Artificial Intelligence

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Frans Johannsen
4 år siden
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1 Artificial Intelligence Entailment and Algorithms [1]

2 Computational Complexity in a Quick Ride Entailment and Algorithms [2]

3 Turing Machine (A. Turing, 1937) Entailment and Algorithms [3]

4 Turing Machine (A. Turing, 1937) 0 Entailment and Algorithms [4]

5 Decisions and decidability (automation) K : <I, S> K c : S S S {0, 1} K K : I {0, 1} I wff wff I Entailment and Algorithms [5]

6 Decisions and decidability (automation) K Entailment and Algorithms [6]

7 An aside: The Halting Problem I H M I halt loop M Machine M Input I H halt / loop Entailment and Algorithms [7]

8 An aside: The Halting Problem I H M I halt loop M Machine M Input I H halt / loop H K halt halt H loop H Machine M Input I H loop? YES NO K halt do loop! Entailment and Algorithms [8]

9 An aside: The Halting Problem I H M I halt loop M Machine M Input I H halt / loop H K halt halt H loop H Machine M Input I H loop? YES NO K halt do loop! K K K K Entailment and Algorithms [9]

10 Computational complexity, wff Entailment and Algorithms [10]

11 Classes P, NP and NP-complete - The SAT problem P NP P( ) n O(P(n)) P P P NP Entailment and Algorithms [11]

12 Classes P, NP and NP-complete - The SAT problem NP-complete NP NP-complete NP K NP-complete NP K NP-complete K M(K) J K M(J) M(K) M(J) M(K) M(J) NP-complete P = NP P NP Entailment and Algorithms [12]

13 Entailment as a Decision Problem Entailment and Algorithms [13]

14 Transforming problems: entailment as satisfiability? {} W w() w() w({}) {, } w({}) = w( {}) = w() w({}) w( {}) = = Entailment and Algorithms [14]

15 Transforming problems: entailment as satisfiability {} W w() w() w({}) {, } w({}) = w( {}) = w() w({}) w( {}) = = {} wff ( {}) {} wff wff {} {} Entailment and Algorithms [15]

16 Satisfiability and decidability (in L P ) wff Entailment and Algorithms [16]

17 Exhaustive (Tree) Search Entailment and Algorithms [17]

18 Satisfiability and decidability (in L P ) (B D (A C)) B 1 0 D D A A A A C C C C C C C C Entailment and Algorithms [18]

19 Satisfiability and decidability (in L P ) (B D (A C)) B 1 0 D D A A A A C C C C C C C C (B D A C) Entailment and Algorithms [19]

20 Satisfiability and decidability (in L P ) (B D (A C)) B 1 0 D D A A A A C C C C C C C C (B D A C) O(2 n ) n Entailment and Algorithms [20]

21 Semantic Tableaux Entailment and Algorithms [21]

22 Semantic Tableau, alpha and beta rules wff L P wff Entailment and Algorithms [22]

23 Semantic Tableau, alpha and beta rules wff L P wff wff (a1) (a2) (a3) (a4) () ( ) ( ),,, (b1) (b2) (b3) (b4) (b5) ( ) ( ),,,, Entailment and Algorithms [23]

24 Semantic Tableau - a working example A (B C) B (A C)? {} A, B, A, C closed A (B C), (B (A C)) (a4) A (B C), B, (A C) (a4) A (B C), B, A, C B, B, A, C (b3) (B C), B, A, C (b3) C, B, A, C (a4) ( ), (b3) closed closed Entailment and Algorithms [24]

25 Semantic Tableau - a working example A (B C) B (A C)? {} A (B C), (B (A C)) B, (A C) A, C A (B C) x B C (a4) ( ), (b3) x x wff Entailment and Algorithms [25]

26 Semantic Tableau - algorithm recap {} wff Entailment and Algorithms [26]

27 Semantic Tableau - (required) algorithm properties ST ST ST Entailment and Algorithms [27]

28 Semantic Tableau - (required) algorithm properties ST ST Entailment and Algorithms [28]

29 Which method is faster? O(2 n ) A B C A 2 3 = 8 3 (A B) (A B) (A B) (A B) 2 2 = = Entailment and Algorithms [29]

30 Resolution Entailment and Algorithms [30]

31 Inference rule: Resolution,,, Entailment and Algorithms [31]

32 Normal forms wff wff 1 2 n i ( 1 2 n ) j (B D) (A C) C (B A C) (D A C) 1 2 n i 1 2 n j Entailment and Algorithms [32]

33 Conjunctive Normal Form,, ( ) ( ) ( ) ( ) (( ) ) (( ) ( )) (B D) (A C) B D (A C) B D A C (B D) (A C) (B D) (A C) (B D) (A C) (B A C) (D A C) Entailment and Algorithms [33]

34 Clausal Forms wff wff 1 2 n { 1, 2,, n } (B D) (A C) C {(B D), (A C), C} 1 2 n { 1, 2,, n } {{B, D}, {A, C}, {C}} A set of literals: ordering is irrelevant no multiple copies Entailment and Algorithms [34]

35 Resolution by refutation {} {} {} {} {C 1,C 2 } C C {C 1,C 2 } C { } Entailment and Algorithms [35]

36 Resolution by refutation B D A C, B C, A D, B D B D A C, B C, A D, B, D {B, D, A, C}, {B, C}, {A, D}, {B}, {D} {B, D, A, C} {B, C} {B} {A, D} {D} {C} {B, D, A} {D, A} {D} { } Refutation graph: shows relevant resolutions only Entailment and Algorithms [36]

37 Resolution by refutation B D A C, B C, A D, B D B D A C, B C, A D, B, D {B, D, A, C}, {B, C}, {A, D}, {B}, {D} {B, D, A, C} {B, C} {B} {A, D} {D} {D, A, C} {B,A, C} {C} {A} {D, C} {B, D, C} {B, D, A} {C} { } {B, C} {B} { } {B, D} {D, A} {D} { } Refutation graph: shows relevant resolutions only (but there are more) Entailment and Algorithms [37]

38 Resolution by refutation n O(2 n ) {} Entailment and Algorithms [38]

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