Institute for Applied Information Processing and Communications (IAIK) Secure & Correct Systems. Decidability

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1 Decidability and the Undecidability of Predicate Logic IAIK Graz University of Technology 1

2 Fork of ways Brainteaser: Labyrinth Guards One to salvation One to perdition Two guards One always lies One always tells the truth Can t tell them apart One question What to ask? 2

3 But Beware! Image source: 3

4 Game: The Meta Game 1. Two players A, B 2. A starts, turns alternate 3. Always ends (Win or Draw) Examples: Tic-Tac-Toe, Connect-Four, but not Chess Meta-Game One Turn: Player picks a game Game is played (other player starts) Winner gets one point (both if Draw) Ends at 5 points Game? 4

5 Gödel s Incompleteness Theorem Kurt Gödel Jedes hinreichend mächtige formale System ist entweder widersprüchlich oder unvollständig. 5

6 Notions of Completeness Theory in Predicate Logic System of Axioms E.g. Theory of Arithmetic Proof System E.g. Natural Deduction 6

7 Completeness of a Proof System All True Sentences Provable and Nothing else provable Natural Deduction for Predicate Logic 7

8 Completeness of a Theory All Sentences Φ Provable (Φ is True ) or Negation provable (Φ is False ) (sufficiently complex) Theories in Predicate Logic 8

9 Enumerable set Σ Decision Problem Function f: Σ, Example: Σ = PPPPPPPPPPPPP FFFFFFFF, ii σ ii sssssssssss f σ = ooooooooo 9

10 Decidability Decision Problem Σ, f, Algorithm A: Decidable A always halts A computes f σ for all σ Σ Semi-Decidable Iff f σ = A halts A outputs For f σ = A may not halt 10

11 Example: Decidability Propositional SAT Problem Σ = PPPPPPPPPPPPP FFFFFFFF, ii σ ii sssssssssss f σ = ooooooooo Decidable e.g. A = DDDD Always halts Correct answer 11

12 Halting Problem Image source: Does program P halt? Decision problem Example programs: while(true){ }; print( Hello World ); while(n!=1) { if(n%2==0) n=n/2; else n=3*n+1; }? 12

13 Undecidability of HALT Proof by Contradiction Assume A A decides HALT P (A 0 (P) outputs iff P HALTs) Weird Program: weird() { if(a 0 (wwwww)) while(true){ } else exit(); } A 0 : wrong answer for weird() A A decides HALT 13

14 By Alan Turing (1936) Turing Machine Model of computation Formal notion of Algorithm Subsumes (modern) computers Infinite Memory ( Tape ) Universal Church-Turing Thesis 14

15 Formal Definition of a Turing Machine Set of States Q Input Alphabet Σ Tape Alphabet Γ Γ Σ Γ Transition Function δ: Q Γ Q Γ L, R Start State q o Q Accept State q aaa Q Reject State q rrr Q 15

16 Configuration of a Turing Machine State q Position of Head Tape Content uv u q v 16

17 Decision Problems Problem Reduction f: Σ, g: Δ, f redudes to g f g Use g to solve f h: Σ Γ h effectively computable f σ = g h σ 17

18 Problem Reduction & Decidability f g, g decidable f decidable Reduce it to g f g, f not decidable g not decidable Deciding f via g not possible 18

19 Decidability of Predicate Logic Claim: Halting Problem (for Turing Machines) (Validity of Formulas in) Predicate Logic Consequence: HALT not decidable Predicate Logic not decidable 19

20 Summary Predicate Logic is not decidable Predicate Logic is semi-decidable True statements can be proven Proofs can be checked Try all possible proofs 20

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