Alternation

1
Alternation
Giorgi Japaridze Theory of
Computability
Section 10.3
2
Alternating Turing machines
10.3.a
Giorgi Japaridze Theory of Computability
Definition 10.16 An alternating Turing machine
is a nondeterministic TM with an additional
feature. Its states, except for the accept and
reject states, are divided into universal states
and existential states. When we run an
alternating TM on an input string, we label each
node of its nondeterministic computation tree
with ? or ?, depending on whether the
corresponding configuration contains a universal
or existential state. We determine acceptance by
designating a node to be accepting if it is
labeled with ? and all of its children are
accepting or if it is labeled with ? and at
least one of its children is accepting.
?
?
?
?
?
?
?
?
?
?
?
reject
?
?
?
reject
reject
accept
reject
reject
accept
reject
nondeterministic computation tree
alternating computation tree

3
ATIME and ASPACE defined
10.3.b
Giorgi Japaridze Theory of Computability
ATIME(t(n)) def L L is decided by an
O(t(n)) time alternating TM ASPACE(t(n)) def
L L is decided by an O(t(n)) space alternating
TM We further define AP, APSPACE and AL to be
the classes of languages that are decided by
alternating polynomial time, alternating
polynomial space, and alternating logarithmic
space TMs, respectively.
Example 10.19 Here is an alternating polynomial
time algorithm for the UNSATISFIABILITY problem
for Boolean formulas On input lt?gt 1.
Universally select all assignments to the
variables of ?. 2. For a particular
assignment, evaluate ?. 3. If ? evaluates to
0, accept otherwise reject.
4
MIN-FORMULA is in AP
10.3.c
Giorgi Japaridze Theory of Computability
Example 10.20 This example features a language
in AP that isnt known to be in NP or coNP. Two
Boolean formulas are said to be equivalent iff
they evaluate to the same value on all
assignments to their variables. A minimal formula
is one that has no shorter equivalent. Let
MIN-FORMULA lt?gt ? is a
minimal Boolean formula. The following is an
alternating polynomial time algorithm for this
language. On input lt?gt 1. Universally
select a formula ? that is shorter than ?. 2.
Existentially select an assignment to all
relevant variables. 3. Evaluate both ? and ?
on this assignment. 4. Accept if the formulas
evaluate to different values. Reject otherwise.
5
Main theorems
10.3.d
Giorgi Japaridze Theory of Computability
Theorem 10.21 a) For f(n) n we have
ATIME(f(n)) ? SPACE(f(n)) ? ATIME(f2(n)) .
b) For f(n) log n we
have ASPACE(f(n)) TIME(2O(f(n))).
Corollary
AL P AP
PSPACE
APSPACE EXPTIME

6
The polynomial time hierarchy
10.3.e
Giorgi Japaridze Theory of Computability
Alternating machines provide a way to define a
natural hierarchy of problems within the
class PSPACE.
Definition 10.27 Let i be a natural number. A
?i-alternating TM is an alternating TM that
contains at most i runs of universal or
existential steps, starting with existential
steps. A ?i-alternating TM is similar except
that it starts with universal steps.
?iTIME(f(n)) is defined as the class of
languages that a ?i-alternating TM can decide in
O(f(n)) time. Similarly for ?iTIME(f(n)). Simi
larly for ?iSPACE(f(n)) and ?iSPACE(f(n)). The
polynomial time hierarchy is the collection of
classes ?iP ?
?iTIME(nk) and ?iP ? ?iTIME(nk)
k
k
PH is defined as ? ?iP, which can be seen to be
the same as ? ?iP.
i
i
Clearly NP ?1P and coNP ?1P. Also,
MIN-FORMULA? ?2P.