Artificial Intelligence (AI) is at the heart of IT-research and pioneers automated problem-solving. A recent breakthrough that could shift the AI-field was reported by Silver and colleagues of the famous Google Deep mind developer team around Demis Hassabis. They have reported that a new adaption of their in-house generalized neural network (NN) machine learning algorithm, termed AlphaZero, has achieved to outperform the world’s best-rated chess engine Stockfish, already after four hours of self-learning, and starting only with the rules of chess. AlphaZero originates from the superhuman Alpha Go program and could beat the best chess engine via blank slate tabula rasa self-play reinforcement machine learning, i.e. only by learning from many games played against itself. This worldwide strongest AI-performance claim has been drawn from a 100-game match between AlphaZero and Stockfish engines and has attracted much attention by the media, especially in the world of chess, which has been historically a key domain of AI. AlphaZero did not lose one game and won 28 times, while the remainders of the 100 games were draws. General reinforcement learning is very promising for many applications of mathematical solution finding in complexity. However, the requirement to independently verify if this breakthrough AI claim can be made poses some major difficulties and raises some inevitable doubts if a final proof has been given. Machine and method details are not available and only10 example games were given. This research starts with a reproducibility testing of all 10 example games and reveals that AlphaZero shows signs of human openings and might have outperformed Stockfish due to an irregular underperformance of Stockfish8, like post-opening novelties or subperfect game moves, like in game moves and post-opening novelites. At this juncture, the testing revealed that AI quiescence searches could be improved via multiple roots for both engines, which could boost all future AI performances. In light of a lack of tournament conditions and an independent referee, comparability challenges of software and hardware configurations such as AlphaZero’s TFLOP super-calculation-power, this work suggests that a final best AI-engine-claim requires further proof. Overclaim biases are found in all sciences of today due to the publishing imperatives and wish to be first.

Keywords: Artificial intelligence, ai, best, chess, engine, stockfish, AlphaZero, google, deepmind, elo, bias, machine, learning, tflops, root, tree, algorithm

AI, Artificial Intelligence; CPU, Central Processing Unit; Depth, The deepness of the calculation often given in ply as the sequence of moves calculated ahead; ECO, ECO codes is a classification system of chess openings. ECO abbreviates Encyclopaedia of Chess Openings code. E.g. see free online versions of the Encyclopaedia;7 FLOPs, Floating Point Operations per second, a measure of computing performance; KPI, Key Performance Indicator; MTCS, Monte-Carlo Tree Search; NN, Neural Network; Ply, Half move in chess of either black or white, e.g. 1. e4 is one ply and 1. …e5 is a second play of the first move (here 1.); Root, the root of the search tree is the ply in which the calculation is started at a depth of zero; TPU, Tensor Processing Unit; TFLOP, Terra FLOPs

Games are an ideal model system to research and develop artificial intelligence (AI). In chess, information is perfect and symmetric, and decision-making is logical and ranges from boolean to fuzzy logic, and is alternately challenged by sequential complex puzzles and non-trivial positions throughout the iterative game. This provides an ideal IT assay model system to develop and test AI and machine learning in the field of decision-making and machine learning via continuous improvement. It represents an ideal and major challenge for the development of AI software and general reinforcement self-learning programs due to the simultaneously high levels of game complexity and the vast amount of branching positions and a lower bound of about 10¹²⁰ possible games according to the Shannon number. Hence, engines and players must somehow find the best ways and algorithms to manage certainty and uncertainty in their evaluations. A recent report by Silver and colleagues¹ has claimed a big major breakthrough that has been widely perceived as such by most of the noticed experts in the field and by the media. Many speak of a major milestone that has been reached by developing a pure machine self-learning algorithm that can outperform even the best conventional chess engine, which is an equally significant and relevant benchmark. According to the authors, this world-breakthrough has been achieved by simply advancing and adopting the Alpha Go program to Chess, next to Shogi, a Japanese Chess form, and Go, games where it also won outstandingly.¹ In the game of chess, AlphaZero clearly outperformed Stockfish8 in 100 games after only a short period of self-training, starting with the rules of chess and by pure reinforcement learning algorithms and simulated self-play. In the field of AU, this would represent a striking world-breakthrough of efficient and effective machine learning.

The claim has been made that already after 4 hours and 300k steps, AlphaZero would have been able to outperform Stockfish8, basically by starting and learning from scratch.¹ Hereby, AlphaZero would have only searched 80.000 positions per second, while Stockfish8 would have searched 70 million.¹ Astoundingly, this is 875-times fewer calculations per second due to self-learning that would indicate another novelty of AI condensing game information into a human-like concluding, learning and intuition, as if variation pruning is optimized game by game. This would be the source code that everybody in the AI field was searching for, but has it actually been achieved in all relevant ways and not in a different way, e.g. via accumulated book-like knowledge? This work will re-evaluate if the claim of the best AI-chess-engine can really already be made by testing for reproducibility.

In a 100-game-match the new AlphaZero algorithm was shown to beat and obliterate Stockfish8, one of the world’s best chess engines,² which won the 2016 TCEC computer chess championship. Today, an updated version of Stockfish8 has been released, called asmFish,³ which is again in December 2017 became the best engine of today, also in a direct comparison of all latest versions and other engines, and “in an official engine tournament”. In the original publication, AlphaZero has not been subjected to an official independent testing¹ like on CCRL or tournaments, or elsewhere to the author’s knowledge. Only 10 of the 100 games were made available,¹ which are reanalyzed in this work by multiple re-evaluations of every move played by Stockfish8.

Science and publishing have become very biased today,^4–6 especially since 2000, according to most estimates. It could be possible that an overclaim and many other systemic biases are inherent to all or almost all publications. Thus, as a first objective, this hypothesis shall be tested in the field of AI, a very clear-cut field of unambiguous IT and mathematics, by reevaluating if the best-engine-claim can already be made, or not, while also having a very constructive question in mind of how to improve AI and machine learning, as the second objective.

Materials

Chess Engines: Stockfish7 64 bit and Stockfish 8 64 bit 4 CPUs and Fritz Chess software was used to recalculate the individual positions of the 10 example games,¹ as this was the only information that we have yet received from the publication. The engine AlphaZero in its latest version was not available to the author and all downloadable forms were not in the latest version. Opening book: Fritz11.ctg 06.11.2007 11:00 CTG-file 263.816 KB; Personal Computer: RAM: 3,44 GB usable; 64-bit operating system, x64-based processor, AMD A6-6310 APU Radeon R4 Graphics 1,80 GHz, HP Notebook; System: Windows 10 Home Edition; Chess Software: Fritz Twelve Beginner Edition 2011 (USK 0 years) ChessBase GmbH ISBN-978-3-86681-248-2; Screenshot and Office Software and further irrelevant tools; Chess games database were derived from 365Chess.com⁷ and from Chessify with 6M chess positions.

Methods

The 10 example games that were published by Silver and colleagues of the outstanding Google Deep Mind Machine Learning Artificial Intelligence team¹ were transferred into PGN format using chessbase.com and uploaded into the Fritz Twelve Beginner Edition software. Chess Engines, Stockfish7 and Stockfish8 were uploaded to Fritz Twelve Beginner Edition software and used with the following parameters: hash table size of 2139 MB, 64-bit versions, permanent brain to enable in some cases pre-calculations, which is using the calculations that were performed during the opponent’s and own time; Further, maybe less relevant Stockfish8 64-bit version parameters are only mentioned for a further precision: Contempt: O, Threads: 4, Skill Level: 20, Move overhead: 30, Minimum Thinking Time: 20, Slow Mover: 89, nodestime: 0, Syzygy Probe Depth 1, Sysygy Probe Limit: 6, activated Syzygy50MoveRule; Stockfish7 64-bit version original parameters included: Contempt: O, Threads: 4, Skill Level: 20, Move overhead: 30, Minimum Thinking Time: 20, Slow Mover: 84, nodestime: 0, Syzygy Probe Depth 1, Sysygy Probe Limit: 6; activated Syzygy50MoveRule; 1 or 4 CPUs (central processing units); only a few available details about AlphaZero and Stockfish8 parameters were found in, and can be obtained from, the original publication of AlphaZero,¹ which poses another challenge to reproducibility. An automated and manual move-by-move re-evaluation screening was performed with Fritz Twelve and Stockfish8.

AI methods

MCTS is a best first search algorithm with four phases: selection, expansion, simulation, and backpropagation, based on random playouts. The original publication reveals that the Neural-Network parameters of AlphaZero “are adjusted by gradient descent on a loss function l that sums over mean-squared error and cross-entropy losses respectively,”¹

$\left(p,v\right)=f_{\theta }\left(S\right),l={\left(z-v\right)}^2-{\pi }^T\log p+c\parallel \theta {\parallel }^2$

„where c is a parameter controlling the level of L₂ weight regularization. The updates parameters are used in subsequent games of self-play”.¹ This helps achieve AlphaZero’s learning together with root visit counts.¹ Stockfish8 uses advanced alpha-beta searches and alpha-beta pruning that focuses quiescent searches on the most relevant combinations. It begins calculations based on the root of the current position, which is set to root ply zero.² Stockfish’s more static AI is a non-learning algorithm that uses specific evaluation algorithms are used for opening, middle games, and endgame, and special positions.² Still, hash tables bear the potential to enable some side-effect short-term memory learning under permanent brain conditions² that is used as a method in the reevaluation of the example games.

Re-evaluation of AI match statistics

In the game of chess, AlphaZero has recently been reported to outperform Stockfish8, after only 4 hours of learning via self-play. AlphaZero could outperform Stockfish8 in an initial experiment already after 300k steps. The centerpiece of the claim is a 100-game match in which AlphaZero won against Stockfish8. Here the training phase has resulted in about 700k steps of tabula rasa learning under tournament time controls of 1 minute per move, each MTCS was calculated on a single machine with 4 tensor processing units (TPUs), training started with randomly initialized parameters using 5.000 first-generation TPUs to generate self-play games and 64 second-generation TPUs to train the neural networks.¹ TPUs are built and assembled in a specific machine learning hardware configuration of integrated circuits developed by Google (ASICs) and mainly for AI applications.

In this 100-game match, Stockfish8 has been? reported to have lost 28 times, 25 times with black and 3 times with white, while AlphaZero did not lose any game, and won 28 times, and drew 72 times, 25 times with white and 47 times with black (taken from Table 1 in Silver et al.)¹. All of the 100 original games¹ were played in a direct comparison as an engine match of Stockfish8 (official Linux release, 64 CPU threads, hash table size of 1GB, and according to the authors, at their strongest skill level) versus AlphaZero (1.1) 04.12.17 using a single machine with 4 TPUs,¹ which is, in fact, a comparison of very different software and hardware configurations.

Normally, on average, top games of the world’s best 2003 different engines result in a win distribution of white-draw-black (Figure 1) of about 34,3%-40,3%-25,4% favoring white with 54,4% versus 45,6% for black.³ AlphaZero, however, reveals a new, unexpected ‘atypical’ distribution pattern of 25%-3%-72% favoring white by 61% versus black by only 39% from the perspective of AlphaZero. From Stockfish’s perspective, the pattern appears even more ‘atypical’ as 0%-72%-0% (Figure 1), only achieving draws against the learning AlphaZero engine. Put in other words, AlphaZero, as white, would have better exploited the first-mover advantage than normally seen by the sum of top engines (Figure 1). To better make use of a starting position, it would be required to figure out the most forcing lines in the widely ‘unexpectable’ long-run of the game. This would indicate that AlphaZero had learned and abstracted rules from its training phase and derived an understanding-like intuition by calculation higher quality lines and trees, as it is known to perform 875-times “fewer calculations per second”.¹ Where should the performance stem from? Could a better focus on key lines also be achieved by self-learned book hints has AI machine learning gone perfect, human, or both?

Figure 1 Comparison of the 100 AlphaZero vs. Stockfish8 match statistics with an engine database. (Right) CCRL 40/40 chess engine match results of 771.905 games and newest 2003 machines versions, starting from December 2005 till December 2017, compared with 100 published games of AlphaZero versus Stockfish version 8(Table 1, Silver et al.,1 2017). Note the very strong shift from black wins into draws in comparison to the 2003 top engines data. (Left) The black-vs-white-score distribution pattern is fully absent if the statistics are viewed from the perspective of Stockfish8 that did not win one game and drew 72 times. This distribution of white versus black wins is unexpectedly high and would suggest that a starting position could be more exploited than it is today by humans or most engines. This strong exploitation of a beginner advantage can be unexpected for a null-sum game.

Re-evaluation screening for atypical moves

In a next step, all 10 example games of the 100-match engine contest of AlphaZero versus Stockfish8¹ were reanalyzed (Figure 2) (Figure 3) to identify potential causes or reasons for this atypical black-vs-white winning pattern (Figure 1). This reevaluation should reveal if there is a potential reason or bias why the best engine around that time, was unexpectedly outperformed so strikingly and if AlphaZero’s strengths and Stockfish’s weaknesses can be found.

Figure 2 Stockfish 8 re-analysis of every move in example game 1-5. Engine parameters: 64-bit, 4 CPUs, 2139 MB. Re-evaluation profile of every position reveals the balances during the game and decisiveness of the potential biases or blunders. Green: Advantage for white; red: advantage for black; yellow: goes beyond scale; the scale is given in centipawns, which equals 1/100 of a pawn on the vertical axis; move number is given on the horizontal axis; the search depth of the program is given below the horizontal axis. The game notation and re-analysis evaluation of respective moves are found in the supplementary material link. Please note several post-opening weakness novelties that are not very common for human or Stockfish in the games in which Stockfish plays black.

Figure 3 Stockfish8 re-analysis of every move in example game 6-10. Engine parameters: 64-bit, 4 CPUs, 2139 MB. Re-evaluation profile of every position reveals the balances during the game and decisiveness of the potential biases or blunders. Green: Advantage for white; red: advantage for black; yellow: goes beyond scale; the scale is given in centipawns, which equals 1/100 of a pawn on the vertical axis; move number is given on the horizontal axis; the search depth of the program is given below the horizontal axis. The game notation and re-analysis evaluation of respective moves are found in the supplementary material link. Please note several post-opening weakness novelties that are not very common for human or Stockfish in the games in which Stockfish plays black. Machine learning bears very powerful possibilities and still has a very big future, and AlphaZero is at the very heart of these new innovations. Even if the differences, in a 100-game match were only slight or equal, one could still speak of a great breakthrough and milestone in machine learning history and Neural Networks (NN) due to the efficiency and effectiveness of machine learning. But this evidence is elusive until it is reproducible, which bears major implication for all sciences, as the majority of all publications are biased4,5,9 and cannot be reproduced in any way. 90% of scientist agree that there is a reproducibility crisis and only 3% do not see it.10

Using Stockfish8 for the reanalysis of all moves of the 10 example games between Stockfish8 and AlphaZero could identify potential blunders of very few but likely decisive Stockfish8 moves during the 100-games match (Figure 2) (Figure 3) (Table 1). This indicates that Stockfish8 might have unusually underperformed in ca. 1 move per game.

Potential Stockfish8 weaknesses were identified via automated move-by-move evaluation screening of all 10 example games. An evaluation landscape of all 10 example games is given in Figure 2 & Figure 3 in centipawns (cps). These move-by-move cp-evaluation landscapes were then used for manually screening of atypical or weak moves.

While it seems likely that potentially underperforming or biased Stockfish8 moves, which are termed potential Stockfish8 blunders in Table 1 (i.e. potential or debatable Stockfish8 weaknesses or atypical novelties), seem not to be a robust stable first choice of a normal operating Stockfish8 engine under the re-evaluation settings, there is still a remaining small probability that Stockfish8 might have played them under different parameters or conditions, which were not entirely defined in the publication. Still, most moves will likely reproducibly vary also if another re-evaluation would try to reproduce some of the original moves. Most if not all of the highlighted blunder moves in Table 1 raise new inevitable questions about the reproducibility of this 100-match game result and the AI-performance in chess, as at least one move per game could be indicated as a potential blunder or bearing an atypical likelihood, and others potentially also could. At first glance, it seems possible that these unexpected Stockfish blunders or novelties (Table 1) would not have been very typically played in an official engine tournament under controlled engine settings, best engine parameters and full information details of the hardware and software configurations, with an independent referee, and without conflicts of interests or systemic biases.^4,8

Lack of reproducibility that AlphaZero is the best engine

As a result, this work must suggest that it is not finally proven if AlphaZero has already achieved to be the best chess engine worldwide, which should be reassessed in a new and official engine tournament match. Nevertheless, AlphaZero has achieved a very strong play that could be better than Stockfish8 and further interesting findings.¹ Access to the engine and all 100 games, however, is still needed for a best engine claim, as much as an independent tournament engine match, and reproducibility is not assured, like for most scientific publications of today, due to a lack of the incentives. Potential Stockfish8 blunders might represent what was recently termed strategic biases⁹ in all sciences.^4,5

A claim to be the best angine must anyhow be proven under official settings. AlphaZero seems to have the potential but will it win against a normally performing Stockfish or now Asm Fish? The best chess players also have to play more than one match alone at home. Thus, it is absolutely required to make the AlphaZero engine available as an open source, like it has been done with most of the top-rated engines including the latest versions of Stockfish, Houndi, and others, or to go with it to official engine tournaments like the CCRL or alike. Once a first position can be shown i.e. in an independent engine match, a claim can be made, otherwise some doubts will or can prevail (Figure 2) (Figure 3.

Comparability challenges of AI matches

How to compare AI apples with AI pears? This becomes a more technical question that must be resolved in its details in order to ask the right experimental questions. It becomes clear that the engine parameters, hardware, and software configurations are also another key part of these questions. While hardware parameters are at a high level of performance, they could be resource factors in saturation and less relevant for the final engine performance, which has been recently suggested also by chess grandmasters. This can be but could be also different due to the very different approaches used. For example, the speed of the calculation could have an impact on the performing operating chess engine software and its timing of the respective first choice moves at a given time interval. To test IA software ability, self-learning or pre-programmed, it would be important to keep the hardware and all variables constant, in order to test for engine abilities. If this is, however, not possible, one still could compare two systems of software and hardware on the board. But here the KPI calculation/s could potentially bear a different meaning.

How to achieve an equal firepower or equal calculation power between so different approaches? Are we comparing software or hardware or, in fact, always both? AlphaZero uses special hardware developed by Google that was running on a single machine with 4 TPUs, which are Google’s own Tensor Processing Units rather than the available CPUs, Central Processing Units. In the NN case of second-generation TPUs this would correspond to 180 TFLOPs or TeraFLOPs, 10¹² floating point operations per second – at Google, one new cloud TPU delivers up to a 180 TFLOP floating point performance for machine learning engine models - so it is difficult to compare.

On the other hand, the best CPU computers do normally not reach this very high TFLOP-level of computation power and one could assume that Stockfish8 was running on a 100-fold slower hardware, but these details have not been clearly specified in the publication.¹ In light of this, it becomes questionable if AlphaZero really derived and abstracted findings in a “human-like” intuitive manner, as the hardware is probably more than 100-times faster,¹ as compared to a single machine with “64 CPU threads” and a hash size of only 1 GB.¹ This view is further sustained by table 1, which identifies many underperforming moves of Stockfish8 on a much slower laptop pc with a lower-end mass product standard’s minimal configuration (see also materials and methods).

Despite these points, there have been not many or no obvious experts that were allowed to reveal this to the media or to the public. Contemporary censorship of all hidden experts opinions causes extreme biases and dominant one-sided views.^4,5,8 It is almost impossible to publish independent views in today’s journals.

Secondly, from the software side, there are the same potential issues of a lack of computational comparability. If Stockfish8 utilizes 70 Million (M) calculations per second and AlphaZero only 80 thousand positions per second¹ the question arises of how, and how evenhandedly, “positions per second” is defined. What are the 180 TFLOPs good for if it calculates in sum so much less, are calculation results stored on hard drive books? Why do you need a 100-fold better machine if you need 875-times fewer calculations per second? What is the remaining hardware and software doing and calculating? Is this a “more human-like approach” or a lack of comparability?

A closer look at the general differences by comparing the overall hardware and software features of the individual approaches of the two different engines might help to estimate the level of comparability (Table 2).

AlphaZero uses MTCS (Monte-Carlo Tree Search) instead of Stockfish’s alpha-beta searches with domain-specific enhancements¹ that most of all conventional non-learner AI engines are using (Table 2). The search focus is achieved by deep NN for AlphaZero, while it is alpha-beta pruning in Stockfish8 that deprioritizes those by internal score disproven tree branch variations or lines. Non-linear approximating deep NN positional evaluation is used in machine learning while Minimax evaluation is in place for most of the top-level competitive traditional engines like Stockfish8. All further software differences are listed in Table 2 to give an overview. During the learning phase, each MTCS used 800 simulations.¹ Learning happens by selecting moves in proportion to root visit count.¹

But how long did the learning phase last before the 100-game match has started? The abstract reveals it took AlphaZero 24 hours of learning to outperform Stockfish8.¹ In the paper, we also find a Figure 1 that states that it only took 4 hours and 300k steps based on artificial Elo-rankings. Next, it says that “We evaluated the fully trained instances of AlphaZero against Stockfish8 (…)”,¹ which, would indicate that AlphaZero might have trained for longer, for basically an unspecified time at least in theory. It seems to be not fully specified, is it 4h, 24h, or years – against itself, or did it also learn by playing Stockfish? What was the role of the opening books?

Multiple roots can boost artificial intelligence

Did a human-like thinking evolve or develop during this training period is thus an important and intriguing question that is often and widely asked by experts and the media. How does AlphaZero learn top-level chess play so fast? How can it restrict the number of positions per second to 80.000 while Stockfish utilizes 70 million per second? Stockfish8 reduces and extends the search depth by alpha-beta pruning for bad or promising variations, respectively. But what about the search depth of AlphaZero? The answers could depend on search depth that is not clearly stated in the publication.¹ For example, what if calculation means sequence tree length of 50 for one machine and 25 for the other? Are all simulations played to the foreseeable end? Search depth is the number of half-moves, so-called “plies”, the search function nominally looks ahead, between the “root and the horizon node” at the depth of zero where “quiescence searches” are performed. As an analogy, these quiescence searches maybe resemble what is called in the human chess school of thought the “candidate moves”, lines and variations in trees.

Search depth could be thus a very important and crucial aspect of AI and machine learning. It is frequently viewed to be a basic feature of intelligence to focus on most promising facts and moves and to exclude zero-chance opportunities with certainty. Stockfish uses killer-move and counter move heuristics and AlphaZero semi-random tree play out evaluation and scoring at the root and subtrees¹ that might be as crucial as search depth.

If entire games per move are calculated one would reach a maximum tree variation depth but this is only possible for high-quality candidate moves or via super-calculation-power in TFLOPs. For both engines, the quality of candidate moves is crucial, for quiescent searches at the root for Stockfish or root visit scores and subtree for AlphaZero. The centerpiece of quality begins at a root for both engines. The root is the basis for both approaches.

But this would mean that you can improve the performance of artificial intelligence engines by optimizing the candidate moves for the quiescent searches (for Stockfish-type engines) or the tree root visit scores (for NN-engines like AlphaZero). In both cases, Stockfish8 and AlphaZero, the performance could be theoretically optimized by stabilizing the root tree via multiple roots: more roots more second order stability but also not too many, only a few as a test, to not drop calculation power and depth. Whenever an engine searches for candidate moves or top evaluated trees it has to perform a time-consuming computing work in an unpredictable landscape of future move events that bears both very simple and also very difficult variations in a vast amount.

To best navigate through this extraordinary tree landscapes universes the engine must overcome the resistance of these difficulties that require longer computing times. One simple way to increase the stability of the root and trees would be to allow the engines to have multiple roots at once in a testing manner and not just one root, as one root could be less stable and more prone to fall into a future “difficulty trap” that might be seldom but is possible. More roots at the artificial depth of zero (only one root would be truly and contemporarily zero) would stand on multiple feet like a chair or table needs more than one table leg roots. Candidate moves and tree visits would be simply harmonized in a permanent brain environment in order to not oversee a very important variation or line. Seldom, one root might be keeping the machine focus too busy in calculating the many very difficult variations and tree branches. Here, a short second or third root might provide a new additional landscape, a short engine-restart from an adjacent position, to build on and to work within the searches, as navigation through multiple roots might advance solution finding by helping to find the potentially overseen important lines. This assumes that AlphaZero also uses an evaluation function in its play and not only de novo derived chess understanding, feel free to correct the author if this is different, it is just not imaginable for the author that an NN could do this without a live evaluation function. To find empirical indicatory evidence for the theory shown in Figure 4 that could help to boost AI and AI predictive “move searches”, all AlphaZero games were screened and investigated with regard to this question using a permanent brain function combining multiple roots for a position that could serve as a read-out.

Empirical test if multiple roots boost AI

To empirically prove the hypothesis (Figure 4) of the theoretically derived technical AI idea how to improve both AI in chess engines, all games were screened for a good read-out position to test the hypothetical model. Example Game 3 of AlphaZero vs Stockfish¹ was co-incidentally identified re-evaluated by Stockfish8 64-bit 4 CPUs and 2139 MB hash: in move 49 Stockfish8 has originally played Rf8 in the 100-game-match versus AlphaZero.¹ At first glance, this seems not to be the very best move even for a human player because, someone might argue, as it further and unnecessarily crams the king and queen and obviously suffocates their flexibility and options as it might to much limit their potential future moves (Figure 5). But this seems not to be a Stockfish8 blunder but a real Stockfish8 move that drives a loss of the game. In fact, in a 1-minute per move calculation, Stockfish8 could independently reproduce to find Rf8 as the first choice with a significant likelihood of ca. 19% of all cases (N=16), an unexpected reproducibility that has led to a new finding. The obviously better move 49. …Kf8! is not always found by Stockfish8 (64-bit, 4 CPUs, 1 or 2139 MB hash), which has played the right move Kf8 in only about ~50% of cases (Figure 5). Now, if multiple roots are simulated by a permanent brain function and a pre-calculation is performed of the X_n-1 root ply, it yielded a significant improvement in the overall Stockfish8 performance in this position and also in several other games and positions (not shown).

Figure 4 Theoretical model for AI engines (e.g. Stockfish and AlphaZero in the game of chess) to use more than one but a few multiple staggered roots to provide more stability in the calculations stemming from multiple starts that improve the second-order probabilities to identify the best move in some circumstances of future variations. Every AI calculation that starts at a set ply depth of zero can deal with extreme quantities of variations and high levels of unpredictability. In the theoretical model, if multiple roots are used and computing is harmonized subsequently it might become more likely to not oversee a best move and subtree as new screening approaches arise by starting at different levels e.g. for the evaluation and quiescence searches. R: hypothetical resistance of the future landscape of heterogeneous complexity trees with both simple and difficult variations. α: pseudo-adjustment variables for the engine approach whether in a semi-random tree-walk or in handcrafted heuristics or alike. Noteworthy, adjacent roots can be added from the past and future, as both could provide more stability for AI.

Figure 5 Preliminary simulation of multiple roots via permanent brain functions and pre-calculation of the Xn-1 root can yield a significant improvement in Stockfish8 engine performance in a read-out position (49./50.; game 3), as sometimes variations can be disproven by computing subsequent plies. This might suggest new benefits for AI in general if multiple roots are used that could simulate an effect of using multiple cores. (left) pre-calculation of move 49. Rf6 using a permanent brain function can yield an improvement of Stockfish 8 64-bit 4 CPUs moves. (center) board at position 49. Rf6 of game 3.1 (right) multiple roots could slightly help to find the best move (p=+17%) and it also helps to faster disprove a losing move, here Rf8 (p=0%).

Still, this “AI boost effect” could be also simply explained by a longer calculation time and/or by a single root and variation trap in the future (the hypothetical model in Figure 4). To exclude this possibility, a longer calculation time was tested without the use of multiple roots and did not improve the result (N=4; 0% Kf8 2 min., 1 root, no permanent brain): thus it is not the time that has improved the predictive AI search but “multiple roots”.

More calculation time also did not play a role in several additional positions in which Stockfish8 calculated many 0.00-evaluation-type candidate moves and lines (not shown). In this very specific position (move 49. … of game 3), AlphaZero has seemingly uncovered an interesting roughly 50%-weakness of Stockfish8 that might help the developers of Stockfish, AlphaZero, engines and AI, to improve future versions in some way, which cannot be finally proven here. Also, placing the queen earlier on 45. … Qh8 could be questioned. Prolonged indifferences during the calculation (0.00) seem to cause this ca. 50%-level type of a natural Stockfish error. These indifferences stem from an incomplete candidate move portfolio while the quiescence search is trapped in a difficult-to-assess variation and seemingly a wish to delay a rejection of a previous estimate (Figure 4), and multiple roots could be the solution here.

As it is not the time given for the calculation, but the root start of the quiescence search with a potential lack of all candidate moves and lack of all evaluations (often seen as 0.00 values), it is likely that a switching to multiple roots could solve this instability issue of such and other AI searches to approach a difficult tree landscape. This can be shown in both, (a) in the pre-calculation (X_n-1) and it can be also shown in (b) the post-calculation (X_n+1) mode using permanent brain linked roots (Figure 6). Multiple starter simulations of adjacent positions could thus improve the searches. Empirical indicatory evidence was found for the model but it requires further testing.

Figure 6 reveals the benefit of including a future root, Figure 6A shows that move 49. … Kf8 would be the most practical and probably best solution as identified by Stockfish8, a human player, and asmFish, the recently best Stockfish engine. Without any pre- or post-calculation, Stockfish8 can be sometimes indifferent as seen by 0.00 values for all variations and it oversees the best solution, i.e. the Kg8 line. However, if a pre- or post-calculation is added via the permanent brain function, Stockfish8 is almost fully able to see that Rf8(?) is an unfavorable move and increases the likelihood of Kf8 up to 100% (Figure 6B). Hence, it seems to be a quiescence initialization weakness causing an alpha-beta pruning weakness to find the most promising lines. AI using multiple roots at a depth of -1, 0, +1, +2 thus “can much advance the search results”, which might be interesting for Google too. Google has actually already started doing this by integrating our search preferences for advertisements that could be further advances in this AI way using multiple roots. Thus, the model in Figure 4 is helpful, as predicting future moves and pre-calculations could help Stockfish8, AlphaZero, chess and search engines, and AI, in general, to find the best move in an indifferent way: multiple roots can boost the performance, which is supposed to be just a draft not an “overclaim”, but the “overclaim” mechanisms of publishing become apparent. The future will tell if multiple roots can boost AI performance, and only the future can tell if AlphaZero is the best engine one may argue?

Figure 6Post-Calculation might help Stockfish 8 and chess engines (incl. AlphaZero) to boost performance.

Artificial intelligence and the openings

Finally, also the openings of the chess game are crucial and decisive and an interesting case to study AI and searches of the hay needle in vast amounts of complexity and universes of combinations. Table 1, Figure 2 & Figure 3 have revealed that Stockfish8 sometimes had an unexpected opening or post opening weakness. Hence, a final proof is still missing that AlphaZero can beat Stockfish8’s high-performance play, and it would be important to retest this in a reproducible and unbiased fashion in an independent engine match tournament. But what about AlphaZero? How well did AlphaZero perform in these decisive openings? The answer is very well, judging from a human perspective that is built upon a human-centric game database of the best human master games played.

The winning chances of each opening are summarized in Table 3. Interestingly, in the 10 example games given, AlphaZero could best win as black in the C65 openings of the Ruy-Lopez/Berlin Defense, which could indicate a weaker start for white but not necessarily. Of the 8 winning games as white AlphaZero favored the Queen’s Indian Defense (Table 3). What do we know from human games in these opening positions is also given in Table 3. For instance, in game 3, the opening can be viewed up to the move 15., there were 13 games in the human chess games database: 38% were won by white, 46% were a draw, and 15% were won by black. AlphaZero was competing with Stockfish8 as “fully trained instance(s)” (Silver et al.,¹ 2017, page 4), however, officially without any opening book knowledge, as all was learned by self-play reinforcement machine learning of the deep NN.¹ The question arises, how well do the 10 AlphaZero opening choices (Table 3), only derived by AlphaZero e.g. the root visit counts, overlap with the best choices of top-level human openings i.e. the best win-chances know so far?

The answer to this question is given in Figure 7 to some global extend: In the 8 games as white, AlphaZero was clearly selecting the better openings – at least from a 100-years of accumulated human chess knowledge perspective – and also compared to Stockfish8, irrespective of all the post-opening weakness novelties listed in Table 1. Moreover, the heightened opening chances for white and the lower wining chances for black can be shown to be significant (p>0.05, N=8) if compared to all games in the database (N=3.382.780). This shows that AlphaZero would have learned top openings that a century of all human top players had crystallized and precipitate over time. Or were all human games always inspired by engines that cause the overlap? AlphaZero would have reinvented this Gordian wheel of openings in breakthrough time of only hours by playing nobody more intelligent than itself, starting from scratch. One might argue that an evaluation function is still at work and that Stockfish-8’s AI does not even have to train at all to win against maybe all top players. But selectivity for these openings could have many reasons. Still, how can such a human-like book-knowledge choices pattern by Alpha be explained by a self-learning algorithm? Would the same openings be discovered by machines, or has there been an assistance? Or are these openings really the best for both human and engine play? Playing an engine is different than playing a human opponent, no?

Figure 7Comparison of Opening-Chances of AlphaZero games with the database of 3M human book games.

Supposedly, AlphaZero did not use any opening book knowledge and statistic, it has gained an excellent feeling for openings in rather very short time, but how long were the fully trained instances really trained? This is not precisely explained in the publication¹ but 4h or 24h would be very short for this. The question arises whether AlphaZero’s openings in these example games are significantly better than the global book win-statistics.

Figure 7 reveals a striking difference of 45% winning chance for white (AlphaZero) versus only 17,4% winning chances for black (Stockfish8), already after only 10 moves on average. If AlphaZero really did not use any opening book knowledge, this would mean that the 3.3 Million grandmaster games were not played for anything. Both machine learning and all human top games overlap in the most promising openings. But what about all the other known book openings e.g. with good winning chances for white? We cannot say at this point in time. In the learning phase of AlphaZero, there was a growing interest in E00 Queen’s pawn games and a decline in Spanish games (Table2 of Silver et al.¹). Still, engine learning developed a pickiness of promising human openings.

In summary, from an anthropocentric view, AlphaZero did extremely well in the openings and post-opening phase and Stockfish8 was very slightly but decisively underperforming; this could be theoretically a human-bias. Stockfish, for example, in tendency fails to find the best human openings without the use of any book knowledge.

Indications for the hypothesis that all sciences of today are systematically biased could be found. Global biases an blockage of the majority of postdocs predominated every scientific progress of our time in all fields that are led by faculties: physics, biology, chemistry, philosophy, mathematics, medicine, all social sciences, psychology, law, economics and business studies. Bias is a major concern for all progress, the real truth, common sense and thus the future of all sciences.^4,5,8,10

The re-analysis of probably one of the most outstanding breakthrough publications in Artificial Intelligence, in the recent scientific literature since some time, reveals some inevitable concerns about the reproducibility and comparability of the still very promising AI performance. Accessibility of a setting to independently reproduce the engine match and real tournament settings were not fully given, only 10 selected games of 100.¹ Re-evaluation screening of every move has revealed atypical moves of Stockfish8 in almost every example match of AlphaZero and Stockfish8. Hence, this analysis must conclude that a final proof was not undoubtedly given that AlphaZero could or can beat the best chess engines in the world and it clearly requires several further proofs and more details and access. The best chess players in the world also need to prove this more than once in an independent tournament setting, and not only at home experimental conditions without a referee and asymmetric settings information.

Bias arises in uncontrolled settings, reproducibility issues arise systemically but this group is not to be blamed, as most publications do not even show the ambition to enable reproducibility in most of all sciences.^4,5,8,10 Still, reproducibility will secretly remain, and inescapably, an absolute requirement of all science that is not met no more worldwide. Who could still blame an individual working group only based on indications if everyone has gone wrong everywhere already? Hence, this conclusion proposes an overhaul of all sciences: (1) to stop all systematic biases, (2) to better reproducibility, and (3) to provide sustainable and independent career paths to all scientists Postdocs and PhDs without extreme pressures, and prevailing conflicts of interest are lurking around every corner.

A boost of Artificial Intelligence could be achieved if multiple roots are simulated for quiescence searches or any other searches or calculation. Two adjacent eyes see better than one and make a 2D landscape 3D. Similarly, two adjacent roots are better than one and can make the complexity landscapes appear more unbiased and dimensional for the permanent brain to work with. They not only give double information but also different starting positions that are adjacent and slightly shifted, by only very few past and future plys in chess. AI must leave the 2D PCB board to become more three dimensional and multiple roots might more cores for probably very many IT AI applications.

 In opposite to Stockfish, AlphaZero is closer to a better multiple root solution, which might have yielded an AI advantage, as AlphaZero always simulates “multiple games” and not only variations, as compared to Stockfish, for root visit counts, as MTCS assesses series of simulated games that traverses a tree from root s_root to leaf.¹

Hence, roots might be better assessed by AlphaZero and even with less of a bias due to the randomness of play out. This work proposes that roots could be simply simulated before the next root state is reached (Figure 4), and the calculation results of the previous root calculations should also be used. Even if the use of several future roots or predictive roots can be also suboptimal or even false, it sometimes can still yield a higher stability of the search on the zero-ply root due to permanent brain functions, which in a next step could be further advanced, in order to not miss out on a very important variation due to time wasting on what would be maybe called a "false in-depth focus" of an AI exercise.

The performance boost of AI via multiple roots could become a standard for “combinatorial solution finding in complexity” – but who knows if this will be an “over-claim”, and what will be the best AI engine? Only the future of AI that is already approaching us.- Supplementary material is available using the following hyperlink.

I would like to acknowledge the University of Truth and Common Sense that is to be seen as a School of Life and Independence.

The author declares that there is no additional conflict of interest to the conflicts of interest found in all sciences.

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