Definition of Symmetric Turing machines

A Symmetric TM is a TM which has a configuration graph

Configuration graph

Configuration graphs are a theoretical tool used in computational complexity theory to prove a relation between graph reachability and complexity classes.- Definition :...

 that is undirected. That is configuration i yields configuration j if and only if j yields
i. The set of languages that have a symmetric TM deciding it in log space is called SL

SL (complexity)

In computational complexity theory, SL is the complexity class of problems log-space reducible to USTCON , which is the problem of determining whether there exists a path between two vertices in an undirected graph, otherwise described as the problem of determining whether two vertices are in the...

. It was first defined in 1982 by Lewis and Papadimitriou, who were looking for a class in which to place USTCON, which until this time could, at best, be placed only in NL

NL (complexity)

In computational complexity theory, NL is the complexity class containing decision problems which can be solved by a nondeterministic Turing machine using a logarithmic amount of memory space....

, despite seeming not to require nondeterminism. They defined the symmetric Turing machine, used it to define SL. Symmetric Turing machines are kind of Turing machines with limited nondeterministic power. Informally Symmetric computations are reversible in an approximate manner. A symmetric computation is divided into locally deterministic sub computation (segments) between special configuration, where nondeterminism takes place, such that the only special configuration reached from a backward computation is the special configuration started the segment. Thus the segments are locally deterministic in its forward direction and have limited nondeterminism in its backward direction. Also if there is a way from special configuration A1 to special configuration A2, there must be a way back from A2 to A1.

Example of Symmetric Configuration

Symmetric log space complexity

• SSPACE(S(n)) is the class of the languages accepted by a symmetric Turing machine running in space O(S(n))
• SL is the class of problems solvable by a nondeterministic Turing machine in logarithmic space, such that
  1. If the answer is 'yes,' one or more computation paths accept.
  2. If the answer is 'no,' all paths reject.
  3. If the machine can make a nondeterministic transition from configuration A to configuration B, then it can also transition from B to A. (This is what 'symmetric' means.)
  4. It was proved that SL = CoSL.

Idea about why USTCON is SL-complete

The most important Complete Problem is the USTCON, Lewis and Papadimitriou by
their definition showed that USTCON is complete for SL class. They
constructed a nondeterministic machine for USTCON, and they made a lemma for
converting this machine into Symmetric Turing Machine. Then the theorem follows as
any language can be accepted using a symmetric Turing machine 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.