![]() ![]() Some of the major questions facing complex systems researchers are: (1) explaining how complexity arises from the nonlinear interaction of simple components (2) describing the mechanisms underlying high-level aggregate behavior of complex systems (such as the overt behavior of an organism, the flow of energy in an ecology, the GNP of an economy) and (3) creating more » a theoretical framework to enable predictions about the likely behavior of such systems in various conditions. Research models exhibiting complex behavior include spin glasses, cellular automata, and genetic algorithms. Natural systems displaying complex adaptive behavior range upwards from DNA through cells and evolutionary systems to human societies. Research on complex systems-the focus of work at SFI-involves an extraordinary range of topics normally studied in seemingly disparate fields. The fifth annual complex systems summer school brought nearly 60 graduate students and postdoctoral fellows to Santa Fe for an intensive introduction to the field. The Institute`s book series in the sciences of complexity continues to grow, now numbering more than 20 volumes. To date this 1992 work has resulted in more than 50 SFI Working Papers and nearly 150 publications in the scientific literature. Have at it.In 1992 the Santa Fe Institute hosted more than 100 short- and long-term research visitors who conducted a total of 212 person-months of residential research in complex systems. You can ask many follow-ups: can you make this topologically mixing (or some stronger gluing notion), can you use something weaker than CFL, and can you have the language be CFL rather than the forbidden patterns, etc. If $X$ and $Y$ are not conjugate, $A$ detects this, so $B$ also says they are not conjugate, contradicting the previous theorem. Now if $X = Y$, certainly $A$ says they are conjugate, so $B$ says they are the same. If conjugacy were decidable by algorithm $A$, then it would give an algorithm $B$ that accomplishes what we claimed cannot be done: given $X$ and $Y$, if $A$ says they are conjugate, $B$ claims they are the same, and if $A$ says they are not conjugate, $B$ claims they are not conjugate as well. Square.Ĭorollary: Equality and conjugacy of subshifts given by forbidden context-free languages are both undecidable problems. So an algorithm with the properties listed in the statement would solve the halting problem, thus cannot exist. In conclusion, if $M$ never halts on $q_0 0^\omega$, $X = Y$, and if $M$ does, then $X$ contains an isolated periodic point while $Y$ does not, so $X$ and $Y$ are not conjugate. (And if $M$ does halt on $q_0 0^n$, then periodically repeating the computation gives a valid configuration of $X$.) ![]() We see that any configuration not in $Y$ is periodic, and must periodically repeat the accepting computation of $M$ on some $q_0 0^n$, where $n$ is any integer large enough that he computation has time to halt. Another application of this argument shows that the computation from $q_0 0^n$ reaches the halting state $q_f$. But this means there is an occurrence of $q_0 0^n$ between them as well. The last rule then forces an appearance of $q_f$ somewhere between $u_m$ and $u_k$. Since the lengths are the same, the computation must enter a loop starting from any $u_i$, i.e. We will construct two subshifts $X$ and $Y$ over the alphabet $Q \cup A \cup \$. Let $q_0 \in Q$ be the initial state of $Q$ and $q_f$ the unique halting state. Let $M$ be a Turing machine with state set $Q$ and alphabet $A$, suppose $Q \cap A = \emptyset$ and $\#, \notin A \cup Q$, and $0 \in A$. We will forbid a bunch of things to force our configurations to simulate Turing machines. Note that CFL are closed under unions and contain all regular languages, so we can forbid finitely many regular and CFL languages. Note that no requirement, even on halting, is made when they are distinct and conjugate. There is no algorithm that given two context-free languages, says "same" if they define the same subshift, and says "non-conjugate" if they are not conjugate. ![]() For this class, equality and non-conjugacy are recursively inseparable, i.e. ![]() Consider the class of subshifts defined by a forbidden context-free language. ![]()
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