Discrete Mathematics (VI) Yijia Chen Fudan University

Transcription

1 Discrete Mathematics (VI) Yijia Chen Fudan University

2 Review

3 Truth Assignments Definition A truth assignment A is a function that assigns to each propositional letter A a unique truth value A(A) {T, F}.

4 Truth Valuations Example A truth assignment of A and B and the corresponding valuation of (A B). A B (A B) T T T T F T F T T F F F

5 Assignments and Valuations Definition A truth valuation V is a function that assigns to each proposition α a unique truth value V(α) so that its value on a compound proposition is determined in accordance with the appropriate truth tables. More precisely: V(α β) = T iff V(α) = T or V(β) = T. V(α β) = T iff V(α) = T and V(β) = T. V( α) = T iff V(α) = F. V(α β) = T iff V(α) = F or V(β) = T. V(α β) = T iff ( V(α) = T and V(β) = T ), or ( V(α) = F and V(β) = F ).

6 Assignments and Valuations Theorem Given a truth assignment A there is a unique truth valuation V such that V(α) = A(α) for every propositional letter α. Proof. By induction on the structure of α: 1. If α is a propositional letter A, then 2. If α = α 1 α 2, then where {,,, }. V(α) = V(A) = A(A). V(α) = (the truth value of) V(α 1) V(α 2), 3. If α = α 1, then V(α) = V(α 1).

7 Assignments and Valuations Corollary If V 1 and V 2 are two valuations that agree on the support of α, the finite set of propositional letters used in the construction of α, then V 1(α) = V 2(α). Proof. V 1(α) is uniquely determined by V 1(A) for those A in the support of α. V 2(α) is uniquely determined by V 2(A) for those A in the support of α. V 1(A) = V 2(A) for those A in the support of α.

8 Tautologies Definition A proposition σ is said to be valid if for any valuation V we have V(σ) = T. Such a proposition is also called a tautology.

9 Tautologies Example α α is a tautology. α α α α T F T F T T It could happen that V(α) T for any valuation V, e.g., α = A A.

10 Logical Equivalence Definition Two proposition α and β such that, for every valuation V, V(α) = V(β) are called logically equivalent. We denote this by α β.

11 Logical Equivalence Example α β α β. α β α β T T T T F F F T T F F T α β α α β T T F T T F F F F T T T F F T T

12 Consequences Definition Let Σ be a (possibly infinite) set of propositions. We say that σ is a consequence of Σ (and write Σ = σ) if, for any valuation V, V(τ) = T for all τ Σ implies V(σ) = T.

13 Consequences Example 1. Let Σ = { A, A B }, we have Σ = B. 2. Let Σ = { A, A B, C }, we have Σ = B. 3. Let Σ = { A B}, we have Σ = B.

14 Models Definition We say that a valuation V is a model of Σ if V(σ) = T for every σ Σ. We denote by M(Σ) the set of all models of Σ. Example Let Σ = {A, A B}, we have models: 1. A(A) = T and A(B) = T. 2. A(A) = T, A(B) = T, and A(C) = T. 3. A(A) = T, A(B) = T, A(C) = F, and A(D) = F,.

15 Models Definition We say that propositions Σ is satisfiable if it has some model. Otherwise it is unsatisfiable. In case Σ = {α}, then α is invalid. Example 1. { A, A B } is satisfiable. 2. { A, A B, B C, C A } is unsatisfiable. 3. A A is invalid.

16 Proposition Let Σ, Σ 1, Σ 2 be sets of propositions. Let Cn(Σ) denote the set of consequences of Σ and Taut the set of tautologies. 1. Σ 1 Σ 2 implies Cn(Σ 1) Cn(Σ 2). 2. Σ Cn(Σ). 3. Taut Cn(Σ) = Cn(Cn(Σ)). 4. Σ 1 Σ 2 implies M(Σ 2) M(Σ 1). 5. Cn(Σ) = { σ V(σ) = T for all V M(Σ) }. 6. σ Cn({σ 1,..., σ n}) if and only if σ 1 (σ 2... (σ n σ)...) Taut.

17 Deduction Theorem Theorem For any propositions φ and ψ, Σ {ψ} = φ if and only if Σ = ψ φ holds.

18 Tableau Proofs

19 Terminologies Signed propositions. Entries of the tableau. Atomic tableaux.

20 Atomic Tableaux T A F A T (α β) T α T β F (α β) F α F β

21 Atomic Tableaux T (α β) T α T β T ( α) F α F (α β) F α F β F ( α) T α

22 Atomic Tableaux T (α β) F α T β T (α β) F (α β) T α F β F (α β) T α F α T α F α T β F β F β T β

23 An example F (((α β) (γ δ)) (α β)) F ((α β) (γ δ)) F (α β) F (α β) F (γ δ) F α F β F γ F δ F (α β) T α F β

24 Tableaux Definition A finite tableau is a binary tree, labeled with signed propositions called entries, such that: 1. All atomic tableaux are finite tableaux. 2. If τ is a finite tableau, P a path on 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 τ at the end of the path P, then τ is also a finite tableau. If τ 0, τ 1,..., τ n,... is a (finite or infinite) sequence of finite tableaux such that, for each n 0, τ n+1 is constructed from τ n by an application of (2), then τ = τ n is a tableau.

25 Definition Let τ be a tableau, P a path on τ, and E an entry occurring on P. 1. E has been reduced on P if all the entries on one path through the atomic tableau with root E occur on P. 2. P is contradictory if, for some proposition α, T α and F α are both entries on P. P is finished if it is contradictory or every entry on P is reduced on P. 3. τ is finished if every path through τ is finished. 4. τ is contradictory if every path through τ is contradictory.

26 Tableau Proofs Definition 1. A tableau proof of a proposition α is a contradictory tableau with root entry F α. A proposition is tableau provable, written if it has a tableau proof. α, 2. A tableau refutation for a proposition α is a contradictory tableau starting with T α. A proposition is tableau refutable if it has a tableau refutation.

27 A tableau proof of F ((α β) ( α β)). F ((α β) ( α β)) T (α β) F ( α β) F α F β T α T (α β) F α T β

28 Complete Systematic Tableaux Definition Let R be a signed proposition. We define the complete systematic tableau (CST) with root entry R by induction. 1. Let τ 0 be the unique atomic tableau with R at its root. 2. Assume that τ m has been defined. Let n be the smallest level of τ m containing an entry which is unreduced on some noncontradictory path in τ m and let E be the leftmost such entry of level n. 3. Let τ m+1 be the tableau constructed by adjoining the unique atomic tableau with root E to the end of every noncontradictory path of τ m on which E is unreduced. The union of the sequence τ 0, τ 1,..., τ m,... is our desired complete systematic tableau.

29 Theorem Every CST is finished. Proof. Reduce every possible E level by level and there is no E unreduced for any fixed level.

30 Theorem If τ = τ n is a contradictory tableau, then for some m, τ m is a finite contradictory tableau. Thus, in particular, if a CST is a proof, it is a finite tableau. Proof. By König lemma.

31 Definition Let α be a proposition. We define the degree d(α) of α by induction. 1. if α is a propositional letter, then d(α) = if α is β, then d(α) = d(β) if α is β γ, then d(α) = d(β) + d(γ) + 1. The degree of a signed proposition Tα or Fα is the degree of α. If P is a path in a tableau τ, then the degree d(p) of P is the sum of the degree of the signed propositions on P that are not reduced on P.

32 Theorem Every CST is finite. Proof. Let P be a path in τ. Then P = P m, where every P m is the corresponding path in τ m. Assume P m+1 strictly extends P m. Then a straightforward case analysis shows d(p m+1) < d(p m).

33 How to construct CST from premises? Definition 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. 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 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.

34 Tableau proofs 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.

35 Theorem Every CST from a set of premises is finished.