What can be computed What is a computation? by a computer? - PowerPoint PPT Presentation

Theory of Computer Science April 18, 2016 — D1. Turing-Computability Theory of Computer Science D1.1 Computations D1. Turing-Computability D1.2 Reminder: Turing Machines Malte Helmert D1.3 Turing-Computable Functions University of Basel D1.4 Examples April 18, 2016 D1.5 Summary M. Helmert (Univ. Basel) Theorie April 18, 2016 1 / 27 M. Helmert (Univ. Basel) Theorie April 18, 2016 2 / 27 Overview: Course Main Question contents of this course: ◮ logic � ⊲ How can knowledge be represented? Main question in this part of the course: ⊲ How can reasoning be automated? ◮ automata theory and formal languages � What can be computed ⊲ What is a computation? by a computer? ◮ computability theory ⊲ What can be computed at all? ◮ complexity theory ⊲ What can be computed efficiently? M. Helmert (Univ. Basel) Theorie April 18, 2016 3 / 27 M. Helmert (Univ. Basel) Theorie April 18, 2016 4 / 27

Overview: Computability Theory Further Reading (German) Computability Theory ◮ imperative models of computation: Literature for this Chapter (German) D1. Turing-Computability D2. LOOP- and WHILE-Computability D3. GOTO-Computability ◮ functional models of computation: Theoretische Informatik – kurz gefasst D4. Primitive Recursion and µ -Recursion by Uwe Sch¨ oning (5th edition) D5. Primitive/ µ -Recursion vs. LOOP-/WHILE-Computability ◮ Chapter 2.1 ◮ undecidable problems: ◮ Chapter 2.2 D6. Decidability and Semi-Decidability D7. Halting Problem and Reductions D8. Rice’s Theorem and Other Undecidable Problems Post’s Correspondence Problem Undecidable Grammar Problems G¨ odel’s Theorem and Diophantine Equations M. Helmert (Univ. Basel) Theorie April 18, 2016 5 / 27 M. Helmert (Univ. Basel) Theorie April 18, 2016 6 / 27 D1. Turing-Computability Computations Further Reading (English) Literature for this Chapter (English) D1.1 Computations Introduction to the Theory of Computation by Michael Sipser (3rd edition) ◮ Chapter 3.1 Notes: ◮ Sipser does not cover all topics we do. ◮ His definitions differ slightly from ours. M. Helmert (Univ. Basel) Theorie April 18, 2016 7 / 27 M. Helmert (Univ. Basel) Theorie April 18, 2016 8 / 27

D1. Turing-Computability Computations D1. Turing-Computability Computations Computation Formal Models of Computation What is a computation? Formal Models of Computation ◮ intuitive model of computation (pen and paper) ◮ Turing machines ◮ vs. computation on physical computers ◮ LOOP, WHILE and GOTO programs ◮ vs. formal mathematical models ◮ primitive recursive and µ -recursive functions In the following chapters we investigate In the next chapters we will models of computation for partial functions f : N k 0 → p N 0 . ◮ get to know these models and ◮ no real limitation: arbitrary information ◮ compare them according to their power. can be encoded as numbers German: M¨ achtigkeit German: Berechnungsmodelle M. Helmert (Univ. Basel) Theorie April 18, 2016 9 / 27 M. Helmert (Univ. Basel) Theorie April 18, 2016 10 / 27 D1. Turing-Computability Computations D1. Turing-Computability Reminder: Turing Machines Church-Turing Thesis Church-Turing Thesis D1.2 Reminder: Turing Machines All functions that can be computed in the intuitive sense can be computed by a Turing machine. German: Church-Turing-These ◮ cannot be proven (why not?) ◮ but we will collect evidence for it M. Helmert (Univ. Basel) Theorie April 18, 2016 11 / 27 M. Helmert (Univ. Basel) Theorie April 18, 2016 12 / 27

D1. Turing-Computability Reminder: Turing Machines D1. Turing-Computability Reminder: Turing Machines Formal Models of Computation Reminder: Deterministic Turing Machine (DTM) Definition (Deterministic Turing Machine) A deterministic Turing machine (DTM) is given by a 7-tuple M = � Q , Σ , Γ , δ, q 0 , � , E � with: Formal Models of Computation ◮ Q finite, non-empty set of states ◮ Turing machines ◮ Σ � = ∅ finite input alphabet ◮ LOOP, WHILE and GOTO programs ◮ Γ ⊃ Σ finite tape alphabet ◮ primitive recursive and µ -recursive functions ◮ δ : ( Q \ E ) × Γ → Q × Γ × { L , R , N } transition function ◮ q 0 ∈ Q start state ◮ � ∈ Γ \ Σ blank symbol ◮ E ⊆ Q end states M. Helmert (Univ. Basel) Theorie April 18, 2016 13 / 27 M. Helmert (Univ. Basel) Theorie April 18, 2016 14 / 27 D1. Turing-Computability Reminder: Turing Machines D1. Turing-Computability Turing-Computable Functions Reminder: Configurations and Computation Steps How do Turing Machines Work? ◮ configuration: � α, q , β � with α ∈ Γ ∗ , q ∈ Q , β ∈ Γ + D1.3 Turing-Computable Functions ◮ one computation step: c ⊢ c ′ if one computation step can turn configuration c into configuration c ′ ◮ multiple computation steps: c ⊢ ∗ c ′ if 0 or more computation steps can turn configuration c into configuration c ′ ( c = c 0 ⊢ c 1 ⊢ c 2 ⊢ · · · ⊢ c n − 1 ⊢ c n = c ′ , n ≥ 0) (Definition of ⊢ , i.e., how a computation step changes the configuration, is not repeated here. � Chapter C7) M. Helmert (Univ. Basel) Theorie April 18, 2016 15 / 27 M. Helmert (Univ. Basel) Theorie April 18, 2016 16 / 27

D1. Turing-Computability Turing-Computable Functions D1. Turing-Computability Turing-Computable Functions Computation of Functions? Turing Machines: Computed Function Definition (Function Computed by a Turing Machine) How can a DTM compute a function? A DTM M = � Q , Σ , Γ , δ, q 0 , � , E � computes the (partial) function ◮ “Input” x is the initial tape content f : Σ ∗ → p Σ ∗ for which: ◮ “Output” f ( x ) is the tape content (ignoring blanks for all x , y ∈ Σ ∗ : f ( x ) = y iff � ε, q 0 , x � ⊢ ∗ � � . . . � , q e , y � . . . � � at the left and right) when reaching an end state ◮ If the TM does not stop for the given input, with q e ∈ E . (special case: initial configuration � ε, q 0 , � � if x = ε ) f ( x ) is undefined for this input. German: DTM berechnet f Which kinds of functions can be computed this way? ◮ What happens if symbols from Γ \ Σ (e. g., � ) occur in y ? ◮ directly, only functions on words: f : Σ ∗ → p Σ ∗ ◮ What happens if the read-write head is not ◮ interpretation as functions on numbers f : N k 0 → p N 0 : on the first symbol of y at the end? encode numbers as words ◮ Is f uniquely defined by this definition? Why? M. Helmert (Univ. Basel) Theorie April 18, 2016 17 / 27 M. Helmert (Univ. Basel) Theorie April 18, 2016 18 / 27 D1. Turing-Computability Turing-Computable Functions D1. Turing-Computability Turing-Computable Functions Turing-Computable Functions on Words Encoding Numbers as Words Definition (Encoded Function) Let f : N k 0 → p N 0 be a (partial) function. The encoded function f code of f is the partial function f code : Σ ∗ → p Σ ∗ with Σ = { 0 , 1 , # } and f code ( w ) = w ′ iff Definition (Turing-Computable, f : Σ ∗ → p Σ ∗ ) ◮ there are n 1 , . . . , n k , n ′ ∈ N 0 such that A (partial) function f : Σ ∗ → p Σ ∗ is called Turing-computable ◮ f ( n 1 , . . . , n k ) = n ′ , if a DTM that computes f exists. ◮ w = bin ( n 1 ) # . . . # bin ( n k ) and German: Turing-berechenbar ◮ w ′ = bin ( n ′ ). Here bin : N 0 → { 0 , 1 } ∗ is the binary encoding (e. g., bin (5) = 101 ). German: kodierte Funktion Example: f (5 , 2 , 3) = 4 corresponds to f code ( 101#10#11 ) = 100 . M. Helmert (Univ. Basel) Theorie April 18, 2016 19 / 27 M. Helmert (Univ. Basel) Theorie April 18, 2016 20 / 27

D1. Turing-Computability Turing-Computable Functions D1. Turing-Computability Examples Turing-Computable Numerical Functions D1.4 Examples Definition (Turing-Computable, f : N k 0 → p N 0 ) A (partial) function f : N k 0 → p N 0 is called Turing-computable if a DTM that computes f code exists. German: Turing-berechenbar M. Helmert (Univ. Basel) Theorie April 18, 2016 21 / 27 M. Helmert (Univ. Basel) Theorie April 18, 2016 22 / 27 D1. Turing-Computability Examples D1. Turing-Computability Examples Example: Turing-Computable Functions (1) Example: Turing-Computable Functions (2) Example The following numerical functions are Turing-computable: Example ◮ succ : N 0 → p N 0 with succ ( n ) := n + 1 Let Σ = { a , b , # } . � The function f : Σ ∗ → p Σ ∗ with f ( w ) = w # w for all w ∈ Σ ∗ n − 1 if n ≥ 1 ◮ pred 1 : N 0 → p N 0 with pred 1 ( n ) := is Turing-computable. 0 if n = 0 � � blackboard n − 1 if n ≥ 1 ◮ pred 2 : N 0 → p N 0 with pred 2 ( n ) := undefined if n = 0 � blackboard/exercises M. Helmert (Univ. Basel) Theorie April 18, 2016 23 / 27 M. Helmert (Univ. Basel) Theorie April 18, 2016 24 / 27