ITEC 420 - Section 3.3 - Church-Turing Thesis

Algorithm - Informal Definition

• What is an algorithm?


• Informal definition:
• Finite number number of steps
• Each step doable
• Eventually halts
• Could in theory be done with pencil and paper
• If you have enough of each and enough time

Informal Definitions Inadequate

• Informal definition long accepted (eg Euclid's algorithm for greatest common factor)


• But, informal definition not good enough for analysis of what algorithms are possible

Hilbert's Tenth Problem and Algorithm Definition

• Example: Hilbert's Tenth Problem
• Hilbert's Problems: 1900, 23 challenge problems for the 20th century
• Tenth Problem: Find an algorithm to determine if a polynomial has an integral root
• Result (1970): No such algorithm is possible


• Analysis and Proof of 10th problem requires a formal definition of an algorithm

Formal, Equivalent Definitions of Algorithm

• Formal Definitions of algorithms - 1936:
• Alonzo Church: λ calculus (mathematical function)
• Alan Turing: Turing Machines


• λ calculus and Turing Machines proved equivalent
• Other equivalent definitions have been developed

Church-Turing Thesis

• Church-Turing thesis: Informal notion of algorithm is the same as (any of) the formal definition(s)
• Result: Anything that can be computed (in the informal sense) can be computed with a Turing Machine (or with the λ calculus, or with ...)


• This is a Thesis:
• Could it be proved?
• Could it be disproved (ie shown to be false)?

Practical Consequence

• TM can implement any algorithm, so ...
• We can use pseudocode to specify a TM algorithm and we know that a TM can implement it
• Language Wars ???

CT Thesis and Language Wars

• Any language that can simulate a TM has equivalent power
• Called: Turing Complete Languages
• What languages can simulate a TM?
• Language wars???
• Any Turing Complete languages are equivalent in power
• although not in readability/writability
• Of course there are memory constraints, as well

Tenth Problem is NOT Decidable

• It's been proved that no algorithm exists for the tenth problem, which is ...


• Problem: language D = {p| p is a string that represents a polynomial with an integral root}
• D is recognizable, but no decidable


• Machine M:
1. Input is a polynomial p over variable x
2. Evaluate p with x set successively to 0, 1, -1, 2, -2, ...
3. If p evaluates to 0 for any x, accept


• M will halt if p has an integer root but loop otherwise, UNLESS
• we can limit the values of x


• For polynomials over one variable, a limit is known
• Thus M is a decider


• For polynomials over several variables, no limit exists
• Thus M (revised for several variables) is not a decider (but it is a recognizer)

Unrecognizable Languages

• In Chapter 4 we will see:


• That languages that are NOT Turing Recognizable must exist


• Some specific languages that are not Turing recognizable




• In the process we see how to prove that a language is not recognizable