=Paper=
{{Paper
|id=Vol-2862/paper11
|storemode=property
|title=Extracting Physics from Blended Platformer Game Levels
|pdfUrl=https://ceur-ws.org/Vol-2862/paper11.pdf
|volume=Vol-2862
|authors=Adam Summerville,Anurag Sarkar,Sam Snodgrass,Joseph Osborn
|dblpUrl=https://dblp.org/rec/conf/aiide/SummervilleSSO20
}}
==Extracting Physics from Blended Platformer Game Levels==
https://ceur-ws.org/Vol-2862/paper11.pdf
Extracting Physics from Blended Platformer Game Levels
Adam Summerville1 , Anurag Sarkar2 , Sam Snodgrass3 and Joseph Osborn4
1
California State Polytechnic University
2
Northeastern University
3
modl.ai
4
Pomona College
asummerville@cpp.edu, sarkar.an@northeastern.edu, sam@modl.ai, joseph.osborn@pomona.edu
Abstract model that would allow for playing the level? And if so, how
does one extract these latent physics models? Further, while
Several recent PCGML methods have focused on generating working within the domain of a single game might make
game levels and content that blend the properties of multi-
ple games. However, these works ignore the fact that blended
this unnecessary e.g., just use the original Mario physics
levels must in some way have blended physics models that when generating Mario levels – in blended domains, there
enable playable levels. In this work, we present an approach is no ground truth physics model to fall back upon. Re-
for extracting jump physics models for such blended game cently, Sarkar et al. (2020) trained generative models for
domains. We make use of variational autoencoders (VAEs) such blended domains by leveraging a new path and affor-
trained on level data from six platformers, encoded using a dance vocabulary that enabled generation of blended lev-
previously introduced path and affordance vocabulary. Our els with paths and jumps. In this work, we directly extend
results show that the extraction model is able to reasonably this work by leveraging the jumps found in these gener-
recreate the original physics models when given ground truth ated blended levels to extract physics models for different
paths, and is able to produce physics models that can reliably blended domains. We do this by first generating levels tar-
allow an agent to play the generated levels. We also find that
there are promising results for blended physics models behav-
geting specific games and game blends, with special atten-
ing intuitively between physics models of the original games tion to generating paths that encode the directionality of the
being blended. path. We then extract physics models that could reasonably
have created the generated paths. We test this procedure by
comparing the extracted physics to the ground truth physics,
Introduction and examine the physics of blended domains, seeing how the
While methods for procedural content generation via ma- physics alter with respect to the level geometry.
chine learning (PCGML) (Summerville et al. 2018) were
initially motivated by wanting to generate novel content Related Work
in the style of existing games such as Super Mario Bros.
Most prior techniques for procedural content generation via
(Summerville and Mateas 2016; Guzdial and Riedl 2016a;
machine learning (PCGML) (Summerville et al. 2018) have
Snodgrass and Ontañón 2017) and The Legend of Zelda
focused on learning models for a single game. Such meth-
(Summerville and Mateas 2015), a new body of work has
ods have involved using autoencoders (Jain et al. 2016),
emerged that focuses on PCGML techniques that seek to
LSTMs (Summerville and Mateas 2016), GANs (Volz et
leverage trained models to blend existing game domains
al. 2018), Bayes Nets (Guzdial and Riedl 2016a), n-grams
and/or generate new domains altogether. This has produced
(Dahlskog, Togelius, and Nelson 2014) and Markov mod-
works that leverage more creative PCGML approaches such
els (Snodgrass and Ontañón 2017) for learning generative
as domain transfer (Snodgrass and Ontanon 2016; Snod-
models for games such as Super Mario Bros., The Legend
grass 2019), model blending (Guzdial and Riedl 2016b;
of Zelda and Kid Icarus. In an effort to address generaliza-
Sarkar and Cooper 2018), computational creativity (Guz-
tion and lack of data as well as wanting to discover and cre-
dial and Riedl 2018), training on multiple domains to learn
ate new game domains (similar to e.g. the game blending
blended domains (Sarkar, Yang, and Cooper 2019) or a com-
framework of Gow and Corneli (2015)), more recent works
bination of the above (Snodgrass and Sarkar 2020).
have built models that work with multiple games and do-
While some works have included path information, there
mains at the same time. This has included domain trans-
has been no notion of completing the circle i.e., do the
fer (Snodgrass and Ontanon 2016; Snodgrass 2019), game
physics latent within generated paths encode a physics
generation (Guzdial and Riedl 2018) and game blending
Copyright c 2020 for this paper by its authors. Use permitted un- (Sarkar and Cooper 2018; Sarkar, Yang, and Cooper 2019;
der Creative Commons License Attribution 4.0 International (CC Snodgrass and Sarkar 2020). Recent work (Sarkar et al.
BY 4.0). 2020) built on these latter game blending approaches by ex-
here and in the earlier work of Sarkar et al. (2020) is the
representation found here includes not just path information
but also the directionality of the path – the starting and end-
ing position found in a segment have a special representa-
tion. This allows the downstream physics extraction process
to extract the correct physics as paths are not necessarily
bi-directional (e.g., very large falls should be represented as
such, and not very high jumps). See Figure 1 for an example.
To account for differences in sizes and dimensions of the
levels in each game, we used a uniform segment size of
15 ˆ 32 for all games, adding vertical padding as required.
We focused on horizontal sections of levels, thereby ignor-
ing the vertical sections found in Ninja Gaiden, Metroid and
Figure 1: An example of the level representation used (note: Mega Man. After a filtering process to discard duplicate seg-
color is added only for presentation purposes here). Player ments and segments mixing discrete rooms, we ended up
path is represented using + with special handling of start ( 9̀ ), with 1907 segments for Mario (SMB) (referring to both ver-
end (¯), and places where the path overlaps non-empty tiles sion of Mario mentioned above), 504 segments for Ninja
(ě for a moving hazard and ‹ for a collectable). Gaiden, 1833 segments for Metroid, 924 segments for Mega
Man and 775 segments for Castlevania.
tending their domain from two to six games, introducing a Generative Model
path and affordance vocabulary and training on levels an-
notated with A* paths derived from the jump arcs of the For generating levels from which to extract physics, we used
respective games. This enabled generation of blended lev- a Gated Recurrent Unit-Variational Autoencoder (GRU-
els spanning all the games while also containing traversable VAE), implemented using PyTorch (Paszke et al. 2017). The
paths and jumps. In this paper, we utilize the paths and jumps encoder consisted of 3 hidden layers of size 1024 while
in the blended levels generated by this latter approach to ex- the decoder had 2 hidden layers of size 256—both using a
tract physics models for the blended domains. dropout rate of 50%. To help with convergence, the varia-
Such physics models have not been the subject of much tional loss was annealed linearly from 0 to 0.05 times the
prior PCGML work with a majority of prior PCGML re- variational loss over the first 5 epochs before the rest of the
search focusing on learning models of game levels and only training continued at that rate—for a total of 50 epochs using
a few attempting to learn models of game physics and game the Adam optimizer and a learning rate of 10´5 . At decoding
rules. Guzdial, Li, and Riedl (2017) presented an approach time, the decoder is initialized with a latent embedding and
termed game engine search for learning the rules of Super then decodes in an auto-regressive manner with sampling.
Mario Bros. using video gameplay data. Summerville, Os- For each generation, we sampled 10 segments and kept the
born, and Mateas (2017) learned a hybrid automaton de- one with the lowest perplexity (highest likelihood).
scribing the jump physics in Mario. Similarly, Summerville
et al. (2017) used data from a Nintendo Entertainment Sys- Physics Extraction
tem (NES) emulator to learn automata describing the jump To extract the physics, we must first define the “physics” of
physics of a large number of NES platformer games. To our a static level. In part, this seems ridiculous, as a static level
knowledge, our work is the first to extract such physics mod- cannot have a conventional physics model, as there is no no-
els for blended game domains. tion of time. However, while this seems like an intractable
problem, we believe that for several platformer games, there
Level Data and Representation is an implicit correlation between horizontal position and
For our approach, we used six classic NES platformer games time – e.g., a speedrunner of Mario is almost always moving
- Super Mario Bros., Super Mario Bros. II: The Lost Lev- to the right as quickly as they possibly can. In fact, the A˚
els, Ninja Gaiden, Metroid, Mega Man, and Castlevania - all agent that we use to simulate “playing” the levels also oper-
represented using the path and affordance vocabulary intro- ates under this assumption. Thus, we think it is reasonable to
duced in (Sarkar et al. 2020), which in turn was derived us- relax the physics models from a notion of y position versus
ing the Video Game Level Corpus (Summerville et al. 2016) time to a relation of y position to x position – with the un-
and the Video Game Affordance Corpus (Bentley and Os- derstanding that the x position is supposed to be constantly
born 2019). Because these games have disparate vocabular- progressing in the direction of the goal. It is important to
ies of tiles, we need a common language to describe all of note that the “physics” model we are extracting actually sup-
the levels – solidity, climbability, passable, powerup, haz- ports an infinite number of different possible physics mod-
ard, moving, portal, collectable, and breakable. These affor- els – changing the maximal x speed will result in different
dances can be combined – e.g., a breakable brick would be physics models. Some games have much slower horizontal
“breakable+solid” – which leads to 14 unique combinations speeds (Castlevania has a maximal horizontal speed of „3.7
(see (Sarkar et al. 2020) for a more detailed description). tiles per second), while others have much faster speeds (Su-
A key difference between the level representations found per Mario Bros. has a maximal horizontal speed of 10 tiles
Standard Physics Model Extracted Physics Model function S EGMENT E XTRACTION(path)
Parameters Ñ jumps, f alls
Impulse ( By ) By
Impulse ( Bx ) l Ð path[0]
Bt
By
Gravity ( Bt2 ) By
Gravity ( Bx gp Ð onGround(l)
2)
Assumptions yp Ð l.y
Player has control Player takes the jumps Ð rs
over height of jump highest possible jump jumping Ð not gp
jumping Ð not gp
Player can alter horizontal Player is always moving at f alls Ð rs
position during jump maximum horizontal speed seg Ð rls
for l in path[1:] do
Table 1: A comparison between the standard physics mod- g Ð onGround(l)
els as found in platformer games and the extracted physics y Ð l.y
models produced here. seg.appendplq
if g and not gp then Ź Landed
jumping Ð False
per second) – the rest of the games we looked at have speeds f alling Ð False
of around 5.5 tiles per second. If one wished to take these f alls.appendpsegq
extracted physics and use them in a playable game, the dif- seg Ð rls
ferent x speeds would result in different feeling games, but else if not g and y ą yp then Ź In Jump
somewhere in the 4 to 10 tiles per second range would result if not jumping then
in games playable by humans. jumping Ð True
We also note that a large number of platformer games f alling Ð False
allow for the player to control the arc of the jump based seg Ð rsegr´1ss
on how long they hold the jump button – in this work, Su- end if
per Mario Bros., Metroid, and Mega Man all allow for this, else if not g and y ă yp then Ź Falling
while Castlevania and Ninja Gaiden do not – and this no- if jumping then
tion of player control is not contained within the static maps. jumps.appendpsegq
Again, we make the simplifying assumption that higher end if
jumps are preferred – we want to determine the frontier of if not f alling then
what space is reachable. Table 1 describes the “physics” in f alling Ð True
contrast to the standard physics found in the game. jumping Ð False
One final note – many games have different physics mod- seg Ð rsegr´1ss
els when the player is falling in their jump versus when end if
they are in the rising portion of their jump – e.g., in Super end if
Mario Bros. gravity can more than double when the player end for
is falling. As such, we learn a separate gravity value for the end function
rising and falling portions of a jump.
Figure 2: Pseudocode describing the segmentation of jump-
Extraction ing, falling, and being on the ground from path positions
Having defined the physics model of a static level, we
now discuss the process for extracting said physics model.
To determine how the path represents the player’s position rr0, 0s, r0, 1s, r1, 1s, r1, 2ss then the highest recorded posi-
through time, a Breadth-First Search is performed, begin- tions per x value are rr0, 1s, r1, 2ss. This is done for all jumps
ning from the start position and progressing until the end and the statistics for the x positions found across all jumps
position is found. This provides a coarse notion of the pro- are calculated. Jumps are then scored by how many of their
gression of the path. The path is then followed and the algo- y positions agree with the P ´percentile y values across all
rithm described in Figure 2 is used to separate the portions of jumps. P is then a hyperparameter that can be tuned to de-
the path that are (1) grounded, (2) jumping, and (3) falling. termine what one expects to see from the jumps – given that
Once segmented, the segments are filtered to remove 3 of the games have variable height jumps, our inductive
noisy jumps: bias is that jumps higher than the median should be selected
• Any jump or fall of two or fewer data points is removed – given that we wish to find the upper extents of possible
these are too small to derive any useful physics from jumps. We filter jumps that have more than 50% disagree-
ment with the P ´percentile jump. In the next section, we
• Any segment that moves more than 2 tiles in a single step discuss the criterion for the selection of P . Finally, given
is removed – these represent “broken” paths and are likely the filtered jumps and falls, we perform an Ordinary Least
to represent a corruption of the physics Squares regression where the dependent variable is y posi-
For each x position in a jump, the highest correspond- tion and the independent variables are x (corresponding to
ing y value is recorded – e.g. if a jump consists of Impulse) and x2 (corresponding to Gravity).
Game Original Generated covered. Mega Man (light blue) has a slightly different arc,
Castlevania 1.03 4.98 but reaches the exact same height and has the same horizon-
Super Mario Bros 0.71 0.25 tal space. Ninja Gaiden’s (dark blue) extracted jump falls
Metroid 0.93 0.94 short of the true jump both in height (reaching a maximum
Mega Man 0.22 2.94 of 3.8 tiles instead of 4) and in distance (9 tiles instead of
Ninja Gaiden 0.80 1.03 10). Finally, Castlevania’s extracted jump has the same hor-
Total 3.69 10.14 izontal reach, but reaches higher (2.8 tiles instead of 2 tiles).
Generally, these jumps would support much of the same
Table 2: Root Mean Squared Error (RMSE) for y values per gameplay, although the height differences for Super Mario
x value for the physics models extracted for the original lev- Bros. and Ninja Gaiden would need to be bumped up to
els and the generated levels when compared to the actual the nearest whole number of tiles to have the same game-
game physics. We also see Castlevania is difficult to extract, play. With this, we feel satisfied that the physics extraction
in part because its arc does not actually follow a parabola. process works well enough to faithfully extract physics that
would support playing the game, and we turn our attention
to the generated levels, to see how faithfully they are able to
Evaluation/Discussion represent the physics.
To evaluate the extracted physics, there are a number of con-
cerns. Latent Reconstructions
1. Faithfulness of the Generated Physics – Do the paths To evaluate the generated physics, we sampled the gener-
contained in segments generated targeting a specific game ative model to produce level segments that were from the
domain faithfully recreate the physics found in the origi- latent space of the encoding corresponding to each game.
nal segments? To do this, we first obtained the latent encoding for every
level segment from a given game. We then calculated the
2. Validity of the Extraction Process – Is the extraction
mean and standard deviation for these encodings. Finally,
process capable of reconstructing the original jump pa-
we sampled 2000 encodings from a normal distribution with
rameters from the training data (where the paths were gen-
the calculated parameters – these encodings were then de-
erated from an agent using the original parameters)?
coded into level segments. As a note, the level segments
3. Interpretability of Blended Physics – Do the segments were sometimes lacking in a beginning and ending (due to
found in interpolations between the original games result the stochastic nature of the generation process) – these seg-
in physics that are interpolated between the games? ments were excluded from the physics extraction process as
it is impossible to determine the progression of the gener-
Faithfulness to Original Physics ated path – in all, this led to the dropping of 326 segments in
To assess whether the original physics can be extracted, we total (3.26% of the generated segments) with Metroid hav-
first use the extraction process on the training segments – ing the highest proportion of corrupted segments (8.1%).
these should have the physics flawlessly encoded within The physics models were extracted using the same 75th per-
them. We ran a hyperparameter grid search over the P - centile criterion as computed in the original levels (no hy-
percentile to ascertain what percentile leads to the most ac- perparameter search). Table 2 shows the RMSEs between
curate physics model. We assess the accuracy of the physics the physics models extracted from the original and gener-
model by calculating the Sum of Squared Error (SSE) for ated segments when compared with the actual game physics.
y values per x value for the jump models produced by the Both errors are broadly comparable (and is in fact lower for
physics models extracted for the original segments. P “ generated segments in the case of Super Mario Bros.).
75% led to the lowest total SSE summed over all of the As noted above, the aesthetics of the reconstructed jump
games, although different games had differing values (From arcs are as, if not more, important than the errors. Figure 4
Castlevania at 60% to Super Mario Bros. at 80%). How- shows the true jump arcs in comparison with the jump arcs
ever, the mean value of the percentiles was 72.6%, so we extracted from the generated segments. We see that the ex-
feel comfortable with using the 75th percentile jumps for tracted jump for Metroid (orange) reaches the same height,
the physics model (which confirms our inductive bias that but does not reach quite as far horizontally, due to a higher
we wanted jumps higher than the median). fall gravity. We also see that the extracted jump for Mario
Of course, in some sense, the aesthetics of the recon- (red) reaches the same height but extends horizontally.
structed jump arcs are more important than the error – most We note that for the rest of the games, the generated arcs
importantly, do the extracted jumps result in the same ma- show a regression towards the physics of Super Mario Bros..
neuvering and reachability as the true physics? Figure 3 The generated Mega Man and Castlevania arcs are nearly
shows the true jump arcs in comparison with the jump arcs identical to the Super Mario Bros. arc. Finally, we see that
extracted from the original segments. We see that the ex- the generated Ninja Gaiden (dark blue) arc is very similar
tracted jump for Metroid (orange) reaches the same heights, to the one extracted from the original segments reaching not
but does not reach quite as far horizontally, due to a higher quite as high but having the same horizontal duration.
falling gravity. We see that the extracted jump for Mario Again, generally, these physics would support the same
(red) is a bit short in height (reaching only „3.5 tiles high gameplay – in fact Super Mario Bros. would be playable as
instead of 4 tiles in height) but has the same horizontal space is with no intervention with the model extracted from the
Figure 3: The jump arcs for the true physics (solid lines) compared to the extracted training jumps (dotted lines).
Figure 4: The jump arcs for the true physics (solid lines) vs the extracted generated jumps (dash-dotted lines).
generated levels (unlike the model extracted from the orig- average across the games (although a jump of 3.5 in height
inal segments). Also, while the models for Castlevania and and 9 in width would be closer to average), so it seems that
Mega Man are more lenient for the generated extractions most blends actually go through a sort of in-between average
than the true physics, the levels would be playable with the space that just encodes generic platformerness as opposed
extracted models. to any real per-game-pair specific physics. That being said,
Metroid – being the most extreme of the physics – does tend
Blended Physics to have some blends that incorporate its higher and longer
jumps – namely, blends with Castlevania (Figure 5h), Mega
Unlike the above categories, there is no direct comparison Man (Figure 5j) and Ninja Gaiden (Figure 5b).
to see how well the extracted physics recreate the original
physics – instead visual inspection is the best way to as- Conclusion and Future Work
sess the interpolated physics. Figure 5 shows the physics ex-
tracted from interpolations between different games. To get In this paper, we presented a method for extracting “physics”
the interpolations, we take 10 segments from each game, en- from static levels of the kind often used in PCGML level
code them into the latent space, interpolate between all pairs generation. We compared the extractions from both ground
of encodings at 25%-75%, 50%-50%, 75%-25%, and then truth training examples and generations to the ground truth
decode 20 times (since the decoding process is stochastic, physics. In addition, we explored the physics found within
the same encoding can produce different segments). We note blended domains, with some promising examples of blended
that for most pairs, the interpolated physics seems to settle physics.
into a jump that is actually unlike the exemplars, but rela- In the future, we would like to expand this work to explore
tively stable across the blends – a jump that reaches about different level orientations (e.g., vertical levels found in Kid
3 tiles in height and 8 tiles in width (which is actually quite Icarus). We would also like to explore the inverse process –
similar to the jump of Mega Man). This jump is somewhat given a physics model, generate levels that are playable.
(a) Interpolation between SMB and Ninja Gaiden (b) Interpolation between Ninja Gaiden and Metroid
(c) Interpolation between SMB and Metroid (d) Interpolation between Ninja Gaiden and Castlevania
(e) Interpolation between SMB and Castlevania (f) Interpolation between Ninja Gaiden and Mega Man
(g) Interpolation between SMB and Mega Man (h) Interpolation between Metroid and Castlevania
(i) Interpolation between Mega Man and Castlevania (j) Interpolation between Metroid and Mega Man
Figure 5: The jump arcs for the interpolations between games. We note that many of the blended jumps form a sort of “average”
jump (found in (a), (b), (c), (d), (e),(f), (g), and (i)) where the jump reaches about 3 tiles in height and 8 tiles in length. However,
some blends have more interesting, intuitive blends, such as those between Metroid and both Castlevania and Mega Man.
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