Discrete Mathematics

Transcription

1 Discrete Mathematics Yi Li Software School Fudan University April 10, 2017 Yi Li (Fudan University) Discrete Mathematics April 10, / 27

2 Review Atomic tableaux CST and properties Yi Li (Fudan University) Discrete Mathematics April 10, / 27

3 Outline Deduction from premises Syntax and semantics Soundness theorem Completeness theorem Yi Li (Fudan University) Discrete Mathematics April 10, / 27

4 Consequence Definition Let Σ be a (possibly infinite) set of propositions. We say that σ is a consequence of Σ (and write as Σ = σ) if, for any valuation V, (V(τ) = T for all τ Σ) V(σ) = T. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

5 Consequence Yi Li (Fudan University) Discrete Mathematics April 10, / 27

6 Consequence 1 Let Σ = {A, A B}, we have Σ = B. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

7 Consequence 1 Let Σ = {A, A B}, we have Σ = B. 2 Let Σ = {A, A B}, we have Σ = B. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

8 Consequence 1 Let Σ = {A, A B}, we have Σ = B. 2 Let Σ = {A, A B}, we have Σ = B. 3 Let Σ = { A}, we have Σ = (A B). Yi Li (Fudan University) Discrete Mathematics April 10, / 27

9 Deductions from Premises How to construct CST from premises? Yi Li (Fudan University) Discrete Mathematics April 10, / 27

10 Deductions from Premises How to construct CST from premises? Definition (Tableaux from premises) Let Σ be (possibly infinite) set of propositions. We define the finite tableaux with premises from Σ by induction: 1 Every atomic tableau is a finite tableau from Σ Yi Li (Fudan University) Discrete Mathematics April 10, / 27

11 Deductions from Premises How to construct CST from premises? Definition (Tableaux from premises) Let Σ be (possibly infinite) set of propositions. We define the finite tableaux with premises from Σ by induction: 1 Every atomic tableau is a finite tableau from Σ 2 If τ is a finite tableau from Σ and α Σ, then the tableau formed by putting Tα at the end of every noncontradictory path not containing it is also a finite tableau from Σ. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

12 Deductions from Premises Definition (Tableaux from premises(cont.)) 3 If τ is a finite tableau from Σ, P a path in τ, E an entry of τ occurring on P and τ is obtained from τ by adjoining the unique atomic tableau with root entry E to the end of the path P, then τ is also a finite tableau from Σ. If τ 0,..., τ n,... is a (finite or infinite) sequence of finite tableaux from Σ such that, for each n 0, τ n+1 is constructed from τ n by an application of (2) and (3), then τ = τ n is a tableau from Σ. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

13 Tableau proof Definition A tableau proof of a proposition α from Σ is a tableau from Σ with root entry Fα that is contradictory, that is, one in which every path is contradictory. If there is such a proof we say that α is provable from Σ and write it as Σ α. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

14 Property of CST Theorem Every CST from a set of premises is finished. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

15 Syntax & Semantics Give you two Chinese characters 更衣, what s it mean? Yi Li (Fudan University) Discrete Mathematics April 10, / 27

16 Syntax & Semantics Give you two Chinese characters 更衣, what s it mean? It means change clothes in modern Chinese. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

17 Syntax & Semantics Give you two Chinese characters 更衣, what s it mean? It means change clothes in modern Chinese. It means go to washroom in ancient Chinese. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

18 Syntax & Semantics Give you two Chinese characters 更衣, what s it mean? It means change clothes in modern Chinese. It means go to washroom in ancient Chinese. Give an acronym IP, what s it mean? Yi Li (Fudan University) Discrete Mathematics April 10, / 27

19 Syntax & Semantics Give you two Chinese characters 更衣, what s it mean? It means change clothes in modern Chinese. It means go to washroom in ancient Chinese. Give an acronym IP, what s it mean? Internet Protocol in network. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

20 Syntax & Semantics Give you two Chinese characters 更衣, what s it mean? It means change clothes in modern Chinese. It means go to washroom in ancient Chinese. Give an acronym IP, what s it mean? Internet Protocol in network. Integer Programming in operation research. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

21 Syntax & Semantics Give you two Chinese characters 更衣, what s it mean? It means change clothes in modern Chinese. It means go to washroom in ancient Chinese. Give an acronym IP, what s it mean? Internet Protocol in network. Integer Programming in operation research. Interactive proof in complexity. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

22 Syntax & Semantics Give you the following programming segments: Yi Li (Fudan University) Discrete Mathematics April 10, / 27

23 Syntax & Semantics Give you the following programming segments: 1 in C, printf( Hello World! ); Yi Li (Fudan University) Discrete Mathematics April 10, / 27

24 Syntax & Semantics Give you the following programming segments: 1 in C, printf( Hello World! ); 2 in Java, system.print( Hello World! ); Yi Li (Fudan University) Discrete Mathematics April 10, / 27

25 Syntax & Semantics Give you the following programming segments: 1 in C, printf( Hello World! ); 2 in Java, system.print( Hello World! ); 3 in C++, cout<< Hello World! ; Yi Li (Fudan University) Discrete Mathematics April 10, / 27

26 Syntax & Semantics Give you the following programming segments: 1 in C, printf( Hello World! ); 2 in Java, system.print( Hello World! ); 3 in C++, cout<< Hello World! ; All of them just output Hello World! on the screen. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

27 Syntax & Semantics in PL What s syntax? Yi Li (Fudan University) Discrete Mathematics April 10, / 27

28 Syntax & Semantics in PL What s syntax? What s semantic? Yi Li (Fudan University) Discrete Mathematics April 10, / 27

29 Syntax & Semantics in PL What s syntax? What s semantic? What s relationship between them? Yi Li (Fudan University) Discrete Mathematics April 10, / 27

30 Soundness Consider Pierce Law ((A B) A) A. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

31 Soundness Consider Pierce Law ((A B) A) A. Give its tableau proof. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

32 Soundness Consider Pierce Law ((A B) A) A. Give its tableau proof. Give its truth table. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

33 Sign & Noncontradictory Path Given proposition ((A B) (A C)) (B C), there is a truth valuation which make it false. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

34 Sign & Noncontradictory Path Given proposition ((A B) (A C)) (B C), there is a truth valuation which make it false. Consider the tableau with the root as F ((A B) (A C)) (B C) Yi Li (Fudan University) Discrete Mathematics April 10, / 27

35 Soundness Lemma If V is a valuation that agrees with the root entry of a given tableau τ given as τ n, then τ has a path P every entry of which agrees with V. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

36 Soundness(Cont.) Theorem (Soundness) If α is tableau provable, then α is valid, i.e. α α. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

37 Soundness of deductions from premises Lemma If a valuation V makes every α Σ true and agrees with the root of a tableau τ from Σ, then there is a path in τ every entry of which agrees with V. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

38 Soundness of deductions from premises Theorem If there is a tableau proof of α from a set of premises Σ, then α is a consequence of Σ, i.e. Σ α Σ α. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

39 Completeness Given proposition ((A B) (A C)) (B C), there is a truth valuation which make it false. Observe the non-contradictory path of the tableau with the root as F ((A B) (A C)) (B C) Yi Li (Fudan University) Discrete Mathematics April 10, / 27

40 Completeness Lemma Let P be a noncontradictory path of a finished tableau τ. Define a truth assignment A on all propositional letters A as follows: Yi Li (Fudan University) Discrete Mathematics April 10, / 27

41 Completeness Lemma Let P be a noncontradictory path of a finished tableau τ. Define a truth assignment A on all propositional letters A as follows: 1 A(A) = T if TA is an entry on P. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

42 Completeness Lemma Let P be a noncontradictory path of a finished tableau τ. Define a truth assignment A on all propositional letters A as follows: 1 A(A) = T if TA is an entry on P. 2 A(A) = F otherwise. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

43 Completeness Lemma Let P be a noncontradictory path of a finished tableau τ. Define a truth assignment A on all propositional letters A as follows: 1 A(A) = T if TA is an entry on P. 2 A(A) = F otherwise. If V is the unique valuation extending the truth assignment A, then V agrees with all entries of P. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

44 Completeness(Cont.) Theorem (Completeness) If α is valid, then α is tableau provable, i.e. α α. In fact, any finished tableau with root entry Fα is a proof of α and so, in particular, the complete systematic tableaux with root Fα is such a proof. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

45 Completeness of deduction from premises Lemma Let P be a noncontradictory path in a finished tableau τ from Σ. Define a valuation V as the last section, then it agrees with all entries on P and so in particular makes every proposition β Σ true. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

46 Completeness of deduction from premises Theorem If α is consequence of a set Σ of premises, then there is a tableau deduction of α from Σ, i.e., Σ α Σ α. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

47 Hilbert Proof System Definition The axioms of Hilbert system are all propositions of the following forms: 1 (α (β α)) 2 ((α (β γ)) ((α β) (α γ))) 3 ( β α) (( β α) β) Yi Li (Fudan University) Discrete Mathematics April 10, / 27

48 The Rule of Inference Definition (Modus Ponens) From α and α β, we can infer β. This rule is written as follows: α α β β Yi Li (Fudan University) Discrete Mathematics April 10, / 27

49 Hilbert Proof System Definition Let Σ be a set of propositions. 1 A proof from Σ is a finite sequence α 1, α 2,..., α n such that for each i n either: 1 α i is a member of Σ. 2 α i is an axiom; or 3 α i can be inferred from some of previous α j by an application of a rule of inference. 2 α is provable from Σ, Σ H α, if there is a proof α 1, α 2,..., α n from Σ where α n = α. 3 A proof of α is simply a proof from the empty set 0; α is provable if it is provable from 0. Yi Li (Fudan University) Discrete Mathematics April 10, / 27

50 Next Class Compactness Application Yi Li (Fudan University) Discrete Mathematics April 10, / 27