Abstract
This chapter begins with Turing’s model of computable functions, called Turing Machines, and presents them as upgraded pushdown automata. We give informal arguments that Turing machines can perform any computation that a modern computer can compute. We then discuss two different computational models: partial recursive functions and \(\lambda \)-calculus, which are shown as powerful as Turing machines. Next, we return to Turing machines, which play a central role in describing unsolvable problems. Finally, we focus on solvable problems and discuss the complexity issue—how much time and space are required to solve them.
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Notes
- 1.
The definition of completeness in this section is called “syntactic/maximal/negation completeness”. By contrast, the definition of completeness in previous sections only requires that all valid formulae can be derived, and we call it “semantic completeness”.
- 2.
The Latin phrase “ignoramus et ignorabimus” means that “we do not know and will not know”. It represents the idea that scientific knowledge is limited.
- 3.
Means “We must know, we shall know.” in English. Perhaps ironically, these are the words engraved on Hilbert’s tombstone.
- 4.
At that time, Turing was only 23 and had not obtained his PhD degree yet.
- 5.
By contrast, we assume that PDAs are nondeterministic.
- 6.
For example, 2 is written as 11 and 5 as 11111.
- 7.
Most say in 1934.
- 8.
This is an equivalent statement of Church’s thesis.
- 9.
Not just the above three computational models, but all the models for capturing “effectively computable” functions that people have developed since the 1930s, and many of them are of drastically different forms. See examples given below.
- 10.
The process of encoding components of functions into natural numbers is called Gödelisation, which is a technique Gödel used to prove his incompleteness theorems.
- 11.
Recall in Sect. 5.1 that this assumption does not reduce computational power.
- 12.
Clearly, if a problem is not recursively enumerable, it is (even harder than) undecidable. But in this case, we will be explicit and call it non-recursively enumerable (non-RE). The literature often refers to non-recursive but recursively enumerable problems as undecidable/semi-decidable problems.
- 13.
Theorem 5.5 indicates that the halting problem of TMs corresponds to the question “does a \(\lambda \)-term have normal form?”.
- 14.
It is called “recursively enumerable” because we can go through the input in the order of natural numbers and enumerate each input that is in the language. But given an arbitrary input, we may not be able to give an answer.
- 15.
\(\mathcal {P}\) versus \(\mathcal {NP}\) has a million-dollar prize funded by the Clay Mathematics Institute.
- 16.
Some may ask, can quantum computers solve NP-complete problems efficiently? If we go with our definition of “solving a problem efficiently” means solving it in polynomial time, then to the best of our knowledge, the answer is no. For example, with Grover’s search algorithm, a quantum computer can provide a worse-case quadratic speed-up than the best-known classical algorithm. However, \(O(\sqrt{c^n})\) is still exponential time. Confusions arise when the integer factorisation problem is involved. Shor’s algorithm, the best-known quantum algorithm for this problem, does run in polynomial time, and there is no known efficient classical algorithm for this problem. However, integer factorisation is not known to be NP-complete, although it is in \(\mathcal {NP} \).
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Hou, Z. (2021). Turing Machines and Computability. In: Fundamentals of Logic and Computation. Texts in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-030-87882-5_5
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