computer science

Symmetric Turing machine

Symmetric Turing machine is a computer science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Symmetric Turing machine rather than just read about it. In short: A symmetric Turing machine is a Turing machine that has a configuration graph that is undirected (that is, configuration i yields configuration j if and only if j yields i). Definition of symmetric Turing machines Formally, we define a variant of Turing machines with a set of transitions of the form ⁠ ( p , a b , D , c d , q ) {\displaystyle (p,ab,D,cd,q)} ⁠, where p,q are states, ab,cd are pairs of symbols and D is…

Key takeaways

Symmetric Turing machine belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Symmetric Turing machine to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Symmetric Turing machine from memory before moving on to harder problems.

Reference excerpt

A symmetric Turing machine is a Turing machine that has a configuration graph that is undirected (that is, configuration i yields configuration j if and only if j yields i).

Definition of symmetric Turing machines Formally, we define a variant of Turing machines with a set of transitions of the form ⁠ ( p , a b , D , c d , q ) {\displaystyle (p,ab,D,cd,q)} ⁠, where p,q are states, ab,cd are pairs of symbols and D is a direction. If D is left, then the head of a machine in state p above a tape symbol b preceded by a symbol a can be transitioned by moving the head left, changing the state to q and replacing the symbols a,b by c,d. The opposite transition ⁠ ( q , c d , − D , a b , p ) {\displaystyle (q,cd,-D,ab,p)} ⁠ can always be applied. If D is right the transition is analogous. The ability to peek at two symbols and change both at a time is non-essential, but makes the definition easier. Such machines were first defined in 1982 by Harry R. Lewis and Christos Papadimitriou, who were looking for a class in which to place USTCON, the problem asking whether there is a path between two given vertices s and t in an undirected graph. Until this time, it could be placed only in NL, despite seeming not to require nondeterminism (the asymmetric variant STCON was known to be complete for NL). Symmetric Turing machines are a kind of Turing machine with limited nondeterministic power, and were shown to be at least as powerful as deterministic Turing machines, giving an interesting case in between. ⁠ S T I M E ( T ( n ) ) {\displaystyle {\mathsf {STIME}}(T(n))} ⁠ is the class of the languages accepted by a symmetric Turing machine running in time ⁠ O ( T ( n ) ) {\displaystyle O(T(n))} ⁠. It can easily be proved that ⁠ S T I M E ( T ) = N T I M E ( T ) {\displaystyle {\mathsf {STIME}}(T)={\mathsf {NTIME}}(T)} ⁠ by limiting the nondeterminism of any machine in ⁠ N T I M E ( T ) {\displaystyle {\mathsf {NTIME}}(T)} ⁠ to an initial stage where a string of symbols is nondeterministically written, followed by deterministic computations.

SL=L

SSPACE(S(n)) is the class of the languages accepted by a symmetric Turing machine running in space ⁠ O ( S ( n ) ) {\displaystyle O(S(n))} ⁠ and SL=SSPACE(log(n)). SL can equivalently be defined as the class of problems logspace reducible to USTCON. Lewis and Papadimitriou showed this by first constructing a nondeterministic machine for USTCON with properties that they showed are sufficient to make a construction of an equivalent symmetric Turing machine possible. Then, they observed that any language in SL is logspace reducible to USTCON, as from the properties of the symmetric computation we can view the special configuration as the undirected edges of the graph. In 2004, Omer Reingold proved that SL=L by showing a deterministic algorithm for USTCON running in logarithmic space, for which he received the 2005 Grace Murray Hopper Award and (together with Avi Wigderson and Salil Vadhan) the 2009 Gödel Prize. The proof uses the zig-zag product to efficiently construct expander graphs.

Notes

References Lecture Notes :CS369E: Expanders in Computer Science By Cynthia Dwork & Prahladh Harsha Lecture Notes Sharon Bruckner Lecture Notes Deterministic Space Bounded Graph connectivity Algorithms Jesper Janson

Worked examples

Example 1 — a first encounter with Symmetric Turing machine

Start with the simplest possible case. Write down what Symmetric Turing machine claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Symmetric Turing machine before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Symmetric Turing machine ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Symmetric Turing machine

In research: Symmetric Turing machine appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Symmetric Turing machine in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Symmetric Turing machine is common in secondary-school and first-year university syllabi. It links to neighbouring topics Alan Turing, Computational complexity theory, Turing machine, so understanding it makes those chapters shorter.
In everyday life: Look for Symmetric Turing machine outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Symmetric Turing machine in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Symmetric Turing machine means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Symmetric Turing machine out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Symmetric Turing machine in simple terms?

A symmetric Turing machine is a Turing machine that has a configuration graph that is undirected (that is, configuration i yields configuration j if and only if j yields i). Definition of symmetric Turing machines Formally, we define a variant of Turing machines with a set of transitions of the for…

Why does Symmetric Turing machine matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Symmetric Turing machine?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Symmetric Turing machine.