A Case Study of Expressively Constrainable Level Design Automation Tools for a Puzzle Game

A Case Study of
Expressively Constrainable
Level Design Automation Tools
for a Puzzle Game

Adam M. Smith⌘⌥ (presenter),
Erik Andersen⌥, Michael Mateas⌘, Zoran
Popovid⌥
expressiveintelligencestudio centerforgamescience

⌘ UC Santa Cruz ⌥ University of Washington
amsmith@soe.ucsc.edu

FDG 2012 – Raleigh, USA

Refraction (deployed game)

Over 495,000 plays on Kongregate.
expressiveintelligencestudio UC Santa Cruz

Towards Infinite Refraction

 Automatically designed levels & progressions

 Customization for individual players

 Educational soundness


Requirements for Design Automation

 Solves hard combinatorial search problems in
reasonable amounts of time

 Small amount of code

 Trustable output
 Hard constraints
 Global optimization


Refraction gameplay


Roadmap
 Design automation problems
 Original solutions
 Introduction to answer set programming
(ASP)
 New solutions
 Comparisons: size, speed, style
 Lessons learned


Three Design Automation Problems
 Synthesis
 Mission Generation
 Grid Embedding
 Analysis
 Puzzle Solving


Problem: Mission Generation
Expressions: “((1/2)/2)”, “(1/2)+(1/4)”, “(1/4)+(1/4)”,
Parameters: 24 blockers, 7 benders, 7 distractors


Problem: Grid Embedding

Mission

Space

Example: Puzzle Solving

S2
B
x
S2
S2
x x x x x x x
S2

Tgt B
x x x x x
3/4 B
B
x x
E2
B
x
C2

Tgt Src S2
1/2 1/4 C2
S2
x
S2

Src B
x x x x
1/1 B

B
x x x

Puzzle Solution


Six Design Automation Tools
Problem Original Approach New Approach

Mission Generation Custom, feed-forward ASP (AnsProlog + Lua)
algorithm (Java)

Grid Embedding Depth-first search (Java) ASP (AnsProlog)

Puzzle Solving Depth-first search (Java) ASP (AnsProlog + Lua)
S2
B
x
S2
S2
x x x x x x x
S2

Tgt B
x x x x x
3/4 B
B
x x
E2
B
x
C2

Tgt Src S2
1/2 1/4 C2
S2
x
S2

Src B
x x x x
1/1 B

B
x x x


Original Mission Generator
 Custom feed-forward algorithm:
 Several stages that iteratively expand graph
 Expression translation
 Opportunistic connection
 Target completion
…

 Note:
 Common PCG strategy
 Output meets requirements by construction


Original Grid Embedder
 Depth-first search:
 Transfer pieces from mission to spots on grid one-
by-one

 Notes:
 This is a tough combinatorial search problem!
 DFS is complete on finite problems.
 A* would be faster, but more complex.
 We want to maintain less code!


Original Puzzle Solver
 Depth-first search:
 Place pieces from tray one-by-one

 Note:
 Harder: math not resolved, no solution template
 A* ... we really want more control over
solutions!


Brief intro to ASP (not Prolog!)
 ASP (“answer set programming”)
 … is a declarative logic programming paradigm
focused on complex (NP and NPNP) combinatorial
search an optimization problems.

 Overview:
 Designer defines problem (in AnsProlog syntax).
 Grounder transforms problem into SAT-like.
 High-performance solver enumerates solutions.


Answer Set Program Design
 Guess a solution so that
none of its deduced properties are forbidden,
optionally optimizing a metric.


Answer Set Program Design
 “Which two numbers that multiply to 42 have
the smallest sum?”
% guess two numbers
2 { picked(1..42) } 2.

% deduce if they multiply to 42
product_ok :- picked(A), picked(B), 42==A*B.

% forbid lack of proper product
:- not product_ok.

% optimize (weighted) sum
#minimize [ picked(A)=A ].


New Grid Embedder
 Answer Set Program:
 Guess:
 Absolute positions for pieces (x;y)
 Directions for laser ports (n;e;w;s)
 Deduce:
 Relative positions from absolute positions
 Free paths from relative positions
 Beam production from free paths
 Expression instantiation from beams
 Forbid:
 Overlapping pieces
 Lack of embedding for mission edges
 Illegal port configurations

 Notes:
 How do I place and configure the pieces so that the apparent
mission subsumes the required mission?
 This tool involves 75 lines of code.


New Mission Generator
 Guess node and edge descriptions so that
required expressions and piece counts are
satisfied.

 Notes:
 “What” vs. “How” – clarifies problem
 Now easy to drop in additional constraints
without redesign.
 Concatenate this with embedder to make
monolithic generator!

New Puzzle Solver
 Guess placement of player pieces so that all
targets are powered correctly.

 Notes:
 100% faithful model of game
 “How many pieces are used in the simplest
solution that does not cross any beams or use an
expander?”


Code Size Comparison

Problem Original Solution New Solution

Mission Generation 1,145 Java 194 AnsProlog + 38 Lua

Grid Embedding 987 Java 75 AnsProlog + 0 Lua

Puzzle Solving 988 Java 83 AnsProlog + 61 Lua

Improvements:
- Much smaller than even simple custom algorithms
- No procedural commitments
- Freely include extra constraints in input


Search Time Comparison

Problem Java Search ASP Search

Mission Generation < 1 ms. < 1 ms.

Grid Embedding 650 ms.* 110 ms.

Puzzle Solving No solutions within 350 ms.
one hour*
* Including an advanced feature borrowed from the answer set solver.

Problems derived from running example:
(representative of late-stage gameplay)


Embedding Style Examples


Lessons Learned
 We rebuilt our design automation tools for a
real, deployed game on a substrate that
natively understood hard constraints and
global optimization…

 … and they were smaller and went a bit faster.

 … and they transformed the way we’re
designing our new game.


Design Process Impact
 Got out of the search algorithm design business
 Got into the expressive authoring business

 We clarified our design automation problems

 Found new uses for tools:
 Mission Gen. + Grid Embed. = Unified Puzzle Synthesizer
 Solver = Puzzle Analysis Power Tool
 Discovered new constraints:
 Stylistic
 Educational

 Gained rapid prototyping tools


Thanks!

Presenter:
Adam M. Smith
amsmith@soe.ucsc.edu

A Case Study of Expressively Constrainable Level Design Automation Tools for a Puzzle Game

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