Complexity Theory has two kinds of meaning: one is Computational Complexity, and another is the study of Complex System. In this section, I will briefly introduce these two fields and their relation to puzzle game.
A. Computational Complexity
Computational Complexity is the study of theoretical computer science and mathematics that focus on how efficiency to handle a problem (M. Sipser, 1997;
Sanjeev Arora & Boaz Barak, 2009). For example, there have three famous type of computational efficiency problem NP, NP-Complete and NP-Hard, indicate whether it
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can be solved within linear time; furthermore, there also exists the problem about space efficiency: PSPACE, PSPACE-Complete and PSPACE-Hard, indicate whether it can be solved with limited space. Give an overview, there have some research may like: Reduce Time Complexity By an Algorithm for solving a puzzle(R. E. Korf, M.
Reid, & S. Edelkamp, 2001), analysis Complexity of Search a Graph (N. Megiddo, S.
L. Hakimi, M. R. Garey, D. S. Johnson, & C. H. Papadimitriou, 1988) and a reduction method for handle games(R. A. Hearn, 2006).
Puzzle is very suitable for further study in this field, because it require player to choose a sequence of action in order to solve it that has many interesting feature for calculate model. Quote from Robert Aubrey Hearn (2006), in his research,
Computational Complexity of a puzzle can classify into following category: ―If a
game is a one-player puzzle with a bounded length, odds are it is NP-Complete.” and
“Indeed, unbounded puzzles are often PSPACE-Complete.”
Bounded and unbounded puzzle means whether it has a restrict length to solve it.
In unbounded puzzle we can always go back to previous state, therefore it has no restrict length. Both of them need exponential time to compute a solution, but they are different in whether we can use polynomial space to verify a specific action sequence is correct. Because Savitch‘s (1970) theorem had proofed that NP-SPACE = PSPACE, therefore we can solve any puzzle problem with polynomial space. The main research direction in this filed is how to solve a puzzle more computational and space
efficiently. Is computational effort relate to complexity of puzzle and can use for sorting purpose? I think it is not a good idea, because Computational Time and Space problem, your machine will run a long time or crash due to memory lacking when compute a very complex puzzle.
B. Complex System and Emergence
What is complex system? Although this filed has been studied in modern computer science for a long time, but it is one of profound problem that people tends to understand in past several thousand years. Aristotle (384 BC – 322 BC), a noted Greek philosopher, who first organized the concept in his questions about
Metaphysica: “The whole is more than the sum of its parts.”, that actually indicate the
most important property of complex system.Jeremy Campbell (1982) looks this ―whole phenomenon‖ from the aspect of information, language, and DNA, says that when system beyond a ―complex barrier‖,
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entirely new principle will come into play. The principle, call emergence, may allow a system to self-organizing, replicating, learning, or adaptive itself to environment.
Penny Sweetser (2007) has summarized some common property for a complex system: Elements, Interactions, Formation, Diversity, Environment, and Activities. In other word, if there exist a set of elements, that will inter-interact with a set of rule in an environment for specific purpose, their interaction process has large state space, element will reorganize itself over time changed, and then it is a complex system. The first deep exploration about emergence is from John Holland‘s (1999) book
―Emergence: From Chaos to Order‖, shows many example about how emergence arise from complexity.
When a system is emergence, it means we can‘t predict it high level behavior or structure of system from observer lower level. But, not all of system is complex.
Christopher Langton (1995) provides four level of complexity for understand system:
Fixed, Periodic, Complex, and Chaotic. The boundary between Periodic and Complex is complexity barrier; between Complex and Chaotic is edge of chaos. Beyond
complexity barrier means system will have emergence phenomenon, but if it complex reaches chaotic level, this phenomenon will disappear. Following two “Cross Block”
puzzle levels in Figure 4 shows complex and chaotic level of puzzle:
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It shows emergence phenomenon in the “Cross Block” puzzle, that both choice and dead ends will increase when it beyond complexity barrier and dead ends will decrease when reach chaotic level.
The study has widespread research in many fields, such as Information Complexity on Communication System (C. E. Shannon, 1948), Artificial Life (Adamatzky. Andrew, 2010; Christopher G. Langton, 1995), Biological System
(Gerald M. Edelman & Joseph A. Gally, 2001), Economic System and Human Society (Holling, 2001)…etc. We can‘t survey all of those fields here for understand what is complexity, since it will diverse our discussion to focus on puzzle game. With a general idea, quote from Penny Sweetser (2007), we simply define complexity as following meaning :“Complexity is a measure of the difficulty involved in
understanding a system.”
What means to understanding the puzzle? If someone can solve a puzzle level, we say he/she understand it. How do we measure complexity of a puzzle? From previous discuss, we know insight is important skill to solve a puzzle, and there has two components will affect it: choice and dead ends. But, because they are emergence phenomenon in the puzzle, therefore we can‘t directly control it. How do we calculate it? From computational complex theory we discuss, it will fail when we want to expand search space in a puzzle. In chapter 4.1, I will introduce our approximate method.
(a) Chaotic Level (a) Complex Level
Figure 4 Emergence Phenomenon Example
(a) Complex level of puzzle that has high complexity. (b) Chaotic level of puzzle that with no complexity (no dead ends) that every square can interaction with each other to form a basic element that can be canceled by player.
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