default
Generalized Hex and logical characterizations
of p olynomial space
a; b;
A A Arratia Quesada and I A Stewart
a
Department of Mathematics University of Wisconsin Madison
Madison WI U S A
b
Department of Mathematics and Computer Science Leicester University
Leicester LE RH U K
submitted to Information Processing Letters
We answer a question p osed by Makowsky and Pnueli and show
that the logic HEX FO where HEX is the op erator i e
s
uniform sequence of Lindstr om quanti ers corresp onding to the
well known PSPACE complete decision problem Generalized Hex
collapses to the fragment HEX FO and moreover that this logic
s
has a particular normal form which results in the problem HEX
b eing complete for PSPACE via quanti er free pro jections with
successor HEX is the rst natural problem to b e shown to have
this prop erty Our pro of of this normal form result is remark
ably similar to Immerman s original pro of that transitive closure
logic TC FO has such a normal form which is surprising
s
given that HEX FO captures PSPACE and TC FO
s s
captures NL We also show that HEX FO do es not capture
PSPACE and that this logic do es not have a corresp onding normal
form
keywords Key words Computational complexity Descriptive
complexity Finite mo del theory Logical characterizations of
p olynomial space Completeness via logical reductions
We consider a particular logical characterization of the complexity class
PSPACE using rst order logic with a built in successor relation extended
Partially supp orted by EPSRC Grant GR K
Most of this work was done whilst the author was visiting the University of Leicester
with an op erator corresp onding to the well known PSPACE complete deci
sion problem Generalized Hex that is the logic HEX FO It was shown
s
by Makowsky and Pnueli see also that any problem in PSPACE
can b e de ned by a sentence of the logic HEX FO and conversely that
s
any problem de nable by a sentence of this logic is in PSPACE
There are numerous other similar logical characterizations of complexity
classes that is using logics obtained by extending rst order logic with suc
cessor using op erators or more precisely uniform sequences of Lindstr om
quanti ers corresp onding to problems the rst such b eing Immerman s char
acterization of NL as those problems de nable in the now well studied tran
sitive closure logic TC FO To our knowledge for all of these other
s
characterizations more information is forthcoming that is the logics involved
p ossess normal forms and these normal forms yield strong complexity theoretic
completeness results However Makowsky and Pnueli s logical characterization
of PSPACE failed to establish such a normal form for the logic HEX FO
s
and they left it as an op en problem as to whether the normal form existed In
Theorem of this note we establish such a normal form for HEX FO
s
which yields as an immediate corollary that HEX is complete for PSPACE via
quanti er free pro jections also called pro jection translations with successor
Other problems have b een shown to b e complete for PSPACE via quanti er
free translations with successor in However these problems are rather
unnatural b eing based around the logical characterization of PSPACE as
partial xed p oint logic with successor in the sense that rst order logic
was augmented with a contrived op erator to try and mimic the application of
the partial xed p oint construct On the other hand the normal form results
for the logics in hold in the absence of a successor relation which was
the whole p oint of the research in those pap ers A complete problem for
PSPACE via quanti er free pro jections with successor was also exhibited in
although it was not explicitly stated there as b eing so but again this
problem was unnatural b eing based around a characterization of PSPACE
using a di erent inductive construct Our result that HEX is complete for
PSPACE via quanti er free pro jections or rst order translations for that
matter with successor is the rst such completeness result involving what
could b e called a natural problem
Not withstanding the preceding paragraph to our mind our actual pro of of
Theorem is the most interesting asp ect of this note given that it is essentially
identical to Immerman s pro of in that transitive closure logic or as was
proven there the p ositive version TC FO has a normal form except that
s
in the combinatorial construction we replace an edge in one of Immerman s
digraphs with a particular gadget see the pro of of Theorem This fact
encourages one to view the problem HEX as a game theoretic counterpart to
TC We intend to investigate this phenomenon more closely in future and hop e
to obtain criteria under which one can automatically transform a normal
form result for some logic FO which might capture NL for example
s
to the logic formed using the game theoretic version of which might
capture PSPACE for example
The complexity class PSPACE do es have logical characterizations in which
a successor relation or any other built in relation do es not app ear and con
sequently we have b een very careful ab ove in detailing when the successor
relation is present in our logics However as yet no problem natural or
otherwise has b een shown to b e complete for PSPACE via restricted logical
reductions in the absence of the successor relation Our nal result in this note
is that in the absence of the successor relation b oth the normal form and the
logical characterization in Theorem fail to hold Note that although the log
ics in have normal form results these logics do not capture PSPACE
in the absence of a successor relation
Given that nite mo del theory and descriptive complexity theory are now
rmly established in logic and theoretical computer science rather than give
de nitions here we simply refer the reader to the pap er for all de ni
tions and concepts regarding logics of the form FO and their relation
s
to complexity classes to the pap er for generalized Ehrenfeucht Fra ss e
games and their applicability to logics without built in relations and to the
b o ok for background issues
Now for our results Let the signature hE C D i where E is a binary
relation symbol and C and D are constant symbols our signatures never
contain function symbols The problem HEX consists of those structures
S for which Player has a winning strategy in the game of Generalized Hex
on S where the game of Generalized Hex is played as follows Starting with
Player two players take it in turns to colour previously uncoloured vertices
S S S
of the graph describ ed by E apart from C and D with Player using the
colour blue and Player using the colour red If at the end of the play there
S S
is a path from the source C to the sink D consisting entirely of blue
coloured vertices then Player wins otherwise Player wins The notion of
Player having a winning strategy should b e clear The problem HEX is well
known to b e complete for PSPACE via logspace reductions see for more
details
The problem TC consists of those structures S for which there is a path
S S S
in the digraph describ ed by E from the source C to the sink D
Theorem HEX FO PSPACE and every problem in PSPACE
s
can be de ned by a sentence of the form
HEX x y x y max
where jxj jy j k for some k is a quanti er free projection with suc
cessor and resp max is the constant symbol resp max repeated k
times
PROOF The result that HEX FO PSPACE is due to Makowsky
s
and Pnueli see also
Like the pro of of Theorem we pro ceed by induction on the complexity
of a sentence HEX FO The induction step assumes that every well
s
formed sub formula of is logically equivalent to a formula of the desired form
and then treats the di erent ways in which can b e built from its maximal
sub formulae in turn
Consider the case in the pro of of Theorem when is of the form
z TC x y x y max
where jxj jy j k for some k and is a quanti er free pro jection with z
amongst its free variables but di erent from those of x and y Let the under
lying signature of b e and let S b e some structure of size n The construc
tion in the pro of of Theorem takes copies of the digraphs D describ ed
z
S
by x y z where the vertices are k tuples over jS j f n g and
where z is given a value from jS j and strings them together to form the di
graph D by including an edge from the vertex max of D to the vertex of
z
D for each z f n g the vertex of D resp max of D is
z n
denoted as the source resp sink of the resulting digraph D Consequently D
has a path from its source to its sink i for each z f n g D has a
z
path from its vertex to its vertex max What is more it is shown in the
pro of of Theorem that the digraph D can b e describ ed in terms of S
uniformly by a quanti er free pro jection so that the source is and the sink
is max with the length of these tuples as dictated by the logical description
When dealing with the op erator HEX as opp osed to TC it is not enough
to simply rep eat the ab ove construction However by utilizing the following
gadget Immerman s construction can b e made to work
Let G b e some undirected graph with source a and sink b Let G b e
obtained from copies of G namely fG i g by amalgamating
i
the sources of G G G and G to form the vertex a
the sources of G G G and G to form the vertex a
the sinks of G G G and G to form the vertex b
the sinks of G G G and G to form the vertex b
We say that G has two sources a and a and two sinks b and b The
graph G can b e visualized as in Fig where each graph G is represented
i
as a b old line
a a 1 G 2 G4 5 GG 1 2 G 7 G8
GG6 3
b 1 b 2
Fig The graph G
By the game of Generalized Hex on G we mean the following Either Player
or Player starts with b oth players colouring vertices as usual except that
they can also colour the sources and the sinks of G Player has a winning
strategy if he has a strategy which ensures a path of blue coloured vertices
from at least one of the sources to at least one of the sinks in G note
that the source and the sink must b e coloured blue also
Lemma If Player has a winning strategy in the game of Generalized Hex
on G then Player has a winning strategy in the game of Generalized Hex on
G no matter whether Player or Player starts If Player has a winning
strategy in the game of Generalized Hex on G then Player has a winning
strategy in the game of Generalized Hex on G no matter whether Player
or Player starts
PROOF Supp ose that Player has a winning strategy in the game of Gen
eralized Hex on G and that Player starts the game of Generalized Hex on
G W l o g we may assume that Player plays in the copy G of G in G
a If Player s rst move has coloured the sink b resp source a red then
Player colours the sink b resp source a blue w l o g supp ose that b
has b een coloured red and b has b een coloured blue If Player replies by
colouring a vertex of G or G red then Player colours the source a blue
otherwise Player colours the source a blue W l o g we may assume that
a and b have b een coloured blue and no other vertex in G or G has b een
coloured If Player colours a vertex of G red then Player plays in G
according to his winning strategy on G and if Player do es not colour a
vertex of G red then Player plays in G according to his winning strategy
on G In any event Player wins the game of Generalized Hex on G
b If Player s rst move has coloured a vertex of G but not a or b then
Player colours the source a blue If Player replies by colouring a vertex
in G or G then Player colours the sink b blue otherwise Player colours
the sink b blue Player then wins as he did in the preceding paragraph
Reasoning similarly for the case when Player has a winning strategy in the
game of Generalized Hex on G and Player starts the game of Generalized
Hex on G yields that Player wins the game of Generalized Hex on G
The second part of the lemma follows similarly 2
Let b e as in the statement of the theorem and have underlying signature
Let S b e a structure of size n in which we interpret Let H b e de ned as
z
S
the undirected graph describ ed by x y z for each z f n g
with the vertex resp max b eing the source resp sink Build the graph
H by stringing together the graphs f H z n g similarly to as
z
was done ab ove to obtain the digraph D except by including edges joining
b oth sources of H to b oth sinks of H for z n and
z z
amalgamating the two sources resp sinks of H resp H to form
n
the source resp sink of H Note that H is undirected with one source and
one sink whereas D is a digraph The graph H can b e pictured as in Fig
. . . source sink
() H () H
() H0 1 n-1
Fig The graph H
Supp ose that Player has a winning strategy in the game of Generalized
Hex on H for each z f n g By Lemma Player has a
z
winning strategy in the game of Generalized Hex on the graph H for
z
each z f n g In the game of Generalized Hex on H Player s
winning strategy is to play according to his winning strategy on each of H
z
for z n as follows Player b egins by playing according to his
winning strategy on H in fact any H can b e adopted as the graph
z
in which Player plays rst In general if Player plays in H for some
z
z f n g then Player replies according to his winning strategy
on H note that Player has a winning strategy regardless of whether he
z
plays rst or not
Conversely supp ose that Player has a winning strategy in the game of Gen
eralized Hex on some graph H for z f n g By Lemma Player
z
has a winning strategy in the game of Generalized Hex on the graph H
z
That is Player has a sequence of moves so that when Player plays rst
in H and no matter how Player plays Player can not obtain a blue
z
coloured path from some source to some sink recall the source and sink must
b e coloured blue as well In the graph H Player simply plays this sequence
of moves in the subgraph H of H No matter how Player plays in H he
z
will never b e able to obtain a blue coloured path from some source of H
z
to some sink of H and consequently from the source of H to the sink of
z
H Hence Player has a winning strategy in the game of Generalized Hex on
the graph H
Consequently Player has a winning strategy for the game of Generalized
Hex on H i he has a winning strategy for the game of Generalized Hex on
H for each z f n g
z
In general the construction of G from a graph G of size n can b e describ ed
by a quanti er free pro jection The vertices of G are indexed by tuples
over f n g Roughly the rst comp onents of the tuple denote which
one of the copies G G G of G the vertex given by the th comp onent
resides with denoting G n denoting G n denoting
G and so on However the sources of G G G and G and of G G G
and G are amalgamated to form the sources of G and the sinks of G G
G and G and of G G G and G are amalgamated to form the sinks of
G We denote the sources of G by and n n n
i e the sources of G and G and the sinks of G by n and
n n n n i e the sinks of G and G The old sources and
sinks of G G G G G and G are left as isolated vertices The edges of
G can clearly b e de ned by a quanti er free pro jection For example the
edges in G emanating from a source are de ned by the formula
x x x x y y y max E x y
and the resulting quanti er free pro jection is just a disjunction of similar for
mula
Note that when we describ e our constructions using logical formula we often
introduce a number of isolated vertices The addition of isolated vertices to a
graph do es not make any di erence to the winner of the game of Generalized
Hex for b oth G or G
The description of H from the graphs f H z n g is then
z
done similarly with an extra comp onent added to the indexing tuples so as to
de ne which copy of H a particular vertex b elongs to The source of H is
z
obtained by amalgamating the two sources of H as ab ove and calling it
and the sink of H is obtained by amalgamating the two sinks of H as
n
ab ove and calling it max The reader is referred to for example where
some quanti er free pro jections are given explicitly
Hence as the notion of quanti er free pro jection is transitive a result due to
Immerman see Prop osition of the problem de ned by the sentence
z HEX x y x y max
can b e de ned by a sentence of the form
HEX x y x y max
where jx j jy j k for some k and is a quanti er free pro jection
The other cases for the construction of in the pro of of Theorem
except with HEX replacing TC can all b e cop ed with by mimicking Im
merman s construction except using the gadget depicted in Fig as is done
ab ove we leave this as an exercise Consequently the result follows 2
Corollary The problem HEX is complete for PSPACE via quanti er free
projections with successor 2
In the absence of a built in successor relation a result of Dawar and Gr adel
Theorem tells us that the logic HEX FO do es not capture PSPACE
as it has a law However this do es not rule out a normal form result for
HEX FO analogous to that in Theorem For de nitions relating to the
following theorem see
Prop osition There are problems de nable in HEX FO which can not
be de ned by a sentence of HEX FO in which the operator HEX does not
appear within the scope of the quanti er
PROOF By Theorem it su ces to show that for all p ositive integers
m there exist structures S and T such that
m m
S HEX and T HEX
m m
m
S T that is S and T satisfy exactly the same sentences of quan
m m m m
FO
ti er rank at most m
l
Fix m Let G a b b e the graph with vertex set fa b i l g fa bg
i i
and edge set
f a a b b i l g f a b i l g
i i i i i l i
f a b i l g f a a a b b b b a g
i l i l l
l
The graph G a b is essentially a ladder with diagonal rungs to which the
vertices a and b are joined at each end Let S and T each consist of
m m
l
disjoint copies of G a b except that the source of S is vertex a of the rst
m
copy and the sink of S is vertex b of the rst copy whereas the source of T
m m
is vertex a of the rst copy and the sink of T is the vertex b of the second
m
m
copy furthermore set l
Clearly S HEX but T HEX A simple induction shows that Duplicator
m m
has a winning strategy in the m p ebble Ehrenfeucht Fra ss egame on S and
m
m
T and so S T see for example Theorem Hence the result
m m m
FO
follows 2
References
S Abiteb oul and V Vianu Fixp oint extensions of rst order logic and datalog
like languages Proc th IEEE Symp on Logic in Computer Science
E Dahlhaus Skolem normal forms concerning the least xp oint Lecture Notes
in Computer Science Springer Verlag
A Dawar and E Gr adel Generalized quanti ers and laws Proc th IEEE
Ann Symp on Logic in Computer Science
H D Ebbinghaus and J Flum Finite Model Theory Springer Verlag
A Ehrenfeucht An application of games to the completeness problem for
formalized theories Fund Math
R Fra ss e Sur quelques classi cations des syst ems de relations Publ Sci
Univ Alger
M R Garey and D S Johnson Computers and Intractability A Guide to the
Theory of NP Completeness Freeman
M Grohe Complete problems for xed p oint logics J Symbolic Logic
H Imhof Fixed Point Logics and Generalized Quanti ers in Descriptive
Complexity Ph D Thesis Alb ert Ludwigs Universit at
N Immerman Languages which capture complexity classes SIAM J Comput
J A Makowsky and Y B Pnueli Oracles and quanti ers Lecture Notes in
Computer Science Vol Springer Verlag
J A Makowsky and Y B Pnueli Computable quanti ers and logics over
nite structures in M Krynicki M Mostowski and L W Szczerba eds
Quanti ers Generalizations Extensions and Variants of Elementary Logic
Kluwer Academic Publishers to app ear
I A Stewart Comparing the expressibility of languages formed using NP
complete op erators J Logic Computat
I A Stewart Metho ds for proving completeness via logical reductions
Theoret Comput Sci
I A Stewart Context sensitive transitive closure op erators Annals Pure App
Logic
I A Stewart Completeness of path problems via logical reductions
Inform Computat
I A Stewart Logics with zero one laws that are not fragments of b ounded
variable in nitary logic Math Logic Quart to app ear