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Recall that NFAs are ‘essentially memoryless’, whilst NPDAs are equipped with memory in the form of a stack. To find the right kinds of machines for the top two Chomsky levels, we need to allow more general manipulation of memory. A Turing machine essentially consists of a finite-state control unit, equipped with a memory tape, infinite in both directions. Each cell on the tape contains a symbol drawn from a finite alphabet
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Linear bounded automataTuring machines The limits of computability: Church-Turing thesis
Informatics 2A: Lecture 29
John Longley
School of Informatics University of Edinburgh [email protected]
25 November, 2011
Linear bounded automataTuring machines The limits of computability: Church-Turing thesis
(^1) The Chomsky hierarchy: summary
2 Turing machines
3 Linear bounded automata
(^4) The limits of computability: Church-Turing thesis
Linear bounded automataTuring machines The limits of computability: Church-Turing thesis
What’s the motivation for the restriction |α| ≤ |β| in context-sensitive rules? Idea: in a context-sensitive derivation S ⇒ · · · ⇒ · · · ⇒ s, all the sentential forms are of length at most |s|. This means that if L is context-sensitive, and we’re trying to decide whether s ∈ L, we only need to consider possible sentential forms of length ≤ |s|. So intuitively, we have the problem under control, at least in principle. By contrast, without the length restriction, there’s no upper limit on the length of intermediate forms that might appear in a derivation of s. So if we’re searching for a derivation for s, how do we know when to stop looking? Intuitively, the problem here is wild and out of control. We’ll see this intuition made more precise as we proceed.
Linear bounded automataTuring machines The limits of computability: Church-Turing thesis
Recall that NFAs are ‘essentially memoryless’, whilst NPDAs are equipped with memory in the form of a stack. To find the right kinds of machines for the top two Chomsky levels, we need to allow more general manipulation of memory. A Turing machine essentially consists of a finite-state control unit, equipped with a memory tape, infinite in both directions. Each cell on the tape contains a symbol drawn from a finite alphabet Γ.
finite control
..... (^) : a c 3 a b 5 $ − ..... read, write, move L/R
Linear bounded automataTuring machines The limits of computability: Church-Turing thesis
To use a Turing machine T as an acceptor for a language over Σ, assume Σ ⊆ Γ, and set up the tape with the test string s ∈ Σ∗ written left-to-right starting at the read position, and with blank symbols everywhere else. Then let the machine run (maybe overwriting s), and if it enters the final state, declare that the original string s is accepted. The language accepted by T (written L(T )) consists of all strings s that are accepted in this way. Theorem: A set L ⊆ Σ∗^ is generated by some unrestricted (Type 0) grammar if and only if L = L(T ) for some Turing machine T. So both Type 0 grammars and Turing machines lead to the same class of recursively enumerable languages.
Linear bounded automataTuring machines The limits of computability: Church-Turing thesis
A Turing machine T consists of: A set Q of control states An initial state i ∈ Q A final (accepting) state f ∈ Q A tape alphabet Γ An input alphabet Σ ⊆ Γ A blank symbol − ∈ Γ − Σ A transition function δ : Q × Γ → Q × Γ × {L, R}. Numerous minor variations possible — differences unimportant. Can then define formally what’s meant by an accepting computation for a string s ∈ Σ∗^...
Linear bounded automataTuring machines The limits of computability: Church-Turing thesis
At the bottom level of the Chomsky hierarchy, it makes no difference: every NFA can be simulated by a DFA. At the top level, the same happens. We’ve defined a ‘deterministic’ version of Turing machines — but any ‘non-det Turing machine’ can be simulated by a det one. At the context-free level, there is a difference: we need NPDAs to account for all context-free languages. (Example: Σ∗^ − {ss | s ∈ Σ∗} is a context-free language whose complement isn’t context-free, see last lecture. However, if L is accepted by a DPDA then so is its complement — can just swap accepting and non-accepting states.) What about the context-sensitive level? Are NLBAs strictly more powerful than DLBAs? This is still an open question! (Can’t use the same argument because it turns out that CSLs are closed under complementation — only shown in 1988.) (^) 10 / 13
Linear bounded automataTuring machines The limits of computability: Church-Turing thesis
Suppose T is an LBA. How might we detect that s is not in L(T )? Clearly, if there’s an accepting computation for s, there’s one that doesn’t pass through exactly the same machine configuration twice (if it did, we could shorten it). Since the tape is finite, the total number of machine configurations is finite (though large). So in theory, if T runs for long enough without reaching the final state, it will enter the same configuration twice, and we may as well abort. Note that on this view, repeated configurations would be spotted not by T itself, but by ‘us watching’, or perhaps by some super-machine spying on T. For Turing machines with unlimited tape space, this reasoning doesn’t work. Is there some general way of spotting that a computation isn’t going to terminate ?? See next lecture...
Linear bounded automataTuring machines The limits of computability: Church-Turing thesis
The CT Thesis is a somewhat informal statement insofar as the general notion of a mechanical procedure isn’t formally defined (although we have a pretty good idea of what we mean by it). Although a certain amount of philosophical hair-splitting is possible, the broad idea behind CTT is generally accepted. At any rate, anything that can be done on any present-day computer (even disregarding time/memory limitations) can in principle be done on a TM. So if we buy into CTT, theorems about what TMs can/can’t do can be interpreted as fundamental statements about what can/can’t be accomplished by mechanical computation in general. We’ll see some examples of such theorems next time.
