## AI helps you reading Science

## AI Insight

AI extracts a summary of this paper

Weibo:

# Lower Bounds for Existential Pebble Games and k-Consistency Tests

Logical Methods in Computer Science, no. 4 (2013): 25-34

EI

Full Text

Weibo

Keywords

Abstract

The existential k-pebble game characterizes the expressive power of the existential-positive k-variable fragment of first-order logic on finite structures. The winner of the existential k-pebble game on two given finite structures can easily be determined in polynomial time, where the degree of the polynomial is linear in k. We show that ...More

Code:

Data:

Introduction

- For two finite relational structures A and B the homomorphism problem asks if there is a mapping from the domain A of A to the domain B of B that preserves all relations.
- If Duplicator chooses a rule r not applicable to p and plays on RD(r), she will be penalized, because Spoiler wins immediately from position {(xi, xip(i)) | i ∈ [k]} (Lemma 9(iii)) by pebbling on yi for some i ∈ Tr(p).

Highlights

- For two finite relational structures A and B the homomorphism problem asks if there is a mapping from the domain A of A to the domain B of B that preserves all relations
- The existential k-pebble game characterizes the expressive power of the existential-positive k-variable fragment of first-order logic on finite structures
- We show that there is no O(n(k−3)/12)-time algorithm that decides which player can win the existential k-pebble game on two given structures
- As pointed out by Feder and Vardi [7] this problem is equivalent to the constraint satisfaction problem (CSP) where the variables correspond to the domain of A, the values correspond to the domain of B and the constraints are encoded in the relations of A and B
- One well-known method introduced in the context of constraint satisfaction is the procedure of establishing strong k-consistency, which can be implemented by an O(n2k)-time algorithm
- The lower bound applies to the k-pebble game that characterizes the expressive power of the existential k-variable fragment

Results

- For every rule r = (u, v, w, c, d) and position p : [k] → [n] the following holds in the existential (k + 1)-pebble game on RD(r): (i) If r ∈ appl(p), Spoiler can reach {(yi, yri(p)(i)) | i ∈ [k]} from {(xi, xip(i)) | i ∈ [k]}.
- The authors will see later that Spoiler has to restart the game, that is, he has to pick up all pebbles and start playing on the initialization gadget, if he reaches a position that is contained in a restart strategy.
- Assume that Spoiler reaches a critical position on the switch where Duplicator plays the input strategy, say M 1.
- In the existential (k + 1)-pebble game on Cm, (i) for every p : [k] → [n] Spoiler can reach one of the positions {((yl)i,ip(i)) | i ∈ [k]} for l ∈ [m] from {(xi, xip(i)) | i ∈ [k]}, and (ii) for every l ∈ [m], p : [k] → [n], T ⊆ [k], Duplicator has a winning strategy C(lp,T ) with boundary function {(xi, xi(p,T )(i)) | i ∈ [k]} ∪ {((yq)i,i0) | i ∈ [k], q ∈ [m] \ {l}} ∪ {((yl)i,i(p,T )(i)) | i ∈ [k]}.
- If Spoiler pebbles output critical positions in these strategies, Duplicator switches the strategies in the same way as the positions in the KAI-game change.
- If Spoiler plays incorrectly in the sense that he pebbles a restart critical position at the switches, Duplicator moves to a corresponding restart strategy.
- If Spoiler pebbles an output-critical position on M S(r), Duplicator can switch to strategy Dr(s) where position p = r(s) is on the y-vertices.

Conclusion

- To settle the complexity of deciding whether Spoiler has a winning strategy in the existential k-pebble game on σ-structures for fixed finite signatures σ, the authors use the following construction to switch from colored simple graphs to directed graphs.
- Lk-equivalence as well as Ck-equivalence is complete for polynomial time [9], but it is an open problem whether the problems are complete for EXPTIME when k is part of the input

Related work

- Kolaitis and Panttaja [11] proved that for every fixed k ≥ 2 the problem of determining the winner of the existential k-pebble game is complete for PTIME under LOGSPACE reductions. Furthermore, they established that the problem is complete for EXPTIME when k is part of the input. It follows that there is no algorithm for this problem whose running time is polynomial in the size of the structures as well as in the number of pebbles. Parameterized by the number of pebbles k, the problem is known to be W[1]-hard. This follows directly from the fact that a graph G contains a k-clique if and only if Duplicator has a winning strategy for the existential k-pebble game on the complete graph on k vertices and G. Thus, the existence of an algorithm of running time f (k)nc for some computable function f and constant c would imply W[1] = FPT, an unlikely event in parameterized complexity theory. However, since we do not know whether W[1] = FPT it is consistent with our previous knowledge that there exists an O(2kn2) algorithm determining the winner of the existential k-pebble game on two relational structures. Thus, for every fixed k, it was possible that there exists a quadratic time algorithm deciding if strong k-consistency can be established.

Reference

- Akeo Adachi, Shigeki Iwata, and Takumi Kasai. Some combinatorial game problems require Ω(nk) time. J. ACM, 31, March 1984.
- Albert Atserias, Andrei A. Bulatov, and Vıctor Dalmau. On the power of k -consistency. In Proc. ICALP’07, pages 279–290, 2007.
- Christoph Berkholz. Lower bounds for existential pebble games and k-consistency tests. In Proc. LICS’12, pages 25–34, 2012.
- Martin C. and Cooper. An optimal k-consistency algorithm. Artificial Intelligence, 41(1):89 – 95, 1989.
- Vıctor Dalmau, Phokion G. Kolaitis, and Moshe Y. Vardi. Constraint satisfaction, bounded treewidth, and finite-variable logics. In Proc. CP’02, pages 310–326, 2002.
- Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Springer-Verlag, 1999.
- Tomas Feder and Moshe Y. Vardi. The computational structure of monotone monadic snp and constraint satisfaction: A study through datalog and group theory. SIAM Journal on Computing, 28(1):57–104, 1998.
- Serge Gaspers and Stefan Szeider. The parameterized complexity of local consistency. In Proc. CP’11, pages 302–316, 2011.
- Martin Grohe. Equivalence in finite-variable logics is complete for polynomial time. In In Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, pages 264–273, 1996.
- Takumi Kasai, Akeo Adachi, and Shigeki Iwata. Classes of pebble games and complete problems. SIAM J. Comput., 8(4):574–586, 1979.
- Phokion G. Kolaitis and Jonathan Panttaja. On the complexity of existential pebble games. In Proc. CSL’03, pages 314–329, 2003.
- Phokion G. Kolaitis and Moshe Y. Vardi. On the expressive power of datalog: Tools and a case study. J. Comput. Syst. Sci., 51(1):110–134, 1995.
- Phokion G. Kolaitis and Moshe Y. Vardi. Conjunctive-query containment and constraint satisfaction. J. Comput. Syst. Sci., 61:302–332, October 2000.
- Phokion G. Kolaitis and Moshe Y. Vardi. A game-theoretic approach to constraint satisfaction. In Proc AAAI/IAAI’00, pages 175–181, 2000.

Tags

Comments

数据免责声明

页面数据均来自互联网公开来源、合作出版商和通过AI技术自动分析结果，我们不对页面数据的有效性、准确性、正确性、可靠性、完整性和及时性做出任何承诺和保证。若有疑问，可以通过电子邮件方式联系我们：report@aminer.cn