Editing A New Kind of Science (section)

== Contents ==
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=== Computation and its implications===
The thesis of ''A New Kind of Science'' (''NKS'') is twofold: that the nature of [[computation]] must be explored experimentally, and that the results of these experiments have great relevance to understanding the [[physical world]].<ref>[https://serendipstudio.org/local/scisoc/emergence/DBergerNKS.pdf The World According to Wolfram]</ref>

===Simple programs===
The basic subject of Wolfram's "new kind of science" is the study of simple abstract rules—essentially, elementary [[computer program]]s. In almost any class of a computational system, one very quickly finds instances of great complexity among its simplest cases (after a time series of multiple iterative loops, applying the same simple set of rules on itself, similar to a self-reinforcing cycle using a set of rules). This seems to be true regardless of the components of the system and the details of its setup. Systems explored in the book include, among others, cellular automata in one, two, and three dimensions; [[Mobile automaton|mobile automata]]; [[Turing machine]]s in 1 and 2 dimensions; several varieties of substitution and network systems; recursive functions; nested [[recursion (computer science)|recursive functions]]; [[combinator]]s; [[tag system]]s; [[register machine]]s; and [[Palindromic number|reversal-addition]]. For a program to qualify as simple, there are several requirements:
# Its operation can be completely explained by a simple graphical illustration.
# It can be completely explained in a few sentences of [[human language]].
# It can be implemented in a [[computer language]] using just a few lines of code.
# The number of its possible variations is small enough so that all of them can be computed.

Generally, simple programs tend to have a very simple abstract framework. Simple cellular automata, Turing machines, and combinators are examples of such frameworks, while more complex cellular automata do not necessarily qualify as simple programs. It is also possible to invent new frameworks, particularly to capture the operation of natural systems. The remarkable feature of simple programs is that a significant proportion of them can produce great complexity. Simply enumerating all possible variations of almost any class of programs quickly leads one to examples that do unexpected and interesting things. This leads to the question: if the program is so simple, where does the complexity come from? In a sense, there is not enough room in the program's definition to directly encode all the things the program can do. Therefore, simple programs can be seen as a minimal example of [[emergence]]. A logical deduction from this phenomenon is that if the details of the program's rules have little direct relationship to its behavior, then it is very difficult to directly engineer a simple program to perform a specific behavior. An alternative approach is to try to engineer a simple overall computational framework, and then do a [[brute-force search]] through all of the possible components for the best match.

Simple programs are capable of a remarkable range of behavior. Some have been proven to be [[universal computer]]s. Others exhibit properties familiar from traditional science, such as [[thermodynamics|thermodynamic]] behavior, [[continuum mechanics|continuum]] behavior, conserved quantities, [[percolation]], [[sensitive dependence on initial conditions]], and others. They have been used as models of [[traffic]], material fracture, [[crystal growth]], biological growth, and various [[sociology|sociological]], [[geology|geological]], and [[ecology|ecological]] phenomena. Another feature of simple programs is that, according to the book, making them more complicated seems to have little effect on their overall [[complexity]]. ''A New Kind of Science'' argues that this is evidence that simple programs are enough to capture the essence of almost any [[complex system]].

===Mapping and mining the computational universe===
In order to study simple rules and their often-complex behavior, Wolfram argues that it is necessary to systematically explore all these computational systems and document what they do. He further argues that this study should become a new branch of science, like [[physics]] or [[chemistry]]. The basic goal of this field is to understand and characterize the computational universe using experimental methods.

The proposed new branch of scientific exploration admits many different forms of scientific production. For instance, qualitative classifications are often the results of initial forays into the computational jungle. On the other hand, explicit proofs that certain systems compute this or that function are also admissible. Some forms of production are also in some ways unique to this field of study—for example, the discovery of computational mechanisms that emerge in different systems but in bizarrely different forms.

Another type of production involves the creation of programs for the analysis of computational systems. In the ''NKS'' framework, these themselves should be simple programs, and subject to the same goals and methodology. An extension of this idea is that the human mind is itself a computational system, and hence providing it with raw data in as effective a way as possible is crucial to research. Wolfram believes that programs and their analysis should be visualized as directly as possible, and exhaustively examined by the thousands or more. Since this new field concerns abstract rules, it can in principle address issues relevant to other fields of science. But in general, Wolfram's idea is that novel ideas and mechanisms can be discovered in the computational universe, where they can be represented in their simplest forms, and then other fields can choose among these discoveries for those they find relevant.

===Systematic abstract science===
While Wolfram advocates simple programs as a scientific discipline, he also argues that its methodology will revolutionize other fields of science. The basis of his argument is that the study of simple programs is the minimal possible form of science, grounded equally in both [[abstraction]] and empirical experimentation. Every aspect of the methodology ''NKS'' advocates is optimized to make experimentation as direct, easy, and meaningful as possible while maximizing the chances that the experiment will do something unexpected. Just as this methodology allows computational mechanisms to be studied in their simplest forms, Wolfram argues that the process of doing so engages with the mathematical basis of the physical world, and therefore has much to offer the sciences.

Wolfram argues that the computational realities of the universe make science hard for fundamental reasons. But he also argues that by understanding the importance of these realities, we can learn to use them in our favor. For instance, instead of [[reverse engineering]] our theories from observation, we can [[enumeration|enumerate]] systems and then try to match them to the behaviors we observe. A major theme of ''NKS''  is investigating the structure of the possibility space. Wolfram argues that science is far too ad hoc, in part because the models used are too complicated and unnecessarily organized around the limited primitives of traditional mathematics. Wolfram advocates using models whose variations are enumerable and whose consequences are straightforward to compute and analyze.

===Philosophical underpinnings===

====Computational irreducibility====
Wolfram argues that one of his achievements is in providing a coherent system of ideas that justifies computation as an organizing [[Philosophy of science|principle of science]]. For instance, he argues that the concept of ''[[computational irreducibility]]'' (that some complex computations are not amenable to short-cuts and cannot be "reduced"), is ultimately the reason why computational models of nature must be considered in addition to traditional [[mathematical models]]. Likewise, his idea of intrinsic randomness generation—that natural systems can generate their own randomness, rather than using chaos theory or stochastic perturbations—implies that computational models do not need to include explicit randomness.

====Principle of computational equivalence====
Based on his experimental results, Wolfram developed the '''principle of computational equivalence'''<!--boldface per WP:R#PLA--> ('''PCE'''): the principle says that [[systems]] found in the [[natural environment|natural world]] can perform [[computation]]s up to a [[maximum|maximal]] ("universal") level of [[computational power]]. Most systems can attain this level. Systems, in principle, compute the same things as a computer. Computation is therefore simply a question of translating [[Input/output|input and outputs]] from one system to another. Consequently, most systems are computationally equivalent. Proposed examples of such systems are the workings of the human brain and the evolution of weather systems.

The principle can be restated as follows: almost all processes that are not obviously simple are of equivalent sophistication. From this principle, Wolfram draws an array of concrete deductions that he argues reinforce his theory. Possibly the most important of these is an explanation of why we experience [[randomness]] and [[complexity]]: often, the systems we analyze are just as sophisticated as we are. Thus, complexity is not a special quality of systems, like the concept of "heat", but simply a label for all systems whose computations are sophisticated. Wolfram argues that understanding this makes possible the "normal science" of the ''NKS'' paradigm.

===Applications and results===
''NKS'' contains a number of specific results and ideas, and they can be organized into several themes. One common theme of examples and applications is demonstrating how little complexity it takes to achieve interesting behavior, and how the proper methodology can discover this behavior.

First, there are several cases where ''NKS'' introduces what was, during the book's composition, the simplest known system in some class that has a particular characteristic. Some examples include the first primitive recursive function that results in complexity, the smallest [[universal Turing machine]], and the shortest [[axiom]] for [[propositional calculus]]. In a similar vein, Wolfram also demonstrates many simple programs that exhibit phenomena like [[phase transition]]s, [[conserved quantity|conserved quantities]], continuum behavior, and [[thermodynamics]] that are familiar from traditional science. Simple [[computational model]]s of natural systems like [[shell growth in estuaries|shell growth]], [[fluid turbulence]], and [[phyllotaxis]] are a final category of applications that fall in this theme.

Another common theme is taking facts about the computational universe as a whole and using them to reason about fields in a [[holistic]] way. For instance, Wolfram discusses how facts about the computational universe inform [[evolutionary theory]], [[SETI]], [[free will]], [[computational complexity theory]], and philosophical fields like [[ontology]], [[epistemology]], and even [[postmodernism]].

Wolfram suggests that the theory of computational irreducibility may explain how free will is possible in a nominally [[deterministic]] universe. He posits that the computational process in the [[brain]] of the being with free will is so [[Complexity|complex]] that it cannot be captured in a simpler computation, due to the principle of computational irreducibility. Thus, while the process is indeed deterministic, there is no better way to determine the being's will than, in essence, to run the experiment and let the being exercise it.

The book also contains a number of results—both experimental and analytic—about what a particular automaton computes, or what its characteristics are, using some methods of analysis.

The book contains a new technical result in describing the [[Turing completeness]] of the [[Rule 110]] cellular automaton. Very small Turing machines can simulate Rule 110, which Wolfram demonstrates using a 2-state 5-symbol [[universal Turing machine]]. Wolfram conjectures that a particular [[Wolfram's 2-state 3-symbol Turing machine|2-state 3-symbol Turing machine]] is universal. In 2007, as part of commemorating the book's fifth anniversary, Wolfram's company offered a $25,000 prize for proof that this Turing machine is universal.<ref>{{cite web | title=The Wolfram 2,3 Turing Machine Research Prize | url=http://www.wolframscience.com/prizes/tm23/ | access-date=2011-03-31| archive-url= https://web.archive.org/web/20110515032324/http://www.wolframscience.com/prizes/tm23/| archive-date= 15 May 2011 | url-status= live}}</ref> Alex Smith, a computer science student from [[Birmingham]], UK, won the prize later that year by proving Wolfram's conjecture.<ref>{{cite web | title=The Wolfram 2,3 Turing Machine Is Universal! | url=https://www.wolframscience.com/prizes/tm23/solution_news.html | access-date=2007-10-24}}</ref><ref>{{cite web | title=Technical Commentary [on Wolfram 2,3 Turing machine universality proof] | url=https://www.wolframscience.com/prizes/tm23/solution_technical.html | access-date=2007-10-24}}</ref>