Some time back I tried implementing a Principal Variation Search , but as I wrote about in my post PVS – Another Fast Fail , the results were not good. At the time I concluded that if PVS is not a win, it must mean that the cost of the researches is outweighing the nodes saved by doing zero width searches. For that to be the case, it must mean that too often the first move is not the best move, which points to move ordering.

Since then move ordering has certainly improved, as documented in this post on Move Ordering . So, I decided to give PVS another try. In my first attempt, it appeared to be another loss. Then, I decided to not do PVS at the root node, and now it appears to be a very small win.

A win is a win, so I’m merging the changes in, but I think there is more to do here. My suspicion is that, as move ordering improves, the benefits of PVS will increase. The most obvious way to improve move ordering is to add a depth preferred hash table (the current strategy is a very naive “always replace”).

It seems like PVS at the root should work as well, if the program can reliably predict the best move often enough. I know a lot of programs put extra effort into ordering the moves at the root. I remember reading that Bob Hyatt’s Crafty does a quiescence search at the root. So, this is on the backlog as well, and once complete I will revisit the idea of PVS at the root.

For now, it is on to the next thing – Late Move Reductions. I’m hopeful that will yield a significant ELO increase, perhaps finally putting P4 on par with P3.

In my recent post on move ordering, I identified a potential area of improvement to the criteria for deciding if a capture is “good” or “bad.” As I wrote in that post, a capture is good if:

1. It is a promotion (technically even non-capturing promotions are included)
2. The value of the captured piece is greater than the value of the capturing piece
3. The Static Exchange Evaluator (SEE) score is non-negative.

The issue is with knights and bishops. They are roughly the same value (which one is more valuable really depends on the position), but in Prophet the bishop has a slightly higher value. A knight has a material value equal to 3 pawns, but a bishop has 3.2 pawns. The consequence of that is that a simple Bishop x Knight capture would be categorized as “bad” and not tried until all non-captures have been tried.

I don’t have the link handy but I read an older post on talkchess.com where Tord Romstad, the author of Glaurang (pre-cursor to Stockfish), mentioned that he used different piece values for the purposes of move ordering than he did in the evaluation. He said he used 1, 3, 3, 5 and 10. That means Bishop x Knight captures, as well as Knight x Bishop captures would both be categorized as “good.” Also, by giving the queen a value of 10, it means that giving two rooks for a queen would be considered equal by the SEE, where giving a queen + pawn for two rooks would have a negative score.

Sure enough, that simple change seems to be worth about 6 ELO.

As suspected the change to move ordering to separate good captures from bad captures has already paid off. Moving bad captures to the bottom of the move order list made it trivial to skip bad captures in the quiescence search altogether. This is an idea I first read about in a discussion on r.g.c.c. between Bob Hyatt, Feng-Hsiung Hsu and others at https://groups.google.com/g/rec.games.chess.computer/c/H6XjY2L13eQ . Hyatt claimed a small improvement, though Hsu was skeptical. Time has proven Hyatt correct though; I believe this is something most strong programs do. In any event, it seems to be worth about 10 ELO for Prophet4 so it’s a keeper.

Improving move ordering has been on the radar of a while now. I started to suspect that move ordering needed some work when my initial attempts at a PVS search and aspiration windows both failed. I reasoned that, if move ordering is subpar, researches would occur too often causing an overall increase in node counts.

To know, you have to measure, so I added data to the logfiles and wrote a Python script to aggregate (1) time to depth, (2) effective branching factor, and (3) the % of nodes in which we get a fail high on the 1st move and first four moves.

Once I was able to measure, I changed the move ordering scheme FROM:

PV move -> Hash move -> All captures in MVV/LVA order -> Killer 1 -> Killer 2 -> noncaptures in the order they are generated

TO:

PV move -> Hash Move -> “Good captures” in MVV/LVA order -> Killer 1 -> Killer 2 -> noncaptures -> bad captures in SEE order.

A “good capture” is one that is a promotion, a capture in which the value of the captured piece is at least that of the capturing piece, or one with a non-negative SEE value.

Incidentally, while researching this I came across a post I myself had made many years ago: http://talkchess.com/forum3/viewtopic.php?f=7&t=15198&hilit=defer+losing+captures&sid=5543f43760ce939ed16deacd59710e06

The change doesn’t seem to add more than just a couple of ELO directly. However, in non-tactical test suites all the measured metrics were improved: time to solution is down, effective branching factor is lower, the percentage of fail highs on the first move is improved, and the percentage of fail highs in the first four moves is dramatically improved. Additionally, the number of nodes required to reach a solution is cut by about a third, with only a 7% or so decrease in speed (nodes per second). I believe this sets the stage to take another stab at PVS and aspiration windows. First though, I’m going to take a stab at pruning bad captures from the quiescence search.

One potential (probable) area of improvement is that knights and bishops have slightly different values (bishops being the more valuable). For the purposes of determining if a capture is “good” when classifying the captures, they should probably be considered equal.

Prophet4 finally has a proper testing rig! A few weeks ago, I purchased a Dell Alienware system — an 8 core (16 logical) AMD Ryzen 7 5800 with 32 GB of RAM and an AMD Radeon RX600XT graphics card.

This replaces the single core laptop I have been using. This is pretty exciting as it will allow P4 testing to go at 8x the speed it did before. Just to break the machine in, I ran the first ever gauntlet with P4. Here are the results:

Rank	Name	Elo	Games	Score	Draws
1	plisk 0.2.7d	103	29785	66%	19%
2	tjchess 1.3	81	29787	63%	22%
3	jazz 840	52	29786	58%	21%
4	myrddin 0.87	44	29786	57%	21%
5	Horizon 4.4	15	29785	52%	19%
6	jumbo 0.4.17	-5	29783	49%	21%
7	p3-20181124	-8	29786	48%	25%
8	p4-20210407	-86	17211	36%	32%
9	prophet2_ja	-94	16000	35%	18%
10	tcb 0052	-182	29785	23%	15%

This closely matches the results I obtained a few years ago when I announced Prophet3 20180811 is released . One notable exception is TCB – it seems to have done much worse on this machine, for whatever reason.

So, it seems P4 is already on par with P2, and is within 78 elo or so from P3. This is encouraging, as the P4 rewrite is still not complete. I’m feeling pretty confident that when it is, it will be at least as strong as P3, and then the real work of improving can begin. And, with the fast feedback from a proper testing rig, I don’t think that should be all that difficult. My goal is to achieve +100 elo over P3 before doing a release.

Note– the versions of the chess engines used in this gauntlet are quite old by now; many of them likely have updates that could be significantly stronger. During the rewrite process, I’ve tried to keep everything “as is” — the goal is to compare P4 to P3, not to other engines. Once the rewrite is complete and P4 grows in strength, new testing engines will be cycled in as the current set is cycled out.

“Tree shaping” is an important component of a strong chess program. There are various ways to shape a search tree. One is by extending lines that seem interesting in some way; particularly those that cannot be fully resolved within the search horizon (see https://en.wikipedia.org/wiki/Horizon_effect). Other methods of shaping the search tree include reducing or even pruning lines that seem less promising. The quiescence search is also a form of tree shaping, as the search becomes more selective in which nodes it expands – typically just captures or captures + checks and check evasions.

One of the simplest, and perhaps the most effective extension is the “check extension.” Whenever a move is made, if the move gives check to the enemy, the search depth is increased by one. That is, instead of this:

apply_move(pos, move)
val = -search(pos, depth-1, -beta, -alpha)

Do this:

apply_move(pos, move)
bool gives_check = in_check(pos)
int ext = gives_check ? 1 : 0
val = -search(pos, depth-1+ext, -beta, -alpha)

Disclaimer: that is probably a little too naive, as it doesn’t really guarantee the line will terminate. There are surely positions that would cause the extension as written above to explode the search tree. But, on average, it’s a win.

I implemented the check extension in Prophet4 and indeed it seems to be a big win.

Wins	Losses	Draws	Pct	Elo	Error
331	183	402	58.1%	56.6	16.8

In an effort to add some sort of terminating criteria to it, I also tried limiting the check extension to capturing moves, and while still a win it didn’t work out as well.

Wins	Losses	Draws	Pct	Elo	Error
3206	3119	4505	50.4%	2.8	5.0

I also tried another form of extensions entirely – promoting moves. However, it had no measurable effect at all and may have even been a small loss, so I abandoned it.

After settling on the “naive check extension” I measured P4’s standing against P3.

Wins	Losses	Draws	Pct	Elo	Error
313	929	758	34.6%	-110.6	12.2

The last time I measured, P4 was -164.29 vs P3, so it’s definitely gained some ground.

Next on the list is to take a careful look at move ordering before moving onto another form of tree shaping – reductions. I expect Late Move Reductions (LMR) in particular to give another large jump. But, the algorithm is very sensitive to good move ordering, so it’s worth taking some time to study that first.

I’ve been pretty busy as of late, but in an effort to get some momentum going I picked some items that have been on my to do list that would take far more processor time than programming time – validating some evaluation terms that have been part of Prophet for many years now. The goal, really, was to validate what I was sure was true- that these evaluation terms do help; if they are removed, surely performance would drop as well. As you’ll see, it didn’t quite work out that way.

Here’s a list of evaluation terms that were tested, with results below.

• Knight tropism – the idea that keeping your knight as close as possible to the enemy’s king is generally a good thing.
• Rooks on open files, or even half open files.
• Passed pawns – pawns with no enemy pawn in front or on an adjacent file should be rewarded, since the likelihood of promotion increases dramatically.
• Isolated pawns – pawns with no friendly pawns on an adjacent file should be penalized, since they are weakened considerably without the supporting pawns.
• Doubled pawns – pawns that occupy the same file as another friendly pawn are a (small) positional weakness.
• Major pieces (rooks, queens) on the 7th rank with the enemy king on the back rank are usually very deadly, especially when “connected.”

Knight Tropism

This term works by penalizing a knight by 2 x distance_from_enemy_king centipawns. Distance from the enemy king is the max of difference in ranks and difference in files.

I actually started out running this one with chess4j, the Java engine, before deciding to use Prophet4 going forward. As explained in a previous post, the overhead of restarting the JVM between matches is just too high, and not restarting the engine between games isn’t great either, so doing these fast tests with a lightweight executable that can be quickly restarted seems preferable. At any rate, the outcome was nearly the same so I’ve combined the results into one table.

Also, since these are validations of existing terms, player A was the “control” player, and player B the “without” player. The hypothesis is that “Player A is better than Player B.”

Wins	Losses	Draws	Pct	Elo	Error
6312	5982	7674	50.8%	5.7	3.8
2343	2122	2766	51.5%	10.6	6.3
5942	5789	8269	50.4%	2.7	3.7

As you can see, the knight tropism is worth a few ELO. Not a game wrecker by any means, but it does have a small positive effect. We’ll check this one off the list.

Rooks on Open Files

Rooks on open files or even half open files are known to be a strategic advantage. It allows the player to move their rooks around easily, projecting strength and penetrating into the opponent position.

Rooks on files with no other pieces are given a 25 centipawn bonus. Rooks on files with just enemy pieces are given a 15 centipawn bonus.

Wins	Losses	Draws	Pct	Elo	Error
573	411	717	54.7%	33.2	12.6

This term is obviously doing the job. Check this one off the list.

Passed Pawns

Passed pawns become promoted pawns. Passed pawns are awarded 20 centipawns.

Wins	Losses	Draws	Pct	Elo	Error
2143	1930	2656	51.6%	11.0	6.5

Another one validated. That’s not to say it’s tuned correctly, but at least we can say it does help.

Isolated Pawns

Isolated pawns don’t have any friendly pawns on adjacent ranks to give them any support. They are a weakness. In Prophet and chess4j, isolated pawns are penalized 20 centipawns.

Wins	Losses	Draws	Pct	Elo	Error
502	641	741	46.3%	-25.7	12.2
1385	1532	1847	48.5%	-10.7	7.7

That is NOT a good result! I was in such disbelief after the first test that I ran a second test, and though the result doesn’t seem quite as bad, it’s still bad. As it stands, the isolated pawn penalty is hurting. I haven’t disabled it, because the major focus right now is rewriting the engine before improving it, and I want to be able to compare apples to apples after the rewrite. However, I have put an item on the backlog to study this in more detail. The heuristic should work. Either the implementation isn’t quite right, or it’s too expensive, or the weights aren’t right. I’ll have to get to the bottom of this.

Doubled Pawns

Doubled pawns also known to be a positional weakness. They are penalized 10 centipawns (note this penalty gets “awarded” to each pawn).

Wins	Losses	Draws	Pct	Elo	Error
593	722	950	47.2%	-19.8	10.9
858	1149	1516	45.9%	-28.8	8.7

Another surprising and disappointing result! Investigating this has also been added to the post-rewrite backlog.

Majors on 7th

This evaluation term awards rooks and queens on the 7th rank when the enemy king is on the back rank. If so, 50 centipawns are awarded. Additionally, if connected to another major piece on the 7th rank, an additional 80 centipawns are awarded – the idea being this is likely a deadly / mating attack.

I can’t remember how long this term has been around, but it’s been a looooong time. Sadly, the results aren’t so good.

Wins	Losses	Draws	Pct	Elo	Error
1019	1137	1542	48.4%	-11.1	8.5
3178	3509	4483	48.5%	-10.3	5.0

Clearly, not a good outcome! I wondered what would happen if I just disabled the “connected” part but left the award for having a major piece on the 7th when the enemy king is on the 8th?

Wins	Losses	Draws	Pct	Elo	Error
5705	6024	8271	49.2%	-5.5	3.7
5691	5973	8336	49.3%	-4.9	3.7

Disabling the “connected” bit may help a little, but still, the heuristic is hurting overall, not helping.

Conclusion

Out of the six evaluation terms to be tested / validated, three of them were found to be helpful and three are actually harmful. The “majors on 7th”, doubled pawn, and isolated pawn heuristics have all been added to the post-rewrite backlog for further study.

The moral of the story here is, do NOT assume anything. Always test! This has been my philosophy for a good while now, but these eval terms were all added at a time that I wasn’t as rigorous in my testing as I am now. (And, frankly, I don’t think many in the computer chess community were before 10-15 years ago.)

If there is any upshot, it’s that there is a guaranteed strength improvement to be had, even if it’s just removing the terms altogether. But, since I know they really should work, I’ve opted to leave them for now. Also, when the Prophet4 rewrite is complete, I really want to be able to run some benchmarks against Prophet3 and be able to make comparisons. The rewritten codebase should “stand on its own.” Improving the evaluator now would cast some doubt on those comparisons.

After the failure with aspiration windows I decided to implement PVS . The idea here is that, with proper move ordering the first move is usually the best. That being the case, it makes sense to search all of the moves after the first with a minimal alpha/beta window, the goal being to prove or disprove the first move was best rather than to get a usable score. If the first move is in fact best, you’ve saved some time. If not, you have to research with the “regular” alpha/beta window.

Wins	Losses	Draws	Pct	Elo	Error
351	487	598	45.3%	-33.0	13.7
533	670	796	46.6%	-23.8	11.8

This result is a little surprising and depressing. I’m not going to give up on PVS, but clearly the time isn’t quite right for it. I’ll have to revisit it later. This leads me to believe I should take a step back and look at move ordering. It seems to me that if PVS is a loss, that could mean the “first move is best” assertion is wrong more than we’d like, meaning move ordering is not where it should be. There are some tasks on my board related to move ordering, such as deferring losing captures until after non-captures and implementing a static exchange evaluator to aid in ordering the quiescence moves. I’ll leave the branch open and revisit this when move ordering has improved.

Earlier this week I implemented aspiration windows to chess4j. It’s a very simple concept. In the iterative deepening loop, the depth 1 iteration is processed as usual, with an alpha bound of “negative infinity” and a beta bound of “infinity.” On subsequent iterations, the alpha bound is initialized to the previous score minus some margin (1/3 pawn in this case), and the beta bound is initialized to the previous score plus some margin (again, 1/3 pawn). If all goes well, and the score is within those bounds, the search should finish faster as there are more cutoffs. However, if the real score is not within that search window, then a research must be done.

My intention was to start with the 1/3 pawn windows, falling back to a “wide open” search if it fails. Once testing confirmed that was a win, I could play with the margins or incrementally expanding the window. Unfortunately, this was a pretty fast fail.

My first run was terminated after 2002 games, and the second after 2965 games.

Wins	Losses	Draws	Pct	Elo	Error
549	689	763	46.5%	-24.5	12.0
886	1026	1053	47.6%	-16.4	10.0

I did expect this to be a win so I’m a little puzzled by the result, but that’s why we test. Perhaps when the search matures a little more and can hit deeper depths the scores between iterations will stabilize ? I’ll just queue this up to revisit later on.

On the plus side, SPRT saved a lot of time by terminating a 20,000 game match after 2-3k games. That’s a big time saver.

On to PVS.

Great things are done by a series of small things brought together. — Vincent Van Gogh.

The last few changes — adding null move pruning, hashing, and a quiescence search — have had major impacts on playing strength, which was expected. But inevitably, not all changes can be big winners, or even big losers. Some will have a very minor effect, either for the good or bad. The change I deal with in this post is one of those. The idea being tested was — should the search iterator be stopped early if, after completing an iteration, more than half the allotted time has been used? The rationale is that, due to the exponential growth of search trees, if you have less than half your time left when starting a new iteration, you’re unlikely to complete the iteration. Perhaps it would be better to save the time for future use. (This would not apply for games with a fixed time-per-move.)

The idea is so simple I hesitate to even show the (pseudo) code below:

        boolean stopSearching = false;
        do {
            ++depth;
            // DO SEARCH
            score = search.search(board, depth ...)
            if (search.isStopped()) {
                break;
            }
            long elapsed = System.currentTimeMillis() - startTime;
            // .... OTHER STOP CONDITIONS ....
            // if we've used more than half our time, don't start a new iteration
            if (earlyExitOk && !skipTimeChecks && (elapsed > maxTimeMs / 2)) {
                stopSearching = true;
            }
        } while (!stopSearching);

Up until this point I’ve been running 2,000 game self-play matches to gauge if a change is good or not. 2,000 games is enough to get an Elo difference with error bars of around 12-13 points, which has been “good enough.” So, my starting point here was the same – 2 matches, running concurrently on my 2 core testing machine, each match being 2000 games.

Wins	Losses	Draws	Pct	Elo	Error
634	589	777	51.1%	+7.82	11.90
634	614	752	50.5%	+3.47	12.02

Judging by the results of this test, it appears likely the change is a good change, albeit a minor one. The problem is the error bars. If the error bars are to be believed, it’s possible the change is actually neutral, or even a minor loss. The only way to tell would be to play more games – a lot more. People that have done the math state that to measure a 5 Elo change, you need something like 10,000 games. To measure a 2 Elo change requires 60,000 games! I decided I would run 2 20,000 game matches concurrently, for a combined 40,000 games. That should be enough to make a decision. The results of that experiment are below.

Wins	Losses	Draws	Pct	Elo	Error
6492	5782	7726	51.8%	+12.34	3.77
6334	6151	7515	50.5%	+3.18	3.80

These results were very confusing. If you take the Elo for each match and use the error bars to create an interval, the two intervals don’t even overlap! I have enough of an appreciation of the statistics to know this is not impossible, but it seems extremely suspect. I ended up creating a post about it on a computer chess forum – http://talkchess.com/forum3/viewtopic.php?f=7&t=74685 . One poster, H.G. Muller gave a possible explanation. Since I am not restarting the engines between matches (chess4j is a Java engine, and the JVM has a”warm up” cost that I’m avoiding by not restarting it), it’s possible that memory (cache) is being mapped in a way that is harmful to one process. Not restarting causes that condition to persist. I don’t know if that’s the case or not, but it does make some sense and raises the question of whether it really is OK to run two matches concurrently (or more generally N matches on an N core machine, especially if you don’t restart between matches). I decided to run another test, but just one match this time, not two.

Wins	Losses	Draws	Pct	Elo	Error
6179	6137	7684	50.1%	+0.73	3.77

My conclusion is that it’s not as stable as I originally thought to run two games concurrently. Restarting between matches could help, but again, being a JVM engine that’s not very appealing either. I could switch to testing first with Prophet as it’s C and could be restarted between matches, but one of the points of having a Java engine was quick prototyping. It’s also possible that a machine with more cores would resolve the issue, if at least one core is left free. I’ll revisit this in the future. For now, it’s one match at a time.

As far as the effect of the change in strength, it appears that it is worth a very small but positive amount of Elo. I think it’s a keeper, but just barely. But, running tens of thousands of games (especially when I’m only able to use a single core!) is not tenable. This would be a good time to look into some statistical methods. Many engine authors use Sequential Probability Ratio Test . If all you want to know is “is this engine better than this engine?” , and not necessarily the Elo difference between the two, SPRT could give a faster result.

The starting point for SPRT is to form a hypothesis. If player PA is my new version, and player PB is the current, best known version, my hypothesis is “PA is stronger than PB”. That is almost complete; that statement is really equivalent to “PA is stronger than PB by at least 1 Elo”.

You also need a “null hypothesis” – the hypothesis that there is no significant difference between players. My null hypothesis is “PA is NOT stronger than PB by at least 5 Elo points.”

Finally, since this is a statistical test you need to determine your tolerance for an error. There are two types of errors to consider. One is the likelihood of rejecting a change that is good (a type I error). The other is the likelihood of accepting a change that is bad (a type II error). I decided to accept a 5% likelihood of error for both.

Putting all that together:

H1: PA is stronger than PB by at least 1 Elo.

Ho: PA is not stronger than PB by at least 5 Elo

A: 0.05

B: 0.05

Note that, SPRT does not necessarily mean the match will terminate early. You can read about the math behind it at https://en.wikipedia.org/wiki/Sequential_probability_ratio_test , but the idea is that, if either H1 or H0 are “accepted” , the match will terminate. If not, it will continue until the maximum number of games is reached.

As it turns out, the program I use to drive the testing process, Cutechess , has the capability to terminate a match using SPRT. I had to experiment with it a little bit, my first attempt ended in an 18,000 game match where H0 was accepted! – not good. But, I had some parameters flipped so the results are invalid. I decided it would be more productive to experiment with a very large and obvious change, like disabling null move. The correct flags are:

-sprt elo0=1 elo1=5 alpha=0.05 beta=0.05

Given the previous results, I am not going to repeat the experiment yet again to see if it would terminate early, as it’s very likely it wouldn’t and I’d end up losing a week of processor time. I will however be using SPRT going forward.

Since the change was a good one in chess4j I’ve ported it to Prophet4 as well, and for confirmation ran a 2,000 game P4 vs P3 match.

Wins	Losses	Draws	Pct	Elo	Error
233	1153	614	27.0%	-172.8	13.4

That is marginally worse than the last run by 1.0%, but well within the margin of error.

Edit: I ran the P4 vs P3 match again out of curiosity:

Wins	Losses	Draws	Pct	Elo	Error
250	1139	611	27.8%	-166.0	13.4

Once again, the error bars are large enough in a 2000 game match that there is a lot of variability between runs, so I’m not making decisions based on these results, but I do like to see how P4 is measuring up against P3 as changes are made.

Next up- aspiration windows.