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: TicTacToe, ConnectFour, but not Chess MetaGame 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 SemiDecidable 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 ChurchTuring 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 semidecidable True statements can be proven Proofs can be checked Try all possible proofs 20
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