Discrete Math. and Logic II. Turing Machines SFWR ENG 2FA3
Ryszard Janicki
Winter 2014
Acknowledgments: Material based on Automata and Computability by Dexter C. Kozen (Chapters 27 and 28).
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 1 / 19 Introduction
In this lecture we introduce the most powerful of the automata: Turing machines (TMs), It is named after Alan Turing, who invented them in 1936 Turing machines can compute any function normally considered computable TMs were invented long before real computers appeared They came at a time when mathematicians were trying to come to grips with the notion of eective computation Several alternative formalisms evolved in the attempt to nail down eectively computable
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 2 / 19 Introduction
These several alternative formalisms are: Turing machines (Alan Turing) handle strings over Σ
Post systems (Emil Post)
µ-recursive functions (Kurt Gödel) handle the natural numbers
A-calculus (Alonzo Church, Stephan C. Kleene) handles terms built from combinator symbols
Combinatory logic (Moses Schön nkel, Haskell B. Curry) handles terms built from combinator symbols
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 3 / 19 Introduction
The one that most closely resembles a modern computer is the Turing machine
Besides the most know model, there are also many variations nondeterministic multitape multidimensional tape two-way in nite tapes
They can all simulate one another
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 4 / 19 Introduction
All these mathematicians were looking at the same thing from dierent angles (Not a coincidence)
They soon came to the realization that the commonality among all these systems must be the notion of eective computability
Alonzo Church gave voice to this thought, and it has since become known as Church's thesis (or the Church-Turing thesis)
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 5 / 19 Introduction
It is not a theorem, but rather a declaration
All these formalisms capture precisely our intuition about what it means to be eectively computable in principle, no more and no less
Very likely that the most compelling development leading to the acceptance of Church's thesis was the Turing machine (model considered readily programmable)
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 6 / 19 Informal Description of Turing Machines
We will be considering a deterministic, one-tape Turing machine
This is the standard model
There are many variations, apparently more powerful or less powerful but in reality not (for more details on this point see textbook Lecture 30)
TM has a nite set of states Q a semi-in nite tape that is delimited on the left end by an endmarker ` and is in nite to the right a head that can move left and right over the tape, reading and writing symbols
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 7 / 19 Informal Description of Turing Machines
a b ba b a . . . .
Two−way, read/write
Q
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 8 / 19 Formal Description of Turing Machines
De nition A deterministic one-tape Turing machine is a 9-tuple s t r , where M = (Q, Σ, Γ, `, [ , δ, , , ) Q is a nite set (the states); Σ is a nite set (the input alphabet); Γ is a nite set (the tape alphabet) containing Σ as a subset; , the blank symbol; [ ∈ Γ − Σ `, the left endmarker; δ : Q × Γ −→ Q × Γ × {L, R}, the transition function; s ∈ Q, the start state; t ∈ Q, the accept state; and r ∈ Q, the reject state, r 6= t.
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 9 / 19 Formal Description of Turing Machines
Intuitively, δ (p, a) = (q, b, d) means, "When in state p scanning symbol a, write b on that tape cell, move the head in direction d, and enter state q.
The symbols L and R stand for left and right, respectively We restrict TMs so that
∀(p | p ∈ Q : ∃(q | q ∈ Q : δ (p, `) = (q, `, R)))
We also require that
∀(b | b ∈ Γ: ∃(c, c0 | c, c0 ∈ Γ: δ (t, b) = (t, c, d) ∧ δ (r, b) = (r, c0, d 0)))
We refer to the state set and transition function collectively as the nite control
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 10 / 19 Formal Description of Turing Machines
Example We are considering a TM that accepts the non-context-free set {anbncn | n ≥ 0}
Q = {s, q1, ··· , q10, t, r}
Σ = {a, b, c}
S Γ = Σ {`, [ , a}
a is just an extra useful symbol in the tape alphabet (NOT a special symbol)
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 11 / 19 Formal Description of Turing Machines
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 12 / 19 Con gurations and Acceptance
We de ne a con guration to be an element of ω y ∗ IN Q × {y [ | ∈ Γ } × A con guration is a global state giving a snapshot of all relevant information about a TM computation at some instant in time The con guration (p, z, n) speci es a current state p of the nite control, current tape contents z, and current position of the read/write head n ≥ 0 We usually denote con gurations by α, β, γ The start con guration on input x ∈ Σ∗ is the con guration s x ω 0 ( , ` [ , ) The last component 0 means that the machine is initially scanning the left endmarker `
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 13 / 19 Con gurations and Acceptance
De nition
A Turing Machine is said to halt on input x if it either accepts x or rejects x (the machine doesn't really grind to a halt in the literal sense)
It is possible that it neither accepts nor rejects, in which case it is said to loop on input x
A Turing machine is said to be total if it halts on all inputs
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 14 / 19 Con gurations and Acceptance
De nition We call a set of strings
recursively enumerable (r.e.) if it is L(M) for some Turing machine M
co-r.e. if its complement is r.e.
recursive if it is L(M) for some total Turing machine M
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 15 / 19 Con gurations and Acceptance
Example Consider the non-CFL {ww | w ∈ {a, b}∗} It is a recursive set, because we can construct a Turing machine that accepts all its strings The TM scans out to the rst blank symbol [ , counting the number of symbols mod 2 to make sure x is of even length and rejecting immediately if not It lays down a right endmarker a, then repeatedly scans back and forth over the input In each pass from right to left, it marks the rst unmarked a or b it sees with ' (prime to the left) In each pass from left to right, it marks the rst unmarked a or b it sees with ` (prime to the right) It continues this until all symbols are marked
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 16 / 19 Con gurations and Acceptance
Example On input aabbaaabba the initial tape contents are
` aabbaaabba [ [ [ ···
1 ` aabbaaabb´a a [ [ [ ···
2 ` `aabbaaabb´a a [ [ [ ··· 3 ´ ` `aabbaaabb´a a [ [ [ ··· 4 ´ ` `a`abbaaabb´a a [ [ [ ··· 5 ´´ ` `a`abbaaabb´a a [ [ [ ··· 6 ` ´´ ` `a`abbaaabb´a a [ [ [ ··· 7 ` ´´ ` `a`abbaa´abb´a a [ [ [ ··· 8 `` ´´ bbaaabba [ aa [ ` `` ´ ´ a [ ···
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 17 / 19 Con gurations and Acceptance
Example ((Continued))
1 `` ´´ ` `a`abba´a´abb´a a [ [ [ ··· 2 `` ´´ ` `a`abb`a´a´abb´a a [ [ [ ··· 3 `` ´´ abba [ abba [ [ [ ` [ ` ` ´ ´ a ··· 4 `` ´´ bba [ [ bba [ [ [ ` [ [ ` ´ a ··· 5 ` ´ ` [ [ [ b`a [ [ [ b´a a [ [ [ ···
6 a [ [ [ [ a [ [ [ ` [ [ [ [ ` ´ a ··· 7 It accepts ` [ [ [ [ [ [ [ [ [ [ a [ [ [ ···
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 18 / 19 Decidability and Semidecidability
De nition (Decidable) A property P of strings is said to be decidable if the set of all strings having property P is a recursive set.
A property P of strings is said to be decidable if there is a total Turing machine that accepts input strings that have property P and rejects those that do not. De nition (Semidecidable) A property P of strings is said to be semidecidable if the set of strings having property P is an r.e. set.
A property P is said to be semidecidable if there is a Turing machine that on input x accepts if x has property P and rejects or loops if not.
Ryszard Janicki Discrete Math. and Logic II. Turing Machines 19 / 19