. Proof. Assume that sentence Φ SC represents the property of being strongly-connected. Define sentences Φ SL, Φ IN and Φ out as follows. Let Φ SL = ( x)( E(x, x)). Let Φ OUT = ( x)( y)( z)(e(x, y) E(x, z) y = z). Let Φ IN = ( x)( y)( z)(e(y, x) E(z, x) y = z). Yi Li (Fudan University) Discrete Mathematics June 12, / 16
31 Limitation(continue) Proof. (Continued) Let Φ = Φ SC Φ SL Φ OUT Φ IN. Thus it describes the class of graphs that are strongly connected, have no self loops and have all vertices of in-degree and out-degree 1. This is clearly the class of cycle graphs (of finite size). By the previous theorem, there must be a infinite graph satisfying Φ. But it is impossible. The problem must be something wrong with Φ SC. So it can not described by predict logic. Yi Li (Fudan University) Discrete Mathematics June 12, / 16
32 Upward Skolem-Löwenheim theorem Theorem If S has an infinite model. Then for every set A there is a model of S which contains at least as many elements as A. Yi Li (Fudan University) Discrete Mathematics June 12, / 16
33 Upward Skolem-Löwenheim theorem Theorem If S has an infinite model. Then for every set A there is a model of S which contains at least as many elements as A. Idea. For each a A let c a be a new constant (i.e. c a L) such that for distinct a, b A. We show that the set S = S { (c a = c b )} of L C where C = {c a a A} is satisfiable. Yi Li (Fudan University) Discrete Mathematics June 12, / 16
34 About Examination 1 Propositional logic and predicate logic. 2 Open exam with two pieces of A4 cheat manuscript. Yi Li (Fudan University) Discrete Mathematics June 12, / 16
35 Next 1 Misc. 2 Q&A. Yi Li (Fudan University) Discrete Mathematics June 12, / 16
.. Discrete Mathematics. Yi Li. June 9, Software School Fudan University. Yi Li (Fudan University) Discrete Mathematics June 9, / 15
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