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<html dir="ltr">
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<title>Verified Null-Move Pruning</title>
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<body bgcolor="#FFFFFF" topmargin="50" leftmargin="100">
<div align="center">
<center>
<table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="580" id="AutoNumber5">
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<td>
<div align="center">
<center>
<table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="AutoNumber2">
<tr>
<td width="33%">
<p align="left"> </td>
<td width="33%">
<p align="center"><font face="Times New Roman" size="2">Verified
Null-Move Pruning</font></td>
<td width="34%">
<p align="right"><font face="Times New Roman" size="2">153</font></td>
</tr>
</table>
</center>
</div>
<font FACE="Times New Roman"><b>
<p ALIGN="center"> </p>
<p ALIGN="center">VERIFIED NULL-MOVE PRUNING</p>
</b><font FACE="Times New Roman" SIZE="2"><i>
<p ALIGN="center">Omid David Tabibi </i><sup>1</sup><i>
Nathan S. Netanyahu </i><sup>2</sup></p>
<p ALIGN="center">Ramat-Gan, Israel</p>
<p ALIGN="center">ABSTRACT</p>
</font></font>
<div align="center">
<center>
<table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" dir="ltr" id="AutoNumber1" width="520">
<tr>
<td><font FACE="Times New Roman" SIZE="2">
<p ALIGN="justify">In this article we review standard null-move
pruning and introduce our extended version of it, which we call <i>
verified null-move pruning</i>. In verified null-move pruning,
whenever the shallow null-move search indicates a fail-high, instead
of cutting off the search from the current node, the search is
continued with reduced depth.</p>
<p ALIGN="justify">Our experiments with verified null-move pruning
show that on average, it constructs a smaller search tree with
greater tactical strength in comparison to standard null-move
pruning. Moreover, unlike standard null-move pruning, which fails
badly in zugzwang positions, verified null-move pruning manages to
detect most zugzwangs and in such cases conducts a re-search to
obtain the correct result. In addition, verified null-move pruning
is very easy to implement, and any standard null-move pruning
program can use verified null-move pruning by modifying only a few
lines of code.</font></td>
</tr>
</table>
</center>
</div>
<p ALIGN="justify"><b><font size="2">1. INTRODUCTION</font></b></p>
<font FACE="Times New Roman" SIZE="2">
<p ALIGN="justify">Until the mid-1970s most chess programs were trying to
search the same way humans think, by generating �plausible� moves. By
using extensive chess knowledge at each node, these programs selected a
few moves which they considered plausible, and thus pruned large parts of
the search tree. However, plausible move generating programs had serious
tactical shortcomings, and as soon as brute-force search programs like
TECH (Gillogy, 1972) and CHESS 4.X (Slate and Atkin, 1977) managed to
reach depths of 5 plies and more, plausible-move generating programs
frequently lost to brute-force searchers due to their tactical weaknesses.</p>
<p ALIGN="justify">Brute-force searchers rapidly dominated the
computer-chess field. Most brute-force searchers of that time used no
selectivity in their full-width search tree, except for some extensions,
consisting mostly of check extensions and recaptures. The most successful
of these brute-force programs were BELLE (Condon and Thompson, 1983a,b),
DEEP THOUGHT (Hsu, Anantharaman, Campbell, and Nowatzyk, 1990), HITECH
(Berliner and Ebeling, 1990; Berliner, 1987; Ebeling, 1986), and CRAY
BLITZ (Hyatt, Gower, and Nelson, 1990), which for the first time managed
to compete successfully against humans.</font></p>
<p ALIGN="justify"><font FACE="Times New Roman" SIZE="2">The introduction
of null-move pruning (Beal, 1989; Goetsch and Campbell, 1990; Donninger,
1993) in the early 1990s marked the end of an era, as far as the
domination of brute-force programs in computer chess is concerned. Unlike
other forward-pruning methods (e.g., <i>razoring </i>(Birmingham and Kent,
1977), GAMMA (Newborn, 1975), and <i>marginal forward pruning </i>(Slagle,
1971)), which had great tactical weaknesses, null-move pruning enabled
programs to search more deeply with minor tactical risks. Forward-pruning
programs frequently outsearched brute-force searchers, and started their
own reign which has continued ever since; they have won all World
Computer-Chess Championships since 1992 (van den Herik and Herschberg,
1992; Tsang and Beal, 1995; Feist, 1999). DEEP BLUE (Hammilton and Garber,
1997; Hsu, 1999) (the direct descendant of DEEP THOUGHT (Hsu <i>et al.</i>,
1990)) was probably the last brute-force searcher. Today almost all top
tournament playing programs use forward-pruning methods, null-move pruning
being the most popular of them (Feist, 1999).</font></p>
<font FACE="Times New Roman" SIZE="1">
<hr noshade color="#000000" align="justify" width="35%" size="1">
<p ALIGN="justify"><sup>1</sup> Department of Computer Science, Bar-Ilan
University, Ramat-Gan 52900, Israel, Email:
<a href="mailto:[email protected]">[email protected]</a>.
<a href="Http://www.cs.biu.ac.il/~davoudo">
Http://www.cs.biu.ac.il/~davoudo</a><sup><br>
2</sup> Department of Computer Science, Bar-Ilan University, Ramat-Gan
52900, Israel, Email: <a href="mailto:[email protected]">
[email protected]</a>, and Center for Automation Research, University of
Maryland, College Park, MD 20742, USA, Email:
<a href="mailto:[email protected]">[email protected]</a>.</font></p>
<font FACE="Times New Roman" SIZE="1"><hr>
<p> </p>
<table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="AutoNumber2">
<tr>
<td width="33%">
<p align="left"><font face="Times New Roman" size="2">154</font></td>
<td width="33%"><font FACE="Times New Roman" SIZE="2">
<p ALIGN="center">ICGA Journal</font></td>
<td width="34%">
<p align="right"><font size="2">September 2002</font></td>
</tr>
</table>
</font><font FACE="Times New Roman" SIZE="2">
<p align="justify">In this article we introduce our new <i>verified
null-move pruning </i>method, and demonstrate empirically its improved
performance in comparison with standard null-move pruning. This is
reflected in its reduced search tree size, as well as its greater tactical
strength. In Section 2 we review standard null-move pruning, and in
Section 3 we introduce verified null-move pruning. Section 4 presents our
experimental results, and Section 5 contains concluding remarks.</p>
<b>
<p align="justify">2. STANDARD NULL-MOVE PRUNING</p>
</b>
<p align="justify">As mentioned earlier, brute-force programs refrained
from pruning any nodes in the full-width part of the search tree, deeming
the risks of doing so as being too high. Null-move (Beal, 1989; Goetsch
and Campbell, 1990; Donninger, 1993) introduced a new pruning scheme which
based its cutoff decisions on dynamic criteria, and thus gained greater
tactical strength in comparison with the static forward pruning methods
that were in use at that time.</p>
<div align="center">
<center>
<table border="1" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" id="AutoNumber3">
<tr>
<td><font FACE="Times New Roman" SIZE="2">
<p ALIGN="LEFT">/* the depth reduction factor */</font><font FACE="Courier" SIZE="2"><br>
#define R 2<br>
int search (alpha, beta, depth) {<br>
if (depth </font><font FACE="CMMI10" SIZE="2"><</font><font FACE="Courier" SIZE="2">=
0)<br>
return evaluate(); </font>
<font FACE="Times New Roman" SIZE="2">/* in practice, quiescence()
is called here */<br>
<font FACE="Courier" SIZE="2"> </font>/* conduct a null-move search if it is legal and desired */<br>
</font><font FACE="Courier" SIZE="2"> if (</font><font FACE="CMR10" SIZE="2">!</font><font FACE="Courier" SIZE="2">in_check()
&& null_ok()) {<br>
make_null_move();<br>
</font>/* null-move search with
minimal window around beta */<br>
<font FACE="Courier" SIZE="2">
value = -search(-beta, -beta + 1, depth - R - 1);<br>
if (value </font>
<font FACE="CMMI10" SIZE="2">></font><font FACE="Courier" SIZE="2">=
beta) </font><font FACE="Times New Roman" SIZE="2">/* cutoff in case
of fail-high */<br>
<font FACE="Courier" SIZE="2">
</font></font><font FACE="Courier" SIZE="2">
return value;<br>
}<br>
</font><font FACE="Times New Roman" SIZE="2">/* continue
regular NegaScout/PVS search */<br>
<font FACE="Courier" SIZE="2"> </font>. . .<br>
<font FACE="Courier" SIZE="2">}</font></font></td>
</tr>
</table>
<p><b>Figure 1</b>: Standard null-move pruning.</p>
</center>
</div>
<p ALIGN="justify">There are positions in chess where any move will
deteriorate the position, so that not making a move is the best option.
These positions are called <i>zugzwang </i>positions. While zugzwang
positions are rare in the middle game, they are not an exception in
endgames, especially endgames in which one or both sides are left with
King and Pawns. Null-move pruning will fail badly in zugzwang positions
since the basic assumption behind the method does not hold. In fact, the
null-move search�s value is an upper bound in such cases. As a result,
null-move pruning is avoided in such endgame positions.</p>
</font><font FACE="Times New Roman" SIZE="1"><hr>
<p ALIGN="justify"> </p>
<table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="AutoNumber2">
<tr>
<td width="33%">
<p align="left"> </td>
<td width="33%">
<p align="center"><font face="Times New Roman" size="2">Verified
Null-Move Pruning</font></td>
<td width="34%">
<p align="right"><font face="Times New Roman" size="2">155</font></td>
</tr>
</table>
</font><font FACE="Times New Roman" SIZE="2">
<p ALIGN="justify">As previously noted, the major benefit of null-move
pruning stems from the depth reduction in the null-move searches. However,
these reduced-depth searches are liable to tactical weaknesses due to the
<i>horizon effect </i>(Berliner, 1974). A horizon effect results whenever
the reduced-depth search misses a tactical threat. Such a threat would not
have been missed, had we conducted a search without any depth reduction.
The greater the depth reduction </font><font FACE="CMMI10" SIZE="2">R</font><font FACE="Times New Roman" SIZE="2">,
the greater the tactical risk due to the horizon effect. So, the saving
resulting from null-move pruning depends on the depth reduction factor,
since a shallower search (i.e., a greater </font>
<font FACE="CMMI10" SIZE="2">R</font><font FACE="Times New Roman" SIZE="2">)
will result in faster null-move searches and an overall smaller search
tree.</p>
<p ALIGN="justify">In the early days of null-move pruning, most programs
used </font><font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 1</font><font FACE="Times New Roman" SIZE="2">,
which ensures the least tactical risk, but offers the least saving in
comparison with other </font><font FACE="CMMI10" SIZE="2">R </font>
<font FACE="Times New Roman" SIZE="2">values. Other reduction factors that
were experimented with were </font><font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 2 </font>
<font FACE="Times New Roman" SIZE="2">and </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3</font><font FACE="Times New Roman" SIZE="2">.
Research conducted over the years, most extensively by Heinz (1999),
showed that overall, </font><font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 2 </font>
<font FACE="Times New Roman" SIZE="2">performs better than the too
conservative </font><font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 1 </font>
<font FACE="Times New Roman" SIZE="2">and the too aggressive </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3</font><font FACE="Times New Roman" SIZE="2">.
Today, almost all null-move pruning programs, use at least </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 2
</font><font FACE="Times New Roman" SIZE="2">(Feist, 1999). However, using
</font><font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
3 </font><font FACE="Times New Roman" SIZE="2">is tempting, considering
the reduced search effort resulting from shallower null-move searches.
(This will be demonstrated in Section 4.) Donninger (1993) was the first
to suggest an adaptive rather than a fixed value for </font>
<font FACE="CMMI10" SIZE="2">R</font><font FACE="Times New Roman" SIZE="2">.
Experiments conducted by Heinz (1999), in his article on adaptive
null-move pruning, suggest that using </font><font FACE="CMMI10" SIZE="2">
R </font><font FACE="CMR10" SIZE="2">= 3 </font>
<font FACE="Times New Roman" SIZE="2">in upper parts of the search tree
and </font><font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 2 </font>
<font FACE="Times New Roman" SIZE="2">in its lower parts can save 10 to 30
percent of the search effort in comparison with a fixed </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 2</font><font FACE="Times New Roman" SIZE="2">,
while maintaining overall tactical strength.</p>
<p align="justify">In the next section we present a new null-move pruning
method which allows the use of </font><font FACE="CMMI10" SIZE="2">R
</font><font FACE="CMR10" SIZE="2">= 3 </font>
<font FACE="Times New Roman" SIZE="2">in all parts of the search tree,
while alleviating to a significant extent the main disadvantage of
standard null-move pruning.</p>
<b>
<p align="justify">3. VERIFIED NULL-MOVE PRUNING</p>
</b>
<p align="justify">Cutoffs based on a shallow null-move search can be too
risky at some points, especially in zugzwang positions. Goetsch and
Campbell (1990) hinted at continuing the search with reduced depth, in
case the null-move search indicates a fail-high, in order to substantiate
that the value returned from the null-move search is indeed a lower bound
on the position. Plenkner (1995) showed that this idea can help prevent
errors due to zugzwangs. However, verifying the search in the middle game
seems wasteful, as it appears to undermine the basic benefit of null-move
pruning, namely that a cutoff is determined by a shallow null-move search.</p>
<p ALIGN="justify">In addition to helping in detecting zugzwangs, the idea
of not immediately pruning the search tree (based on the value returned
from the shallow null-move search) can also help to reduce the tactical
weaknesses caused by the horizon effect, since by continuing the search we
may be able to detect threats which the shallow null-move search has
failed to detect. Based on these ideas, we developed our own
reformulation, which we call<i> verified null-move pruning</i>. At each
node, we conduct a null-move search with a depth reduction of </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3</font><font FACE="Times New Roman" SIZE="2">.
If the returned value from that null-move search indicates a fail-high
(i.e., </font><i><font FACE="CMMI10" SIZE="2">value ≥ β</font></i><font SIZE="2" face="Times New Roman">),
we then reduce the depth by one ply and continue the search in order to
verify the cutoff. However, for that node�s subtree, we use standard
null-move pruning (cutoff takes place upon fail-highs). See Figure 2, for
an illustration.</font></p>
<div align="center">
<center>
<table border="1" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" id="AutoNumber4">
<tr>
<td><img border="0" src="example.gif" width="414" height="171"></td>
</tr>
</table>
<font FACE="Times New Roman" SIZE="2"><b>
<p>Figure 2</b>: Illustration of verified null-move pruning.</p>
</font></center>
</div>
<font FACE="Times New Roman" SIZE="2">
<p align="justify">The basic idea behind verified null-move pruning is
that null-move search with </font><font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 3 </font>
<font FACE="Times New Roman" SIZE="2">constructs a considerably smaller
search tree. However, because of its tactical deficiencies, a cutoff based
on it is too risky. So upon a fail-high, we reduce the depth and continue
the search, using standard null-move pruning</font></p>
<font FACE="Times New Roman" SIZE="1"><hr>
<p> </p>
<div align="center">
<center>
<table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="AutoNumber6">
<tr>
<td width="33%">
<p align="left"><font size="2">156</font></td>
<td width="33%">
<p align="center"><font FACE="Times New Roman" SIZE="2">ICGA Journal</font></td>
<td width="34%">
<p align="right"><font size="2">September 2002</font></td>
</tr>
</table>
</center>
</div>
</font>
<p align="justify"><font FACE="Times New Roman" SIZE="2">(with </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3</font><font FACE="Times New Roman" SIZE="2">)
in that node�s subtree. The search at a node is thus cut off (based on its
null-move search) only if there has been another null-move search
fail-high indication in one of the node�s ancestors (see Figure 2). As the
experimental results in the next section show, verified null-move pruning
constructs a search tree which is close in size to that of standard
null-move pruning with <font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 3</font>, and whose tactical strength is
greater on average than that of standard null-move pruning with
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 2</font>.
This is a smaller search tree with greater tactical strength, in
comparison with standard null-move pruning with
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 2</font>,
which is commonly used nowadays.</font></p>
<p align="justify"><font FACE="Times New Roman" SIZE="2">Since upon a
fail-high indication we do not cut off the search at once, we have the
ability to check whether the returned value is indeed a lower bound on the
position. If the null-move search indicates a cutoff, but the search shows
that the best value is smaller than </font><i>
<font FACE="CMMI10" SIZE="2">β</font></i><font FACE="Times New Roman" SIZE="2">,
this implies that the position is a zugzwang, as the value from the null
move is greater than or equal to the value from the best move. In such
cases, we restore the original depth (which was reduced by one ply after
the fail-high indication), and conduct a re-search to obtain the correct
value.</p>
<p align="justify">Implementation of verified null-move search is a matter
of adding a few lines of code to standard null-move search, as shown in
Figure 3. Regarding the pseudo-code presented, when the search starts at
the root level, the flag <font FACE="Courier" SIZE="2">verify </font>is
initialized to <font FACE="Courier" SIZE="2">true</font>. When the
null-move search indicates a fail-high, the remaining depth is reduced by
one ply, and <font FACE="Courier" SIZE="2">verify </font>is given the
value <font FACE="Courier" SIZE="2">false</font>, which will be passed to
the children of the current node, indicating that standard null-move
pruning will be conducted with respect to the children. Upon a fail-high
indication due to the standard null-move search of these children�s
subtrees, cutoff takes place immediately.</p>
<b>
<p align="justify">4. EXPERIMENTAL RESULTS</p>
</b>
<p align="justify">In this section we examine the performance of verified
null-move pruning, focusing on its tactical strength and smaller
search-tree size in comparison with standard null-move pruning. We
conducted our experiments using the G<font FACE="Times New Roman" SIZE="1">ENESIS<sup>3</sup>
</font>engine. G<font FACE="Times New Roman" SIZE="1">ENESIS </font>is
designed especially for research, emphasizing accurate implementation of
algorithms and detailed statistics. For our experiments we used the N<font FACE="Times New Roman" SIZE="1">EGA</font>S<font FACE="Times New Roman" SIZE="1">COUT</font>/PVS
(Campbell and Marsland, 1983; Reinefeld, 1983) search algorithm, with
history heuristic (Schaeffer, 1983, 1989) and transposition table (Slate
and Atkin, 1977; Nelson, 1985). To demonstrate the tactical strength
differences between the different methods even better, we used one-ply
check extensions on leaf nodes; the quiescence search consisted only of
captures/recaptures. In all test suites used, we discarded positions in
which at least one side had no more than King and Pawns. This was done to
avoid dealing with zugzwang positions, for which verified null-move
pruning obviously fares much better tactically, as explained before.</p>
<p ALIGN="justify">In order to obtain an estimate of the search tree, we
searched 138 test positions from <i>Test Your Tactical Ability </i>by
Yakov Neishtadt (see the Appendix) to depths of 9 and 10 plies, using
standard <font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 1</font>, <font FACE="CMMI10" SIZE="2">R
</font><font FACE="CMR10" SIZE="2">= 2</font>,
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3</font>,
and verified <font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 3</font>. Table 1 gives the total node count
for each method and the size of the tree in comparison with verified
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3</font>.
Table 2 gives the number of positions that each method solved correctly
(i.e., found the correct variation for). Later we will further examine the
tactical strength, using additional test suites.</p>
<div align="center">
<center>
<table border="1" cellpadding="3" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" id="AutoNumber7" height="88">
<tr>
<td align="center" valign="top" height="8">
<font FACE="Times New Roman" SIZE="2">Depth</font></td>
<td align="right" valign="top" height="8">
<font FACE="Times New Roman" SIZE="2">Std
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
1</font></font></td>
<td align="right" valign="top" height="8">
<font FACE="Times New Roman" SIZE="2">Std
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
2</font></font></td>
<td align="right" valign="top" height="8">
<font FACE="Times New Roman" SIZE="2">Std
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
3 </font></font></td>
<td align="right" valign="top" height="8">
<font FACE="Times New Roman" SIZE="2">Vrfd
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
3</font></font></td>
</tr>
<tr>
<td align="center" valign="top" height="30">
<font FACE="Times New Roman" SIZE="2">9</font></td>
<td align="right" valign="top" height="30">
<font FACE="Times New Roman" SIZE="2"> 1,652,668,804<br>
(+267.46%)</font></td>
<td align="right" valign="top" height="30">
<font FACE="Times New Roman" SIZE="2"> 603,549,66<br>
(+34.19%)</font></td>
<td align="right" valign="top" height="30">
<font FACE="Times New Roman" SIZE="2">267,208,422<br>
(-40.58%)</font></td>
<td align="right" valign="top" height="30">
<font FACE="Times New Roman" SIZE="2">449,744,588<br>
-</font></td>
</tr>
<tr>
<td align="center" valign="top" height="30">
<font FACE="Times New Roman" SIZE="2">10</font></td>
<td align="right" valign="top" height="30">
<font FACE="Times New Roman" SIZE="2">11,040,766,367<br>
(+661.64%)</font></td>
<td align="right" valign="top" height="30">
<font FACE="Times New Roman" SIZE="2">1,892,829,685<br>
(+30.57%)</font></td>
<td align="right" valign="top" height="30">
<font FACE="Times New Roman" SIZE="2">862,153,828<br>
(-40.52%)</font></td>
<td align="right" valign="top" height="30">
<font FACE="Times New Roman" SIZE="2">1,449,589,289<br>
-</font></td>
</tr>
</table>
</center><b>
<p align="justify">Table 1</b>: Total node count of standard
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 1</font><font FACE="CMMI10" SIZE="2">;
</font><font FACE="CMR10" SIZE="2">2</font><font FACE="CMMI10" SIZE="2">;
</font><font FACE="CMR10" SIZE="2">3 </font>and verified
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3
</font>at depths 9 and 10, for 138 Neishtadt test positions.</div>
<p ALIGN="justify">The results in Tables 1 and 2 reveal that the size of
the tree constructed by verified null-move pruning is between those of
standard <font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 2 </font>and <font FACE="CMMI10" SIZE="2">R
</font><font FACE="CMR10" SIZE="2">= 3</font>, and that its tactical
strength is greater on average than that of standard
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 2</font>.
These results also show that the use of <font FACE="CMMI10" SIZE="2">R
</font><font FACE="CMR10" SIZE="2">= 1 </font>is impractical due to its
large tree size in comparison with other depth-reduction values. Focusing
on the practical alternatives (i.e., standard <font FACE="CMMI10" SIZE="2">
R </font><font FACE="CMR10" SIZE="2">= 2 </font>and<font FACE="CMMI10" SIZE="2">
R </font><font FACE="CMR10" SIZE="2">= 3</font>, and verified
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3</font>),
we would like to examine the behavior of verified
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3
</font>and find out whether its tree size remains between the tree sizes
associated with <font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 2 </font>and <font FACE="CMMI10" SIZE="2">R
</font><font FACE="CMR10" SIZE="2">= 3</font>, or whether it approaches
the size of one</p>
<font FACE="Times New Roman" SIZE="1">
<hr noshade color="#000000" align="justify" width="35%" size="1">
<p ALIGN="LEFT"><sup>3</sup>
<a href="http://www.cs.biu.ac.il/~davoudo/genesis">
http://www.cs.biu.ac.il/~davoudo/genesis</a></p>
</font></font><font FACE="Times New Roman" SIZE="1"><hr>
<p ALIGN="LEFT"> </p>
<div align="center">
<center>
<table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="AutoNumber8">
<tr>
<td width="33%">
<p align="left"> </td>
<td width="33%">
<p align="center"><font size="2">Verified Null-Move Pruning</font></td>
<td width="34%">
<p align="right"><font size="2">157</font></td>
</tr>
</table>
</center>
</div>
</font>
<p ALIGN="LEFT"> </p>
<div align="center">
<center>
<table border="1" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" id="AutoNumber9">
<tr>
<td><font FACE="Courier" SIZE="2">
<p ALIGN="LEFT">#define R 3 </font>
<font FACE="Times New Roman" SIZE="2">/* the depth reduction factor
*/<br>
/* at the root level, verify = true */</font><font FACE="Courier" SIZE="2"><br>
int search (alpha, beta, depth, verify) {<br>
if (depth </font><font FACE="CMMI10" SIZE="2"><</font><font FACE="Courier" SIZE="2">=
0)<br>
return evaluate(); </font>
<font FACE="Times New Roman" SIZE="2">/* in practice, quiescence()
is called here */<br>
</font><font FACE="Courier" SIZE="2"> </font>
<font FACE="Times New Roman" SIZE="2">/* if verify = true, and depth
= 1, null-move search is not conducted, since<br>
</font><font FACE="Courier" SIZE="2"> </font>
<font FACE="Times New Roman" SIZE="2"> * verification will not
be possible */<br>
</font><font FACE="Courier" SIZE="2"> if (</font><font FACE="CMR10" SIZE="2">!</font><font FACE="Courier" SIZE="2">in_check()
&& null_ok() && (</font><font FACE="CMR10" SIZE="2">!</font><font FACE="Courier" SIZE="2">verify
|| depth </font><font FACE="CMMI10" SIZE="2">> </font>
<font FACE="Courier" SIZE="2">1)) {<br>
make_null_move();</font><font FACE="Times New Roman" SIZE="2"><br>
</font><font FACE="Courier" SIZE="2">
</font><font FACE="Times New Roman" SIZE="2">/* null-move search
with minimal window around beta */<br>
</font><font FACE="Courier" SIZE="2">
value = -search(-beta, -beta + 1, depth - R - 1,<br>
verify);<br>
if (value </font>
<font FACE="CMMI10" SIZE="2">></font><font FACE="Courier" SIZE="2">=
beta)</font><font FACE="CMSY10" SIZE="2"> { </font>
<font FACE="Times New Roman" SIZE="2">/* fail-high */<br>
</font><font FACE="Courier" SIZE="2">
if (verify) {<br>
depth--; </font><font FACE="Times New Roman" SIZE="2">/* reduce the
depth by one ply */<br>
</font><font FACE="Courier" SIZE="2">
</font><font FACE="Times New Roman" SIZE="2">/* turn verification
off for the sub-tree */<br>
</font><font FACE="Courier" SIZE="2">
verify = false;<br>
</font><font FACE="Times New Roman" SIZE="2">/* mark a fail-high
flag, to detect zugzwangs later*/<br>
</font><font FACE="Courier" SIZE="2">
fail high = true;<br>
}<br>
else </font>
<font FACE="Times New Roman" SIZE="2">/* cutoff in a sub-tree with
fail-high report */<br>
</font><font FACE="Courier" SIZE="2">
return value;<br>
}<br>
}<br>
re search: </font><font FACE="Times New Roman" SIZE="2">/* if a
zugzwang is detected, return here for re-search */<br>
</font><font FACE="Courier" SIZE="2"> </font>
<font FACE="Times New Roman" SIZE="2">/* do regular NegaScout/PVS
search */<br>
</font><font FACE="Courier" SIZE="2"> </font>
<font FACE="Times New Roman" SIZE="2">/* search() is called with
current value of �verify� */<br>
</font><font FACE="Courier" SIZE="2"> </font>
<font FACE="Times New Roman" SIZE="2">. . .<br>
</font><font FACE="Courier" SIZE="2"> </font>
<font FACE="Times New Roman" SIZE="2">/* if there is a fail-high
report, but no cutoff was found, the position<br>
</font><font FACE="Courier" SIZE="2"> </font>
<font FACE="Times New Roman" SIZE="2"> * is a zugzwang and has
to be re-searched with the original depth */<br>
</font><font FACE="Courier" SIZE="2"> if(fail_high
&& best </font><font FACE="CMMI10" SIZE="2">< </font>
<font FACE="Courier" SIZE="2">beta) {<br>
depth++;<br>
fail high = false;<br>
verify = true;<br>
goto re search;<br>
}<br>
}</font></td>
</tr>
</table>
<font FACE="Times New Roman" SIZE="2"><b>
<p>Figure 3</b>: Verified null-move pruning.</p>
</font></center>
</div>
<font FACE="Times New Roman" SIZE="2">
<p ALIGN="justify"> </p>
<p ALIGN="justify"> </p>
<div align="center">
<center>
<table border="1" cellpadding="3" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" id="AutoNumber10">
<tr>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">Depth</font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">Std
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
1</font></font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">Std
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
2</font></font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">Std
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
3</font></font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2"><font FACE="CMR10" SIZE="2"> </font>Vrfd
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
3</font></font></td>
</tr>
<tr>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">9</font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">64</font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">62</font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">53</font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">60</font></td>
</tr>
<tr>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">10</font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">71</font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">66</font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">65</font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">71</font></td>
</tr>
</table>
</center><b>
<p ALIGN="justify">Table 2</b>: Number of solved positions using
standard <font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 1</font><font FACE="CMMI10" SIZE="2">;
</font><font FACE="CMR10" SIZE="2">2</font><font FACE="CMMI10" SIZE="2">;
</font><font FACE="CMR10" SIZE="2">3 </font>and verified
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3
</font>at depths 9 and 10, for 138 Neishtadt test positions.</div>
<font FACE="Times New Roman" SIZE="1"><hr>
<p ALIGN="LEFT"> </p>
<div align="center">
<center>
<table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="AutoNumber11">
<tr>
<td width="33%"><font size="2">158</font></td>
<td width="33%">
<p align="center"><font FACE="Times New Roman" SIZE="2">ICGA Journal</font></td>
<td width="34%">
<p align="right"><font size="2">September 2002</font></td>
</tr>
</table>
</center>
</div>
</font>
<p> </p>
<div align="center">
<center>
<table border="1" cellpadding="3" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" id="AutoNumber12">
<tr>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">Depth </font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">Std
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
2 </font></font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">Std
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
3 </font></font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">Vrfd
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
3</font></font></td>
</tr>
<tr>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">9</font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">5,374,275,763<br>
(+10.84%) </font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">2,483,951,601<br>
(-48.76%)</font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">4,848,596,820<br>
-</font></td>
</tr>
<tr>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">10</font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">16,952,333,579<br>
(+17.40%)</font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">7,920,812,800<br>
(-45.14%)</font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">14,439,185,304<br>
-</font></td>
</tr>
<tr>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">11</font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">105,488,197,524<br>
(+106.51%) </font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">24,644,668,194<br>
(-51.75%)</font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">51,080,338,048<br>
-</font></td>
</tr>
</table>
</center><b>
<p ALIGN="justify">Table 3</b>: Total node count of standard
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 2</font>,
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3</font>,
and verified <font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 3 </font>at depths 9, 10, and 11, for 869
ECM test positions.</div>
<div align="center">
<center>
<table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" id="AutoNumber13">
<tr>
<td><img border="0" src="graph.gif" width="420" height="311"></td>
</tr>
</table>
</center><b>
<p align="justify">Figure 4</b>: Tree sizes of standard
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 2</font>,
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3</font>,
and verified <font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 3 </font>at depths 9, 10, and 11, for 869
ECM test positions.</div>
<p ALIGN="justify">of these trees. We therefore conducted a search to a
depth of 11 plies, using 869 positions from the<i> Encyclopedia of Chess
Middlegames </i>(ECM)<sup><font FACE="Times New Roman" SIZE="1">4</font></sup>.
Table 3 provides the total node counts at depths 9, 10, and 11, using
standard <font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 2</font>, <font FACE="CMMI10" SIZE="2">R
</font><font FACE="CMR10" SIZE="2">= 3</font>, and verified
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3</font>.
See also Figure 4. As Figure 4 clearly indicates, for depth 11 the size of
the tree constructed by verified null-move pruning with<font FACE="CMMI10" SIZE="2">
R </font><font FACE="CMR10" SIZE="2">= 3 </font>is closer to standard
null-move pruning with <font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 3</font>. This implies that the saving from
verified null-move pruning will be greater as we search more deeply. This
can be explained by the fact that the saving from the use of
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3
</font>in the shallow null-move search far exceeds the verification cost
of verified null-move pruning.</p>
<p ALIGN="justify">Having studied the effect of verified null-move pruning
on the search tree size, we now take a closer look at the resulting
tactical strength in comparison with standard null-move pruning with
different depth reductions.</p>
<p ALIGN="justify">For this purpose we used 999 positions from the <i>
Winning Chess Sacrifices </i>(WCS) test suite, and 434 positions of �mate
in 4� and 353 positions of �mate in 5� from the test suites of the <i>
Chess Analysis Project </i>(CAP); see the Appendix. The WCS positions were
searched to depths of 8, 9, and 10 plies, using standard
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 2</font>,
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3</font>,
and verified <font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 3</font>. Table 4 provides the total node
counts, and Table 5 gives the number of correctly solved positions for the
WCS test suite. For each position of �mate in 4� we conducted a search to
a depth of 8 plies, and for each �mate in 5� position a search to a depth
of 10 plies. The search was conducted using standard<font FACE="CMMI10" SIZE="2">
R </font><font FACE="CMR10" SIZE="2">= 1</font>,
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 2</font>,
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3</font>,
and verified <font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 3</font>. Table 6 provides the number of
positions that each method solved (i.e., found the checkmating sequence).</p>
<font FACE="Times New Roman" SIZE="1">
<hr noshade color="#000000" align="justify" width="35%" size="1">
<p ALIGN="LEFT"><sup>4</sup> Because of the large number of errors in
ECM�s suggested best moves, we did not check here for number of solved
positions.</p>
<font FACE="Times New Roman" SIZE="1"><hr>
<p ALIGN="LEFT"> </p>
<div align="center">
<center>
<table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="AutoNumber14">
<tr>
<td width="33%"> </td>
<td width="33%"><font FACE="Times New Roman" SIZE="2">
<p ALIGN="center">Verified Null-Move Pruning</font></td>
<td width="34%">
<p align="right"><font size="2">159</font></td>
</tr>
</table>
</center>
</div>
</font>
<p> </p>
<div align="center">
<center>
<table border="1" cellpadding="3" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" id="AutoNumber15">
<tr>
<td align="left" valign="top"><font FACE="Times New Roman" SIZE="2">
<p ALIGN="center">Depth </font><font FACE="CMR10" SIZE="2">
</font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="1">
<font FACE="Times New Roman" SIZE="2">Std </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
2</font></font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="1">
<font FACE="Times New Roman" SIZE="2">Std </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
3</font></font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="1">
<font FACE="Times New Roman" SIZE="2">Vrfd </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
3</font></font></td>
</tr>
<tr>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">8</font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">783,461,647<br>
(-13.55%)</font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">533,282,695<br>
(-41.15%)</font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">906,225,552<br>
-</font></td>
</tr>
<tr>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">9</font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">3,742,064,688<br>
(+47.38%)</font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">1,316,719,980<br>
(-48.14%)</font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">2,539,057,043<br>
-</font></td>
</tr>
<tr>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">10</font></td>
<td align="right" valign="top"><font size="2">11,578,143,939</font><font FACE="Times New Roman" SIZE="2"><br>
(+46.75%) </font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">4,871,295,877<br>
(-38.26%)</font></td>
<td align="right" valign="top">
<font FACE="Times New Roman" SIZE="2">7,889,544,754<br>
-</font></td>
</tr>
</table>
</center><font FACE="Times New Roman" SIZE="2"><b>
<p ALIGN="justify">Table 4</b>: Total node count of standard </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 2</font><font FACE="Times New Roman" SIZE="2">,
</font><font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 3 </font>
<font FACE="Times New Roman" SIZE="2">and verified </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3
</font><font FACE="Times New Roman" SIZE="2">at depths 8, 9, and 10, for
999 WCS test positions.</font></div>
<div align="center">
<center>
<table border="1" cellpadding="3" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" id="AutoNumber16">
<tr>
<td valign="top"><font FACE="Times New Roman" SIZE="2">
<p ALIGN="center">Depth</font></td>
<td valign="top"><font FACE="Times New Roman" SIZE="1">
<font FACE="Times New Roman" SIZE="2">Std </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
2</font></font></td>
<td valign="top"><font FACE="Times New Roman" SIZE="1">
<font FACE="Times New Roman" SIZE="2">Std </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
3</font></font></td>
<td valign="top"><font FACE="Times New Roman" SIZE="1">
<font FACE="Times New Roman" SIZE="2">Vrfd </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
3</font></font></td>
</tr>
<tr>
<td valign="top"><font FACE="Times New Roman" SIZE="2">
<p ALIGN="center">8</font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">762</font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">760</font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">782</font></td>
</tr>
<tr>
<td valign="top"><font FACE="Times New Roman" SIZE="2">
<p ALIGN="center">9</font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">838</font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">812</font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">838</font></td>
</tr>
<tr>
<td valign="top"><font FACE="Times New Roman" SIZE="2">
<p ALIGN="center">10</font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">850</font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">849</font></td>
<td valign="top" align="center">
<font FACE="Times New Roman" SIZE="2">866</font></td>
</tr>
</table>
</center><font FACE="Times New Roman" SIZE="2"><b>
<p ALIGN="justify">Table 5</b>: Number of solved positions using </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 2</font><font FACE="Times New Roman" SIZE="2">,
</font><font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 3 </font>
<font FACE="Times New Roman" SIZE="2">and verified </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3
</font><font FACE="Times New Roman" SIZE="2">at depths 8, 9, and 10 for
999 WCS test positions.</font></div>
<div align="center">
<center>
<table border="1" cellpadding="3" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" id="AutoNumber17">
<tr>
<td align="center" valign="top">
<font FACE="Times New Roman" SIZE="2">Test Suite</font></td>
<td align="center" valign="top">
<font FACE="Times New Roman" SIZE="1">
<font FACE="Times New Roman" SIZE="2">Std </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
1</font></font></td>
<td align="center" valign="top">
<font FACE="Times New Roman" SIZE="1">
<font FACE="Times New Roman" SIZE="2">Std </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
2</font></font></td>
<td align="center" valign="top">
<font FACE="Times New Roman" SIZE="1">
<font FACE="Times New Roman" SIZE="2">Std </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
3</font></font></td>
<td align="center" valign="top">
<font FACE="Times New Roman" SIZE="1">
<font FACE="Times New Roman" SIZE="2">Vrfd </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
3</font></font></td>
</tr>
<tr>
<td align="center" valign="top">
<font FACE="Times New Roman" SIZE="2">�Mate in 4�<br>
Depth 8 plies</font></td>
<td align="center" valign="top">
<font FACE="Times New Roman" SIZE="1">
<font FACE="Times New Roman" SIZE="2">433</font></font></td>
<td align="center" valign="top">
<font FACE="Times New Roman" SIZE="1">
<font FACE="Times New Roman" SIZE="2">385</font></font></td>
<td align="center" valign="top">
<font FACE="Times New Roman" SIZE="1">
<font FACE="Times New Roman" SIZE="2">379</font></font></td>
<td align="center" valign="top">
<font FACE="Times New Roman" SIZE="1">
<font FACE="Times New Roman" SIZE="2">431</font></font></td>
</tr>
<tr>
<td align="center" valign="top">
<font FACE="Times New Roman" SIZE="2">�Mate in 5�<font FACE="Times New Roman" SIZE="2"><br>
Depth 10 plies</font> </font></td>
<td align="center" valign="top">
<font FACE="Times New Roman" SIZE="1">
<font FACE="Times New Roman" SIZE="2">347</font></font></td>
<td align="center" valign="top">
<font FACE="Times New Roman" SIZE="1">
<font FACE="Times New Roman" SIZE="2">292</font></font></td>
<td align="center" valign="top">
<font FACE="Times New Roman" SIZE="1">
<font FACE="Times New Roman" SIZE="2">286</font></font></td>
<td align="center" valign="top">
<font FACE="Times New Roman" SIZE="1">
<font FACE="Times New Roman" SIZE="2">340</font></font></td>
</tr>
</table>
</center><font FACE="Times New Roman" SIZE="2"><b>
<p align="justify">Table 6</b>: Numbers of solved positions using
standard </font><font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 1</font><font FACE="CMMI10" SIZE="2">;
</font><font FACE="CMR10" SIZE="2">2</font><font FACE="CMMI10" SIZE="2">;
</font><font FACE="CMR10" SIZE="2">3 </font>
<font FACE="Times New Roman" SIZE="2">and verified </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3
</font><font FACE="Times New Roman" SIZE="2">for 434 �mate in 4� and 353
�mate in 5� test suites.</font></div>
<font FACE="Times New Roman" SIZE="2">
<p ALIGN="justify">The results in Tables 5 and 6 indicate that verified
null-move pruning solved far more positions than standard null-move
pruning with depth reductions of </font><font FACE="CMMI10" SIZE="2">R
</font><font FACE="CMR10" SIZE="2">= 2 </font>
<font FACE="Times New Roman" SIZE="2">and </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3</font><font FACE="Times New Roman" SIZE="2">.
This demonstrates that not only does verified null-move pruning result in
a reduced search effort (the constructed search tree is closer in size to
that of standard </font><font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 3</font><font FACE="Times New Roman" SIZE="2">),
but its tactical strength is greater than that of standard </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 2</font><font FACE="Times New Roman" SIZE="2">,
which is the common depth reduction value.</p>
<p ALIGN="justify">Finally, to study the overall advantage of verified
null-move pruning over standard null-move pruning in practice, we
conducted 100 self-play games, using two versions of the G</font><font FACE="Times New Roman" SIZE="1">ENESIS
</font><font FACE="Times New Roman" SIZE="2">engine, one with verified
</font><font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
3</font><font FACE="Times New Roman" SIZE="2"> and the other with standard
</font><font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">=
2</font><font FACE="Times New Roman" SIZE="2">. The time control was set
to 60 minutes per game. The version using verified </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3
</font><font FACE="Times New Roman" SIZE="2">scored 68.5 out of 100 (see
the Appendix), which demonstrates the superiority of verified null-move
pruning over the standard version.</font></p>
<font FACE="Times New Roman" SIZE="2"><b>
<p ALIGN="justify">5. CONCLUSION</p>
</b>
<p ALIGN="justify">In this article we introduced a new null-move pruning
method which outperforms standard null-move pruning, techniques, in terms
of reducing the search tree size as well as gaining greater tactical
strength. The idea of not cutting off the search as soon as the shallow
null-move search indicates a fail-high allows verification of the cutoff,
which results in greater tactical accuracy and prevents errors due to
zugzwangs. We showed empirically that verified null-move pruning with a
depth reduction of </font><font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 3 </font>
<font FACE="Times New Roman" SIZE="2">constructs a search tree which is
closer in size to that of the tree constructed by standard </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 3</font><font FACE="Times New Roman" SIZE="2">,
and that the saving from the reduced search effort in comparison with
standard </font><font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 2 </font>
<font FACE="Times New Roman" SIZE="2">becomes greater as we search more
deeply. We also showed that on average, the tactical strength of verified
null-move pruning is greater than that of standard null-move pruning with</font><font FACE="CMMI10" SIZE="2">
R </font><font FACE="CMR10" SIZE="2">= 2</font><font FACE="Times New Roman" SIZE="2">.
Moreover, verified null-move pruning can be implemented within any
standard null-move pruning framework by merely adding a few lines of code.</font></p>
<font FACE="Times New Roman" SIZE="2">
<p ALIGN="justify">We considered a number of variants of standard
null-move pruning. The first variant was not to cut off at all upon
fail-high reports, but rather reduce the depth by 2 plies. We obtained
good results with this idea, but its tactical strength was sometimes
smaller than that of standard </font><font FACE="CMMI10" SIZE="2">R </font>
<font FACE="CMR10" SIZE="2">= 2</font><font FACE="Times New Roman" SIZE="2">.
We concluded that in order to improve the results, the depth should not be
reduced by more than one ply at a time upon fail-high reports. An
additional variant was not to cut off at any node, not even in the subtree
of a node with a fail-high report, but</p>
<font FACE="Times New Roman" SIZE="1"><hr>
<p ALIGN="LEFT"> </p>
<div align="center">
<center>
<table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="AutoNumber18">
<tr>
<td width="33%"><font size="2">160</font></td>
<td width="33%">
<p align="center"><font FACE="Times New Roman" SIZE="2">ICGA Journal</font></td>
<td width="34%">
<p align="right"><font size="2">September 2002</font></td>
</tr>
</table>
</center>
</div>
</font></font></font><font FACE="Times New Roman">
<font FACE="Times New Roman" SIZE="2">
<p ALIGN="justify">merely to reduce the depth by one ply upon a fail-high
report. Unfortunately, the size of the resulting search tree exceeded the
size of the tree constructed by standard </font>
<font FACE="CMMI10" SIZE="2">R </font><font FACE="CMR10" SIZE="2">= 2</font><font FACE="Times New Roman" SIZE="2">.
Still, another variant was to reduce the depth by one ply upon fail-high
reports, and to reduce the depth by two plies upon fail-high reports in
that node�s subtree, rather than cutting off.</p>
<p ALIGN="justify">Our empirical studies showed that cutting off the
search at the subtree of a fail-high reported node does not decrease
tactical strength. Indeed, this is the verified null-move pruning version
that we studied in this article. In contrast to the standard approach
which advocates the use of immediate cutoff, the novel approach taken here
uses depth reduction, and delays cutting off the search until further
verification. This yields greater tactical strength and a smaller search
tree.</p>
<b>
<p align="justify">6. REFERENCES</p>
</b>
<p align="justify">Beal, D.F. (1989). Experiments with the null move. <i>
Advances in Computer Chess 5</i>, (Ed. D.F. Beal) , pp. 65�79. Elsevier
Science Publishers, Amsterdam, The Netherlands. ISBN 0-444-87-159-4.</p>
<p align="justify">Berliner, H.J. (1974). <i>Chess as Problem Solving: The
Development of a Tactics Analyzer</i>. Ph.D. thesis, Carnegie-Mellon
University, Pittsburgh, PA.</p>
<p align="justify">Berliner, H.J. (1987). Some innovations introduced by H</font><font FACE="Times New Roman" SIZE="1">ITECH</font><font FACE="Times New Roman" SIZE="2">.
<i>ICCA Journal</i>, Vol. 10, No. 3, pp. 111�117.</font></p>
<p align="justify"><font FACE="Times New Roman" SIZE="2">Berliner, H.J.
and Ebeling, C. (1990). H</font><font FACE="Times New Roman" SIZE="1">ITECH</font><font FACE="Times New Roman" SIZE="2">.
<i>Computers, Chess and Cognition </i>(Eds. T.A. Marsland and J.
Schaeffer), pp. 79�109. Springer-Verlag, New York, N.Y. ISBN
0-387-97415-6/3-540-97415-6.</p>
<p align="justify">Birmingham, J.A. and Kent, P. (1977). Tree-searching
and tree-pruning techniques. <i>Advances in Computer Chess 1</i>, (Ed.
M.R.B. Clarke), pp. 89�107. Edinburgh University Press, Edinburgh. ISBN
0-852-24292-1.</p>
<p align="justify">Campbell, M.S. and Marsland, T.A. (1983). A comparison
of minimax tree search algorithms. <i>Artificial Intelligence</i>, Vol.
20, No. 4, pp. 347�367. ISSN 0004-3702.</p>
<p align="justify">Condon, J.H. and Thompson, K. (1983a). B</font><font FACE="Times New Roman" SIZE="1">ELLE</font><font FACE="Times New Roman" SIZE="2">.
<i>Chess Skill in Man and Machine</i>, (Ed. P.W. Frey), pp. 201�210.
Springer-Verlag, New York, N.Y., 2nd ed., ISBN
0-387-90790-4/3-540-90790-4.</p>
<p align="justify">Condon, J.H. and Thompson, K. (1983b), B</font><font FACE="Times New Roman" SIZE="1">ELLE
</font><font FACE="Times New Roman" SIZE="2">chess hardware. <i>Advances
in Computer Chess 3</i>, (Ed. M.R.B. Clarke), pp. 45�54. Pergamon Press,
Oxford, ISBN 0-080-26898-6.</p>
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<b>
<p ALIGN="justify">7. ACKNOWLEDGEMENTS</p>
</b>
<p ALIGN="justify">We would like to thank Shay Bushinsky for his interest
in our research, and for promoting the discipline of Computer Chess in our
department. We would also like to thank Dann Corbit for providing the CAP
test positions for our empirical studies, and Azriel Rosenfeld for his
editorial comments. Finally, we are indebted to Jonathan Schaeffer and
Christian Donninger for their enlightening remarks and suggestions.</p>
<b>
<p ALIGN="justify">8. APPENDIX</p>
<p ALIGN="justify">EXPERIMENTAL SETUP</p>
</b>
<p ALIGN="justify">Our experimental setup consisted of the following
resources:</p>
<ul>
<li>
<p ALIGN="justify">138 positions (Diagrams 241 to 378) from: Yakov
Neishtadt (1993). <i>Test Your Tactical Ability</i>, pp. 110�135.
Batsford, ISBN 0-7134-4013-9.<br>
</li>
<li>
<p ALIGN="justify"><font FACE="Times New Roman" SIZE="2">869 positions
from <i>Encyclopedia of Chess Middlegames</i>, and 999 positions from <i>
Winning Chess Sacrifices</i>, as available on the Internet.<font FACE="Times New Roman" SIZE="2"><br>
</font></font></li>
<li>
<p ALIGN="justify"><font FACE="Times New Roman" SIZE="2">434 �Mate in 4�
and 353 �Mate in 5� positions from <i>Chess Analysis Project</i>,
available at <a href="ftp://cap.connx.com/">ftp://cap.connx.com/</a><br>
</font></li>
<li>
<p ALIGN="justify"><font FACE="Times New Roman" SIZE="2">G<font FACE="Times New Roman" SIZE="1">ENESIS
</font>chess engine, with <font FACE="CMR10" SIZE="2">2</font><sup>22</sup><font FACE="CMR7" SIZE="1">
</font>transposition table entries (64MB), running on a 733 MHz Pentium
III with 256MB RAM, with the Windows 98 operating system.</font></li>
</ul>
<p ALIGN="justify">The webpage
<a href="http://www.cs.biu.ac.il/~davoudo/pubs.html">
http://www.cs.biu.ac.il/~davoudo/pubs.html</a> contains additional
information about the test suites, move lists of self-play games, and
detailed experimental results.</font></font></font></td>
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