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Generalized Hex and Logical Characterizations of Polynomial Space

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Generalized Hex and Logical Characterizations of Polynomial Space default Generalized Hex and logical characterizations of p olynomial space a; b; AA ArratiaQuesada and IA Stewart a Department of MathematicsUniversity of WisconsinMadison Madison WI USA b Department of Mathematics and Computer Science Leicester University Leicester LE RH UK 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 ie s uniform sequence of Lindstrom quantiers corresp onding to the wellknown PSPACEcomplete 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 quantierfree 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 Immermans 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 rstorder logic with a builtin successor relation extended Partially supp orted by EPSRC Grant GRK Most of this work was done whilst the author was visiting the University of Leicester with an op erator corresp onding to the wellknown PSPACEcomplete 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 dened by a sentence of the logic HEX FO and conversely that s any problem denable 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 rstorder logic with suc cessor using op erators or more precisely uniform sequences of Lindstrom quantiers corresp onding to problems the rst such b eing Immermans char acterization of NL as those problems denable in the now wellstudied 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 complexitytheoretic completeness results However Makowsky and Pnuelis 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 quantierfree pro jections also called pro jection translations with successor Other problems have b een shown to b e complete for PSPACE via quantier free translations with successor in However these problems are rather unnatural b eing based around the logical characterization of PSPACE as partialxed p oint logic with successor in the sense that rstorder logic was augmented with a contrived op erator to try and mimic the application of the partialxed 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 quantierfree 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 dierent inductive construct Our result that HEX is complete for PSPACE via quantierfree pro jections or rstorder 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 Immermans 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 Immermans 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 gametheoretic 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 builtin 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 denitions here we simply refer the reader to the pap er for all deni tions and concepts regarding logics of the form FO and their relation s to complexity classes to the pap er for generalized EhrenfeuchtFrasse games and their applicability to logics without builtin 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 dened by a sentence of the form HEX x y x y max where jxj jy j k for some k is a quantierfree 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 subformula of is logically equivalent to a formula of the desired form and then treats the dierent ways in which can b e built from its maximal subformulae in turn Consider the case in the pro of of Theorem when is of the form z TCx y x y max where jxj jy j k for some k and is a quantierfree pro jection with z amongst its free variables but dierent 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 quantierfree 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 Immermans 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
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