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Understanding Primitive Recursive Functions and Their Turing Machine Equivalents

This resource delves into the relationship between Primitive Recursive Functions (PRF) and Turing Machines (TM), establishing that every instance of PRF can be effectively represented as an equivalent Turing Machine instance. It details the foundational concepts like base functions, composition, iteration, and bounded minimization. The capabilities of TMs in handling n-ary functions through Turing computations are explained, using formal definitions and operations of base machines and submachines. This work provides valuable insights into computability theory and the transition between different computational models.

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Understanding Primitive Recursive Functions and Their Turing Machine Equivalents

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  1. Primitive Recursive ≤ Turing Machine • Every instance of Primitive Recursive can be replaced by an equivalent instance of Turing Machine • Primitive Recursive • Base functions • Composition • Iteration • Bounded Minimization

  2. Turing Computable • a Turing computation of some n-ary function F is to assume that the machine starts with a tape • containing the n inputs, x1, ... , xn in the form • …01x101x20…01xn0… • and ends with • …01x101x20…01xn01y0… • where y = F(x1, ... , xn).

  3. TM - Base Machine • R -- move right over any scanned symbol • L -- move left over any scanned symbol • 0 -- write a 0 in current scanned square • 1 -- write a 1 in current scanned square

  4. TM - SubMachine • R -- move right to next 0 • L -- move left to next 0 • Ck -- copies k-th preceding value.

  5. PR - SubMachine • Translate -- moves a value left one tape square • Shift -- shift a rightmost value left, destroying value to its left • Rotk -- Rotate a k value sequence one slot to the left

  6. PR - Base Function • Ca(x1,...,xn) = a : constant functions • (R1)aR • Iin(x1,...,xn) = xi : identity functions • Cn-i+1 • S(x) = x+1 : an increment function • C11R

  7. PR - Composition • If G, H1, … , Hk are already known to be Turing computable, then so is F, where • F(x1,…,xn) = G(H1(x1,…,xn), … , Hk(x1,…,xn))‏ • <1> if E(x1,…,xn) is Turing computable then so is E<m>(x1,…,xn, y1,…,ym) = E(x1,…,xn)‏ • Ln+m (Rotn+m)nRn+m E Ln+m+1 (Rotn+m)mRn+m+1 • <2> F can be defined by • H1 H2<1> H3<2> … Hk<k-1> G Shiftk

  8. PR - Minimization • If G is already known to be Turing computable, then so is F, where • F(x1,…,xn) = y (G(x1,…,xn, y) == 1)‏

  9. PR - Iteration • If G, H are already known to be primitive recursive, then so is F, where • F(0, x1,...,xn) = G(x1,...,xn)‏ • F(y+1, x1,...,xn) = H(y, x1,...,xn, F(y, x1,...,xn))‏ • Ln+2 T L T Rn+2 G Ln+3 L--->Rn+4 H Shift1Ln+3 1 L0 Rn+2 • 0 Rn+4

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