Giorgi Japaridze
Theory of Computability
Alternation
Section 10.3
10.3.a
Giorgi Japaridze
Theory of Computability
Alternating Turing machines
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
nondeterministic
computation tree
∨
accept
∧
∨
∨
reject
reject
∧
∨
reject
alternating
computation tree
∨
accept
10.3.b
Giorgi Japaridze
Theory of Computability
ATIME and ASPACE defined
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 <>:
1. Universally select all assignments to the variables of .
2. For a particular assignment, evaluate .
3. If  evaluates to 0, accept; otherwise reject.”
10.3.c
Giorgi Japaridze
Theory of Computability
MIN-FORMULA is in AP
Example 10.20 This example features a language in AP that isn’t 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 = { <> |  is a minimal Boolean formula}.
The following is an alternating polynomial time algorithm for this language.
“On input <>:
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.”
10.3.d
Giorgi Japaridze
Theory of Computability
Main theorems
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
10.3.e
Giorgi Japaridze
Theory of Computability
The polynomial time hierarchy
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)).
Similarly for iSPACE(f(n)) and iSPACE(f(n)).
The polynomial time hierarchy is the collection of classes
 iP = 
iTIME(nk) and iP = k iTIME(nk)
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.