The Set of Virtual Connections in the Game of Hex is PSPACE-Complete - PDF document

The Set of Virtual Connections in the Game of Hex is PSPACE-Complete Stefan Kiefer August 1, 2003 Abstract It is argued in a semi-formal way that the set of virtual connections in hex games is PSPACE-complete. A conclusion is that an exhaustive search for virtual connections in a concrete game situation is (in general) not feasible in polynomial time. This explains theoretically the necessity of heuristics in Hex playing programs that search for virtual connections. This paper is a short version of a comprehensive theoretical analysis of virtual connections which I made as a part of my studies at Stuttgart University (Germany) with the advice of Prof. Dr. Ulrich Hertrampf. Contents 1 Introduction and outline of the problem 2 2 The restricted role of the AND rule 3 3 The strategy for realizing virtual connections 4 4 The PSPACE-hardness 7 5 Conclusions 8

2 1 INTRODUCTION AND OUTLINE OF THE PROBLEM 1 Introduction and outline of the problem The game of Hex is a board game which has very simple rules but requires sharp tactical and strategic skills if played on a high level. For the rules and the history of the game, see [Ans02]. V. V. Anshelevich is the inventor of the concept of virtual connections in Hex games [Ans02]: If a player can force a connection of two cells x and y in the future play even if the opponent moves first, then this player has by definition a (general) virtual connection of x and y . Anshelevich defines in [Ans02] a set of deduction rules (AND deduction rule, OR deduction rule) which can be used to build complex virtual connections starting from the simplest ones, namely the pairs of neighboring cells. This process of building virtual connections is called H-search and defines the set of virtual connections in the stricter sense . This set is a proper subset of the set of (general) virtual connections defined above. In this paper by “virtual connections” we mean the virtual connections in the stricter sense. Anshelevich gives in [Ans02] a straight-forward algorithm performing such an H-search. Unfortunately, no general runtime analysis is given in [Ans02]. 1 However, it is remarked in [Ans02] that in practice heuristics have to be applied to H-search “due to limited computing resources”. In this paper we show that virtual connections are PSPACE-complete. We therefore draw the conclusion that it is highly unlikely – to the best of today’s complexity theory knowledge – that there exists any polynomial time algorithm performing exhaustive H-search. But nevertheless, virtual connections have been proven to be useful in Hex playing programs. Anshelevich himself wrote a superior program, called Hexy , which spends a lot of time calculating virtual connections. If two cells x and y represent the borders belonging to a player and if x and y are virtually connected then this player has a winning strategy. But virtual connections seem to be useful even if x and y do not represent the borders but normal cells. The presence or absence of a virtual connection gives deeper insight into the strategic and tactic situation of a game which then can be utilized for a better evaluation of the position. A better evaluation function, in turn, improves a standard α - β -search [Ans02]. 1 Anshelevich gives a runtime analysis for two special classes of Hex graphs which results in polynomial time concerning these classes.

3 2 THE RESTRICTED ROLE OF THE AND RULE Because the practical value of virtual connections is proven by the superi- ority of the program Hexy, the complexity of their calculation is of great practical interest. For a definition of the AND/OR deduction rules we refer to [Ans02]. In that article it is also argued that the set of virtual connections in the stricter sense is a proper subset of the set of general virtual connections (“incompleteness of the AND/OR deduction rules”). If these sets were equal (but they are not), then it would be obvious that the problem of finding virtual connections cannot be easier than determining the game-theoretical winner of a given Hex position. The latter is PSPACE-complete [Rei81]. 2 The restricted role of the AND rule The goal of this section is to argue that the AND rule can in some cases be omitted without restricting the set of recognizable virtual connections. Keep in mind that the AND rule for virtual semi -connections is the only way to build virtual semi-connections and therefore cannot be omitted. But the part of the AND rule that builds a virtual connection from two smaller virtual connections can in most cases be replaced by instances of the OR rule. A crucial point is the representation of Hex positions in a computer. For calculating virtual connections it is reasonable to model a Hex situation from one player’s point of view. One can model Hex positions as Hex graphs as shown in [Ans02]. Neighboring cells that are both occupied by the player can be regarded as one single cell (sometimes called a group ). Cells occupied by the opponent are not represented. Thus, Hex positions are graphs whose nodes are either marked as free or occupied and whose neighboring nodes are not both marked as occupied. This is roughly the representation which is (probably) used by the Hexy program. One could think of a slight change of the representation: For all occupied cells we add edges between all the neighbours of the occupied cell. This corresponds to an application of the AND rule in the first generation. It can be shown by induction over the generation of virtual connections that any virtual connection can now be deduced by only applying the OR rule for virtual connections and the AND rule for virtual semi-connections. The occupied cells could therefore be removed. If one is interested, however, in a virtual connection whose end is an occupied cell, one should, of course,

4 3 THE STRATEGY FOR REALIZING VIRTUAL CONNECTIONS not remove this cell. But also in this case the application of the AND rule can be restricted as described. We do not propose to omit the AND rule in practice. Consider, for example, the graph in figure 1. x y Figure 1: A chain of two-bridges The cells x and y are virtually connected which can be proven with the AND- rule in an obvious way. The maximal number of virtual semi-connections on the input side of the OR rule is two. If we deleted the two middle occupied cells while adding edges between their neighbors we could build the virtual connection between x and y without applying the AND rule for virtual connections. But we would need six virtual semi-connections on the input side of the last application of the OR rule (one for every empty cell). One heuristic is to restrict the number of virtual semi-connections on the input side of the OR rule. But it seems that one has to pay for the omission of the AND rule by large input sides of the OR rule. If one keeps the “restricted-OR-input” heuristic, severe “blind spots” are probably created. The possibility of omitting the AND rule makes a theoretical analysis eas- ier. Furthermore it can deepen the understanding of which kinds of general virtual connections are recognized by H-search and which kinds are not. In the next section we show, using the result of this section, how the strategy for realizing virtual connections can be described. 3 The strategy for realizing virtual connections With the result of the previous section, we can restrict to Hex graphs with- out any occupied cells. 2 We only apply the AND rule for the building of virtual semi-connections from virtual connections and the OR rule for the building of virtual connections from virtual semi-connections. This leads to the following informal description of a strategy. 2 Some readers might find it easier to assume the ends of a virtual connection to be occupied.

5 3 THE STRATEGY FOR REALIZING VIRTUAL CONNECTIONS Assume, a player (“our player”) has a virtual connection between two cells x and y and the opponent moves first. Our player decides to realize the virtual connection between x and y and announces that he will build a single chain between x and y and that all of his own moves are a part of this chain. Furthermore he announces the set of the fields that he will possibly occupy, the carrier C . Then the opponent occupies some cell. The first move of our player is the occupation of a cell z . According to his announcement, he will connect x with z and z with y . He can now furthermore announce the disjoint carriers A, B ⊆ C that he might need for these connections. Then again, the opponent occupies some cell. Our player occupies a cell z ′ which he either announces to connect with x and z or announces to connect with z and y . In both cases he also announces the carriers. The game is played in this way until our player has completed his chain. Whenever he occupies a cell his chain gets more concrete. Figure 2 shows an illustration of this strategy. We call such strategies direct-connecting strategies . z z ′ x y Figure 2: The strategy for realizing a virtual connection It is quite clear that whenever a player has a virtual connection between x and y he also has a x - y -direct-connecting strategy. The opposite direction is true as well and we will use it in the next section, but we don’t prove it here.