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Talk:Aleatoric music

Aleatory vs Aleatoric
I just checked a couple of reference works and they rather snottily inform me that "The adjective `aleatoric' is a bastard word, to be avoided by those who care for language" (Oxford Dict. Mus.). I think that's probably true, although I'd been making links here rather than to aleatory music because "aleatoric" was a more familiar term to me. I think that in a musical context "aleatoric" is more common than "aleatory" (although in other contexts, this isn't so), so I'm going to leave the article here, and make "aleatory music" a redirect. --Camembert

Why can't it just be redirected to "stochastic music" (of which there is no article despite the "see also section, which just links to the "stochastic" article.)--Brentt 23:18, 6 March 2006 (UTC)

 * You're right, there should be a separate article for stochastic music, but I'm not sure what you're suggesting should be redirected.  There is a remarkable degree of confused terminology in this area, with no clear distinctions between aleatoric, aleatory, or indeterminate music.  However, while all of these are composed or performed with some element of chance, stochastic music is created using probability, etc.  So, stochastic music is its own thing. JonathanDS 23:37, 6 March 2006 (UTC)

 * All stochastic means is "some element or variable is governed by chance" which seems to be the definition for aleatoric music. I don't see the difference, unless stochastic music is something entirely different and just misnamed--or the definition for aleatoric music given in this article is wrong. --Brentt 00:56, 7 March 2006 (UTC)

 * The definition of aleatory music isn't wrong per se, but I think that the structure of the article could be much clearer, with subsections for different degrees of chance in composition and performance. (It could be argued, by this definition, that all notated music is aleatory.)  My dictionary defines "stochastic" as "having a random probability distribution or pattern that may be analyzed statistically but may not be predicted precisely".  A composer of stochastic music, like Xenakis, composes a general structure, with broadly defined formal shape, density, etc, and then uses statistics (usually with the aid of a computer) to generate the details (i.e. the specific pitchs and rhythms) that will create that structure.  This is quite different from Boulez or Haubenstock-Ramati's mobile forms or Cagean nonintentional composition, which both undermine preconceived structure.  Stochastic music does involve chance, but to a comparatively limited degree and on a very local level.  It also results in more traditional, determinate scores, without room for performer choice, and has a mathematical, formalistic approach which is perhaps closer to the aesthetic of integral serialism  (Xenakis did study with Messiaen).  In any case, all stochastic music is aleatory to some degree, but not all aleatory music is stochastic.  I guess it could be included in this article, but it would need a clear subsection to show how it's unique. JonathanDS 11:05, 7 March 2006 (UTC)

 * Common usage decides what a word means. 'Aleatory' hardly counts as a word if it's archaic. I think the page is best here, anyway. Good call. JamesMcGuiggan 22:20, 14 January 2007 (UTC)

Inventor?

 * Contemporary aleatoric music was popularized by John Cage, but probably invented by Henry Brant.

Any effort to pin the "invention" of aleatoric music down to one person is probably doomed to failure. What aleatoric techniques did Brant use, exactly, and when? --Camembert

The article cites various "early" examples of this genre from the 1950's, yet the article for Alan Hovhaness, specifically with regard to his piece Lousadzak, which debuted in 1944, is not mentioned. Wouldn't it be more accurate to cite the earliest examples? --Bitterblogger64.171.242.194 (talk) 17:27, 25 February 2009 (UTC)

Tomatoes
Actually aleatoric means new music, so i guess catsup is the new ketchup - 81.79.133.110 (signed by Hyacinth)

Popular Music
I'm not an expert in this, by any means, but shouldn't there be a section about popular music which has some aleatoric elements? I know that the Beatles experimented with it to a certain extent, and I'm think Zappa did as well (though I'm not a huge Zappa fan, so I can't really say for sure), but are there others? If so, I think it warrants inclusion. Isquitenice 05:17, 27 October 2006 (UTC)

Aleatoric and other concepts; performance criterion
Let's keep it sharp, even if there's no denying that misunderstandings and mixing ups have flourished here. Aleatoric procedures should refer to a certain limited use of chance. Let's not forget the etymology: alea means dice in latin, the number of sides is limited. Some good recent sources that make this clear are Cox: Audio Culture. readings in Modern Music (Continuum, 2004) and Feisst: Der Begriff 'Improvisation' in der neuen Musik (Sinzig, Studio Verlag 1997). Stochastic music, as pointed out below, and also those of chance music and indeterminate music follow distinct, historically different methods.

I am, to say the least, into real doubt how much music, if any, could be called 'aleatoric' without having this realized in the performance process itself. All the important examples stated in the article have this as a main characteristic, and in this way it could be said to be closely related to open form (it is definitely so, in its historical spirit), although of course by no means just the same.

This is also why I don't know examples of popular music using aleatoric procedures. It would be most interesting if such a thing existed. But the use of limited chance procedures to determine the outcome of the composition is not enough, they should also be presented as open to the performers. So the tape running beautifully backwards in Beatles' "In the beginning", collage things of Zappa and Grateful Dead etc. may be very interesting, but belong somewhere else. (Intuitive 10:05, 6 June 2007 (UTC))

 * Hi there. I don't want to weigh in on the distinction between aleatory, stochastic music, etc. but I think it's crucial to this article that the distinction between random elements in the course of composition be distinguished from random elements in performance. That is, both are examples of the composer giving control (over the final musical product) away: the first is to the dice (or the I Ching or whatever) and the second is to the performer. They are separate strands and I think should be shown to be so in the article. I'll do some research and see where we are. JH(emendator) 07:27, 9 July 2007 (UTC)
 * Yes, this was an important distinction made by John Cage, for example, who I believe called these two kinds of "chance" composition "indeterminate with respect to composition" and "indeterminate with respect to performance". This terminology is found in one of the earlier items in Silence—the lecture in which he discusses amongst other things Stockhausen's Klavierstück XI and works by Morton Feldman—if that helps you to narrow your search. I'm sorry I don't have a copy of the book handy at the moment, or I could give you a more precise reference.--Jerome Kohl 18:43, 18 July 2007 (UTC)

Ockeghem
It says that Missa cuiusvis toni by Ockeghem could be considered a precedent for aleatoric composition. I know you can pick what mode you sing it in, but if you have a book of different masses and you just pick out which mass you want to sing, that is a choice too but I do not think it is aleatoric. Or if Ockeghem wrote out his mass in each mode on different pages you would just tell the choir what page to turn to and that wouldn't be very aleatoric either. I only read about it in Reese's book about Renaissance Music and it doesn't say very much there, but I thought you pick what mode you want to sing, and it tells you what key signature and what clefs you have to use to sing it in the mode you picked, and then you just sing what it says. There isn't any choice or any dice is there? This article is not about Renaissance music so I wish it would say some more about why it is like aleatoric music. Teenly (talk) 20:38, 10 October 2008 (UTC)
 * Yes, this is a position held by Konrad Boehmer, as cited at the end of the paragraph (Boehmer 1967, 9–47). The difference between a catholicon and choosing one mass out of several is that in the former case there are different versions of the same composition. The choice is in the version to perform, and this much, at least, could be done with a throw of a die (well, a four-sided die, anyway). I will grant you that this does not really amount to aleatory, and a much better example might be the suites of late-17th-century composers like Marin Marais, who may offer five or six of each type of dance, and a selection of three or four preludes to go with them. The performer must make a selection from among these (it would be absurd to play four preludes in succession, followed by six allemandes, etc.) and, to a limited extent, can also choose the order in which they are played. However, Boehmer does not cite this, and it would be Original Research to put this into the article without a source. It is true that the article is not about Renaissance music, but the section in question is clearly labelled "Early precedents".—Jerome Kohl (talk) 21:54, 10 October 2008 (UTC)

Thank you for answering my note Mr Kohl. I do not have that book by Boehmer but my teacher says he will bring it from the university and read it with me because I can not read German. I did not mean the part about Ockeghem should not be in this article. I just wish it could say something about why the mass by Ockeghem is like aleatoric music, because Ockeghem is not what the article is about, so maybe some people who read an article about modern music do not know very much about Ockeghem. Teenly (talk) 20:17, 14 October 2008 (UTC)
 * You are welcome. Part of the problem stems from the fact that, as the article states, "There has been considerable confusion of the terms aleatory and indeterminate / chance music". The more restrictive definition found in the lede, originally formulated by Meyer-Eppler, is far from the kinds of indeterminacy discussed later on. On the other hand, some parts of the article suggest that aleatory is identical with 'mobile' (or 'open') form. In this sense, the Ockeghem has nothing at all to do with it. However, Boehmer is concerned with a somewhat larger meaning for the term 'open form' which encompasses things approaching Cage's indeterminacy. The other definition for 'open form' (the one found in Wölfflin) has nothing at all to do with mobile form, and even does not necessarily have anything to do with aleatory. It is necessary to bring this into the discussion in order to warn the reader that the term 'open form' is not exclusively used as a synonym for mobile form. There is a heavy-duty discussion (in English—just about) of this that takes issue with Boehmer's definition, by Holger Kaletha, “Decomposition of the Sound Continuum: Serialism and Development from a Genetic-Phenomenological Perspective,” Perspectives of New Music 42, no. 1 (Winter 2004): 84–128.—Jerome Kohl (talk) 22:54, 14 October 2008 (UTC)

Original research?
Ridernyc, using the Friendly" robot, added two tags to this article, one calling for more inline references, the other claiming Original Research. I have deleted the demand for further inline references on grounds that the article is already groaning under the strain of them—any more and there would be grounds for complaint that the article is over-referenced. I have left the "original research" tag in place for the time being, in the hopes that Ridernyc will explain just which claims might fall under this category (since there are one or two sentences that do not currently carry an inline reference). Since this was a bot-driven addition, I suspect mechanical error, and will delete the OR tag as well, unless justification is given within a reasonable time.—Jerome Kohl (talk) 18:38, 20 January 2010 (UTC)

Spirit murmur
Some recent edits point to a fundamental confusion about the term "aleatoric music". The New Grove article on Alan Hovhaness dismisses his "spirit murmur" technique as "hardly aleatory", whereas a freshly added blog article claims his technique is virtually indistinguishable from passages in scores by Witold Lutosławski. I have to say that I find the blog's examples convincing, but this raises at least three problems. The first is that the authors of the New Grove article clearly do not understand "aleatory" in the sense defined by Meyer-Eppler, since they cite control of the resulting sound (an essential feature of Meyer-Eppler's definition of "aleatory") as evidence of non-aleatory technique. The second problem is that, accepting the view of the New Grove article on Hovhaness, we may just as plausibly conclude that Lutosławski's technique does not involve aleatory, as that Hovhaness's does so. On the other hand, the New Grove article on Lutosławski by Charles Bodman Rae uses the term "aleatory" repeatedly to describe just this phenomenon. The third problem is that the blog reference cites a score by Hovhaness dating from only two years prior to the one by Lutosławski, though Hovhaness had actually started using this technique a decade or so earlier. The implication of this article as presently written is that Lutosławski took the technique from Hovhaness without crediting him as the source, naming John Cage instead. Nothing cited so far supports this conclusion (certainly Fisher, the author of the blog does not claim this), and it should perhaps also be made clearer that Cage, Hovhaness, Lutosławski, Stockhausen, Boulez, etc. were all preceded by other composers (particularly Ives and Cowell) who employed similar compositional strategies. This idea was simply "in the air" at the time.—Jerome Kohl (talk) 19:23, 22 June 2010 (UTC)
 * Most probably the New Grove writer (Rosner) meant Hovhaness's technique was not random in the full-blown sense that Cage's and others' pieces could be ... this is a big problem because many published commentators use the terms "indeterminacy" (general variability in a piece) and "aleatory" (variability only at a detailed level) completely interchangeably. I think the Grove article needs revising.  —Preceding unsigned comment added by 87.224.103.226 (talk) 16:15, 23 June 2010 (UTC)

These are precisely my conclusions. The difficulty from a Wikipedia perspective is that claims made in an article in the New Grove remain authoritative until and unless an equally good or better reliable source can be found to contradict it. (It is ironic that Hovhaness appears to be excluded by Rosner and Wolverton precisely for adhering too closely to the narrower definition of "aleatory"!) Blogs will probably always remain at a disadvantage, and for perfectly sound reasons. The underlying problem is, as you say, that the word "aleatory" has been bandied about so loosely now for over fifty years, that it has become virtually meaningless. It seems to me that one purpose of the present article is to rectify this situation, as far as it can be done, but for the moment it looks like we are stuck between the proverbial rock and a hard place. I think the quotation about Hovhaness from New Grove requires too much interpretation (of the sort we are indulging in here) to demonstrate a conflict with Meyer-Eppler's basic definition, without overstepping the boundaries of original research.—Jerome Kohl (talk) 16:56, 23 June 2010 (UTC)

Assessment comment
Substituted at 07:09, 29 April 2016 (UTC)

title: Talk:Aleatoric music


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Tiananmen Square protests (disambiguation)

The 1989 Tiananmen Square protests and massacre were student-led demonstrations in Beijing and the government response in 1989.

Tiananmen Square protests or Tiananmen Square incident may also refer to:
 * "Tank Man", an anonymous protester who blocked tanks leaving Tiananmen Square in 1989
 * Tiananmen Square protest of 4 May 1919, a protest that grew into the May Fourth Movement
 * 1976 Tiananmen Incident, a spontaneous protest of 4–5 April 1976
 * Tiananmen Square self-immolation incident, a 2001 incident allegedly involving Falun Gong practitioners

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Automata theory

Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science with close connections to mathematical logic. The word automata comes from the Greek word αὐτόματος, which means "self-acting, self-willed, self-moving". An automaton (automata in plural) is an abstract self-propelled computing device which follows a predetermined sequence of operations automatically. An automaton with a finite number of states is called a finite automaton (FA) or finite-state machine (FSM). The figure on the right illustrates a finite-state machine, which is a well-known type of automaton. This automaton consists of states (represented in the figure by circles) and transitions (represented by arrows). As the automaton sees a symbol of input, it makes a transition (or jump) to another state, according to its transition function, which takes the previous state and current input symbol as its arguments.

Automata theory is closely related to formal language theory. In this context, automata are used as finite representations of formal languages that may be infinite. Automata are often classified by the class of formal languages they can recognize, as in the Chomsky hierarchy, which describes a nesting relationship between major classes of automata. Automata play a major role in the theory of computation, compiler construction, artificial intelligence, parsing and formal verification.

History
The theory of abstract automata was developed in the mid-20th century in connection with finite automata. Automata theory was initially considered a branch of mathematical systems theory, studying the behavior of discrete-parameter systems. Early work in automata theory differed from previous work on systems by using abstract algebra to describe information systems rather than differential calculus to describe material systems. The theory of the finite-state transducer was developed under different names by different research communities. The earlier concept of Turing machine was also included in the discipline along with new forms of infinite-state automata, such as pushdown automata.

1956 saw the publication of Automata Studies, which collected work by scientists including Claude Shannon, W. Ross Ashby, John von Neumann, Marvin Minsky, Edward F. Moore, and Stephen Cole Kleene. With the publication of this volume, "automata theory emerged as a relatively autonomous discipline". The book included Kleene's description of the set of regular events, or regular languages, and a relatively stable measure of complexity in Turing machine programs by Shannon. In the same year, Noam Chomsky described the Chomsky hierarchy, a correspondence between automata and formal grammars, and Ross Ashby published An Introduction to Cybernetics, an accessible textbook explaining automata and information using basic set theory.

The study of linear bounded automata led to the Myhill–Nerode theorem, which gives a necessary and sufficient condition for a formal language to be regular, and an exact count of the number of states in a minimal machine for the language. The pumping lemma for regular languages, also useful in regularity proofs, was proven in this period by Michael O. Rabin and Dana Scott, along with the computational equivalence of deterministic and nondeterministic finite automata.

In the 1960s, a body of algebraic results known as "structure theory" or "algebraic decomposition theory" emerged, which dealt with the realization of sequential machines from smaller machines by interconnection. While any finite automaton can be simulated using a universal gate set, this requires that the simulating circuit contain loops of arbitrary complexity. Structure theory deals with the "loop-free" realizability of machines. The theory of computational complexity also took shape in the 1960s. By the end of the decade, automata theory came to be seen as "the pure mathematics of computer science".

Automata
What follows is a general definition of an automaton, which restricts a broader definition of a system to one viewed as acting in discrete time-steps, with its state behavior and outputs defined at each step by unchanging functions of only its state and input.

Informal description
An automaton runs when it is given some sequence of inputs in discrete (individual) time steps (or just steps). An automaton processes one input picked from a set of symbols or letters, which is called an input alphabet. The symbols received by the automaton as input at any step are a sequence of symbols called words. An automaton has a set of states. At each moment during a run of the automaton, the automaton is in one of its states. When the automaton receives new input, it moves to another state (or transitions) based on a transition function that takes the previous state and current input symbol as parameters. At the same time, another function called the output function produces symbols from the output alphabet, also according to the previous state and current input symbol. The automaton reads the symbols of the input word and transitions between states until the word is read completely, if it is finite in length, at which point the automaton halts. A state at which the automaton halts is called the final state.

To investigate the possible state/input/output sequences in an automaton using formal language theory, a machine can be assigned a starting state and a set of accepting states. Then, depending on whether a run starting from the starting state ends in an accepting state, the automaton can be said to accept or reject an input sequence. The set of all the words accepted by an automaton is called the language recognized by the automaton. A familiar example of a machine recognizing a language is an electronic lock, which accepts or rejects attempts to enter the correct code.

Formal definition

 * Automaton

 * An automaton can be represented formally by a quintuple $$M = \langle \Sigma, \Gamma, Q, \delta, \lambda \rangle$$, where:
 * $$\Sigma$$ is a finite set of symbols, called the input alphabet of the automaton,
 * $$\Gamma$$ is another finite set of symbols, called the output alphabet of the automaton,
 * $$Q$$ is a set of states,
 * $$\delta$$ is the next-state function or transition function $$\delta : Q \times \Sigma \to Q$$ mapping state-input pairs to successor states,
 * $$\lambda$$ is the next-output function $$\lambda : Q \times \Sigma \to \Gamma$$ mapping state-input pairs to outputs.
 * If $$Q$$ is finite, then $$M$$ is a finite automaton.

 * Input word
 * An automaton reads a finite string of symbols $$a_1a_2...a_n$$, where $$a_i \in \Sigma$$, which is called an input word. The set of all words is denoted by $$\Sigma^*$$.

 * Run
 * A sequence of states $$q_0,q_1,...,q_n$$, where $$q_i \in Q$$ such that $$q_i = \delta(q_{i-1}, a_i)$$ for $$0 < i \le n$$, is a run of the automaton on an input $$a_1a_2...a_n \in \Sigma^*$$ starting from state $$q_0$$. In other words, at first the automaton is at the start state $$q_0$$, and receives input $$a_1$$. For $$a_1$$ and every following $$a_i$$ in the input string, the automaton picks the next state $$q_i$$ according to the transition function $$\delta(q_{i-1},a_i)$$, until the last symbol $$a_n$$ has been read, leaving the machine in the final state of the run, $$q_n$$. Similarly, at each step, the automaton emits an output symbol according to the output function $$\lambda(q_{i-1},a_i)$$.

 * The transition function $$\delta$$ is extended inductively into $$\overline\delta: Q \times \Sigma^* \to Q$$ to describe the machine's behavior when fed whole input words. For the empty string $$\varepsilon$$, $$\overline\delta(q, \varepsilon) = q$$ for all states $$q$$, and for strings $$wa$$ where $$a$$ is the last symbol and $$w$$ is the (possibly empty) rest of the string, $$\overline\delta(q, wa) = \delta(\overline\delta(q,w),a)$$. The output function $$\lambda$$ may be extended similarly into $$\overline\lambda(q,w)$$, which gives the complete output of the machine when run on word $$w$$ from state $$q$$.

 * Acceptor

 * In order to study an automaton with the theory of formal languages, an automaton may be considered as an acceptor, replacing the output alphabet and function $$\Gamma$$ and $$\lambda$$ with
 * $$q_0 \in Q$$, a designated start state, and
 * $$F$$, a set of states of $$Q$$ (i.e. $$F \subseteq Q$$) called accept states.
 * This allows the following to be defined:

 * Accepting word
 * A word $$w = a_1a_2...a_n \in \Sigma^*$$ is an accepting word for the automaton if $$\overline\delta(q_0,w) \in F$$, that is, if after consuming the whole string $$w$$ the machine is in an accept state.

 * Recognized language
 * The language $$L \subseteq \Sigma^*$$ recognized by an automaton is the set of all the words that are accepted by the automaton, $$L = \{w \in \Sigma^* \ |\ \overline\delta(q_0,w) \in F\}$$.

 * Recognizable languages
 * The recognizable languages are the set of languages that are recognized by some automaton. For finite automata the recognizable languages are regular languages. For different types of automata, the recognizable languages are different.

Variant definitions of automata
Automata are defined to study useful machines under mathematical formalism. So the definition of an automaton is open to variations according to the "real world machine" that we want to model using the automaton. People have studied many variations of automata. The following are some popular variations in the definition of different components of automata.

 * Input
 * Finite input: An automaton that accepts only finite sequences of symbols. The above introductory definition only encompasses finite words.
 * Infinite input: An automaton that accepts infinite words (ω-words). Such automata are called ω-automata.
 * Tree input: The input may be a tree of symbols instead of sequence of symbols. In this case after reading each symbol, the automaton reads all the successor symbols in the input tree. It is said that the automaton makes one copy of itself for each successor and each such copy starts running on one of the successor symbols from the state according to the transition relation of the automaton. Such an automaton is called a tree automaton.
 * Infinite tree input : The two extensions above can be combined, so the automaton reads a tree structure with (in)finite branches. Such an automaton is called an infinite tree automaton.

 * States
 * Single state: An automaton with one state, also called a combinational circuit, performs a transformation which may implement combinational logic.
 * Finite states: An automaton that contains only a finite number of states.
 * Infinite states: An automaton that may not have a finite number of states, or even a countable number of states. Different kinds of abstract memory may be used to give such machines finite descriptions.
 * Stack memory: An automaton may also contain some extra memory in the form of a stack in which symbols can be pushed and popped. This kind of automaton is called a pushdown automaton.
 * Queue memory: An automaton may have memory in the form of a queue. Such a machine is called queue machine and is Turing-complete.
 * Tape memory: The inputs and outputs of automata are often described as input and output tapes. Some machines have additional working tapes, including the Turing machine, linear bounded automaton, and log-space transducer.

 * Transition function
 * Deterministic: For a given current state and an input symbol, if an automaton can only jump to one and only one state then it is a deterministic automaton.
 * Nondeterministic: An automaton that, after reading an input symbol, may jump into any of a number of states, as licensed by its transition relation. The term transition function is replaced by transition relation: The automaton non-deterministically decides to jump into one of the allowed choices. Such automata are called nondeterministic automata.
 * Alternation: This idea is quite similar to tree automata but orthogonal. The automaton may run its multiple copies on the same next read symbol. Such automata are called alternating automata. The acceptance condition must be satisfied on all runs of such copies to accept the input.
 * Two-wayness: Automata may read their input from left to right, or they may be allowed to move back-and-forth on the input, in a way similar to a Turing machine. Automata which can move back-and-forth on the input are called two-way finite automata.

 * Acceptance condition
 * Acceptance of finite words: Same as described in the informal definition above.
 * Acceptance of infinite words: an ω-automaton cannot have final states, as infinite words never terminate. Rather, acceptance of the word is decided by looking at the infinite sequence of visited states during the run.
 * Probabilistic acceptance: An automaton need not strictly accept or reject an input. It may accept the input with some probability between zero and one. For example, quantum finite automata, geometric automata and metric automata have probabilistic acceptance.

Different combinations of the above variations produce many classes of automata.

Automata theory is a subject matter that studies properties of various types of automata. For example, the following questions are studied about a given type of automata.

 * Which class of formal languages is recognizable by some type of automata? (Recognizable languages)
 * Are certain automata closed under union, intersection, or complementation of formal languages? (Closure properties)
 * How expressive is a type of automata in terms of recognizing a class of formal languages? And, their relative expressive power? (Language hierarchy)

Automata theory also studies the existence or nonexistence of any effective algorithms to solve problems similar to the following list:

 * Does an automaton accept at least one input word? (Emptiness checking)
 * Is it possible to transform a given non-deterministic automaton into a deterministic automaton without changing the language recognized? (Determinization)
 * For a given formal language, what is the smallest automaton that recognizes it? (Minimization)

Types of automata
The following is an incomplete list of types of automata.

Discrete, continuous, and hybrid automata
Normally automata theory describes the states of abstract machines but there are discrete automata, analog automata or continuous automata, or hybrid discrete-continuous automata, which use digital data, analog data or continuous time, or digital and analog data, respectively.

Hierarchy in terms of powers
The following is an incomplete hierarchy in terms of powers of different types of virtual machines. The hierarchy reflects the nested categories of languages the machines are able to accept.

Applications
Each model in automata theory plays important roles in several applied areas. Finite automata are used in text processing, compilers, and hardware design. Context-free grammar (CFGs) are used in programming languages and artificial intelligence. Originally, CFGs were used in the study of human languages. Cellular automata are used in the field of artificial life, the most famous example being John Conway's Game of Life. Some other examples which could be explained using automata theory in biology include mollusk and pine cone growth and pigmentation patterns. Going further, a theory suggesting that the whole universe is computed by some sort of a discrete automaton, is advocated by some scientists. The idea originated in the work of Konrad Zuse, and was popularized in America by Edward Fredkin. Automata also appear in the theory of finite fields: the set of irreducible polynomials that can be written as composition of degree two polynomials is in fact a regular language. Another problem for which automata can be used is the induction of regular languages.

Automata simulators
Automata simulators are pedagogical tools used to teach, learn and research automata theory. An automata simulator takes as input the description of an automaton and then simulates its working for an arbitrary input string. The description of the automaton can be entered in several ways. An automaton can be defined in a symbolic language or its specification may be entered in a predesigned form or its transition diagram may be drawn by clicking and dragging the mouse. Well known automata simulators include Turing's World, JFLAP, VAS, TAGS and SimStudio.

Category-theoretic models
One can define several distinct categories of automata following the automata classification into different types described in the previous section. The mathematical category of deterministic automata, sequential machines or sequential automata, and Turing machines with automata homomorphisms defining the arrows between automata is a Cartesian closed category, it has both categorical limits and colimits. An automata homomorphism maps a quintuple of an automaton Ai onto the quintuple of another automaton  Aj. Automata homomorphisms can also be considered as automata transformations or as semigroup homomorphisms, when the state space, S, of the automaton is defined as a semigroup Sg. Monoids are also considered as a suitable setting for automata in monoidal categories.

One could also define a variable automaton, in the sense of Norbert Wiener in his book on The Human Use of Human Beings via the endomorphisms $$A_{i}\to A_{i}$$. Then one can show that such variable automata homomorphisms form a mathematical group. In the case of non-deterministic, or other complex kinds of automata, the latter set of endomorphisms may become, however, a variable automaton groupoid. Therefore, in the most general case, categories of variable automata of any kind are categories of groupoids or groupoid categories. Moreover, the category of reversible automata is then a 2-category, and also a subcategory of the 2-category of groupoids, or the groupoid category.
 * Categories of variable automata

title: Automata theory


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Conservation in the Republic of Ireland

Conservation in the Republic of Ireland is overseen by a number of statutory and non-governmental agencies, including those with responsibility for conservation of the built environment and conservation of the natural environment in Ireland. Conservation has sometimes been a contentious issue, with debates impacting its progress since the 1960s. Concrete initiatives are sometimes driven by European Union (EU) heritage protection and environmental policies, including EU environmental law, which – as a member – the Irish government is obliged to adopt and implement.

Heritage conservation
Heritage conservation has been in place in Ireland since the formation as the state, with structures protected under local, national and international legislation.

National legislation
In the 1930s, a national policy was adopted in the form of the National Monuments Act, which established preservation orders, listed national monuments, and outlined standards, prohibitions, and regulations of archaeological objects. The National Monuments Service, an agency of the Office of Public Works, is responsible for National Monuments. As of 2020, the National Monuments Service had responsibility for the conservation of approximately 1,000 individual structures at over 760 different sites.

Local legislation
The Planning and Development Act 2000 establishes that each regional planning authority (administered by county and city councils in Ireland) is responsible for maintaining a Record of Protected Structures. These structures may be protected for conservation from an architectural, historical, archaeological, cultural, social or technical perspective. Owners or occupiers of a listed protected structure are obliged to prevent the structure from becoming endangered through damage or through neglect. As of 2020, the various Records of Protected Structures contained over 40,000 structures nationally.

International conventions
Ireland has been a member of UNESCO, the United Nations Educational, Scientific and Cultural Organization, since 1961. As of 2020, there are two UNESCO World Heritage Sites in the Republic of Ireland. These structures are in the Brú na Bóinne complex in County Meath and on Skellig Michael in County Kerry.

Non-government agencies
A number of non-government and voluntary organisations (including An Taisce (the National Trust for Ireland), the Dublin Civic Trust, the Irish Landmark Trust, and the Mining Heritage Trust of Ireland) undertake architectural conservation projects on a non-statutory basis.

Statutory protections
One of the earliest environmental conservation initiatives in Ireland involved the hosting of an international wildlife seminar in 1970, which was part of its contribution to the European Conservation Year. This seminar played a role in the creation of the Forest and Wildlife Service, which was created in response to the seminar's recommendation for a centralized conservation agency in Northern Ireland. One of the primary functions of the agency involves the identification of sites for conservation purposes.

As of the 21st century, government agencies with responsibility for the protection and conservation of the natural environment include the Department of Communications, Climate Action and Environment, Environmental Protection Agency, and National Parks and Wildlife Service. The Local Government (Planning and Development) Act 1963 also affords some responsibility for environmental protection to local authorities in Ireland, who can control development in areas "outstanding natural beauty and/or special recreational value" by designating them a "Special Amenity Area".

Voluntary organisations
Voluntary and non-governmental agencies undertaking environmental conservation projects in Ireland include the Bilberry Goat Trust, Irish Peatland Conservation Council, and Native Woodland Trust. A number of these organisations are members of the Irish Environmental Network.

title: Conservation in the Republic of Ireland


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Scouting in Michigan

Scouting in Michigan has a long history, from the 1910s to the present day, serving thousands of youth in programs that suit the environment in which they live.

Beginning years (1910–1950)
The YMCA in Michigan was organizing Scout troops based on Scouting for Boys as early as 1909.

In 1910 the first scout troop in West Michigan, Rockford Troop 1 was formed by Merritt Lamb. Shortly after the formation of Rockford Troop 1, Merritt formed Muskegon Troop 1 and the Muskegon Council. Merritt Lamb was the 13th Eagle Scout in the nation, and the first person in Michigan to receive the Bronze Cross for Lifesaving.

The Michigan Forest Scouts were formed by the State of Michigan in 1911. This group was formed in response to a number of late 19th century and early 20th century forest fires, and were effectively "auxiliary fire wardens".

In 1914, St. Stanislaus Boy Scouts of America (BSA) Troop No. 1 obtained its charter as the first organized troop in Bay City Michigan and the first admitted to the Bay City Council (chartered in 1917). Scout activities had been going on for several years prior to this. Father Ladislaus P. Krakowski, pastor of St. Stanislaus Kostka, and his assistant Father Frank Kozak encouraged the official link, per the St. Stanislaus Kostka Church Centennial 1874-1974 book of 1974.

The Forest Scouts crested at 5,000 members and ended in 1916.

In 1929, a group of 8 Eagle Scouts including Gerald R. Ford participate in the first Mackinac Island Honor Guard.

In 1933, the National Order of the Arrow Lodge Meeting was held at the Owasippe Scout Camps outside Whitehall, Michigan.

In 1950, the Gerber baby food company donated the funds to purchase Gerber Scout Reservation (now of the Michigan Crossroads Council) to BSA.

Mackinac Island Honor Guard Program

The Mackinac Island Governor's Honor Guard (Mackinac Island Honor Scouts) program is one of the few elite programs of its kind. Starting in 1929, a select group of eight Eagle Scouts from across the state, including young Gerald Ford, to serve as honor guards in Fort Mackinac. In 1934, as a Civilian Conservation Corps project, Scout Barracks were built just outside the fort walls. The service camp has been known by many names. Originally, it was named to honor the Governor of Michigan at the time. It was also known as the Mackinac Island Eagle Scout Honor Guard until it was opened up to scouts of all ranks. In 1974, the program was expanded to include Girl Scouts. Similar programs are the Utah National Parks Council Honor Guard and Greater Niagara Frontier Council Honor Guard.

The program is popular, selective, and a long-standing tradition. Scouts raise and lower twenty-six flags on the island, serve as guides, and complete volunteer service projects during their stay. Duties on the island include raising and lowering flags each day, serving downtown and in Fort Mackinac as guides, and doing a variety of service projects to better Mackinac Island State Park.

A unique tie that Scouting has with Mackinac Island can be seen overlooking the Mackinac Island harbor. The Mackinac Island Statue of Liberty replica was dedicated in 1950 by the Boy Scouts of America. The statue is one of 200 donated by the BSA in 1950 to celebrate the 40th anniversary of Scouting. The program was called Strengthen the Arm of Liberty and the Mackinac statue is the only one located in Michigan. Due to the unique role that BSA Scouts play on Mackinac Island during the summer, the statue is a lasting testament to the island’s scouting heritage. The statue was restored by the Mackinac Island American Legion in 2013 and was rededicated on July 4, 2014.

Between Memorial Day and Labor day, the island is the summer home to 14 scout troops. Of the six Girl Scout troops, one is from Michigan Shore to Shore Council, two from Heart of Michigan Council, and three from Southeastern Michigan Council. Of the eight BSA councils, Bay-Lakes Council, Southern Shores Field Service Council, and Great Lakes Field Service Council all provide one troop, President Gerald R. Ford Field Service Council is home to two units, while Water and Woods Field Service Council has three honor troops.

Middle years (1950–1990)
The Bass Lake lone troop Scout camp was part of the Owasippe Scout Reservation operated by the Chicago Area Council. The Bass Lake camp was a single camp on a small (approximately 600 ft wide) lake in the Owasippe reservation. A single troop would take over the entire camp, usually for a two-week period, preparing all their own food and overseeing all aspects of the camp life. Active in the 1950s and 1960s, Bass Lake camp is no longer in use.

International Girl Scouts of the USA gatherings named Senior Roundups were held every three years from 1956 until 1965. The first one was held at Milford, Michigan in 1956, attended by 5,000 girls.

As the scouting program grew across the state, local councils began to be consolidated into larger councils to manage the increased number of scouts and units. By the end of the 20th century, Michigan was home to 14 Boy Scout councils and 14 Girl Scout Councils.

The 1986 National Order of the Arrow Conference was held at Central Michigan University in Mount Pleasant.

Recent history (2000–present)
The 2006 and 2012 National Order of the Arrow conferences (NOACs) were held at Michigan State University in East Lansing. The host lodge in 2006 was Gabe-shi-win-gi-ji-kens Lodge #374. 8,003 Arrowmen from around the country came by plane, train, and bus to participate in the Order's 2006 NOAC; Over 600 people from the state of Michigan were present during the 5 day event.

At the beginning of 2010, there were 12 Boy Scout Councils and 14 Girl Scout Councils in Michigan. In 2008, many Girl Scout Councils were merged in the state, resulting in the 5 present day Girl Scouts councils. In 2012, the Boy Scouts of America consolidated nine of its councils across the state into the Michigan Crossroads Council.

In 2020, Michigan Crossroads Council made a decision to merge their Field Service Councils to create one central Council.

Michigan Crossroads Council
Due to a population decline in the 2000s, with a corresponding loss in youth members, the Boy Scouts of America decided that Michigan's councils should consolidate to help save costs and raise membership. In October 2010, the Area 2 Project was launched to develop a sustainable origination for the 21st Century. In 2012 the Area 2 project committee presented a "Crossroads Recommendation" to the Central Region. As a result of the recommendation, nine councils were merged into the Michigan Crossroads Council, which was then divided into four field service councils.

In 2020, Michigan Crossroads Council made a decision to merge their Field Service Councils to create one central Council.
 * Great Lakes Field Service Council serves Scouts in Metro Detroit.
 * President Gerald R. Ford Field Service Council serves Scouts in western and northern Michigan.
 * Southern Shores Field Service Council serves Scouts in Southern Michigan.
 * Water & Woods Field Service Council serves Scouts in eastern and central Michigan.

Bay-Lakes Council
The Bay-Lakes Council is headquartered in Appleton, Wisconsin, and also serves Scouts in Michigan. It is merged with Hiawathaland Council, retaining the Bay-Lakes name.

Camp Hiawatha was opened in 1967 on property formally known as Wolfe's Lodge. There were several small cabins and a larger lodge on the "family side" and little else. The "camp side" was nothing but trees, brush and ferns as tall at the campers. The first campers that summer faced a real challenge. No electricity, telephones, running water and very little shelter.

Today Camp Hiawatha is 800 acre and encircles Bunting Lake, a 60 acre lake in the middle of Hiawathaland National Forest in the heart of the Upper Peninsula of Michigan. The closest towns are Chatham and Munising, Michigan.

Camp Hiawatha is now under the administration of Bay-Lakes Council, becoming Bay-Lakes' third Scouts BSA resident camp, joining Bear Paw Scout Camp and Gardner Dam Camps, and fifth overall resident camp, joining Cub Scout World Camp Rokilio and Twin Lakes Webelos Resident Camp.

La Salle Council
La Salle Council serves Scouts in Indiana and Michigan.

Voyageurs Area Council
Headquartered in Hermantown, Minnesota, Voyageurs Area Council serves Scouts in Minnesota, Wisconsin, and Michigan

Girl Scouts of the USA

There are five Girl Scout councils in Michigan.

Girl Scouts Heart of Michigan
Girl Scouts Heart of Michigan serves 30,000 girls in central Michigan. It was formed on October 1, 2008 by the merger of Glowing Embers Girl Scout Council, Girl Scouts of The Huron Valley Council, Girl Scouts - Irish Hills Council, and Girl Scouts of Michigan Capital Council.

 * Camp Deer Trails, Harrison, MI
 * Camp Linden, Linden, MI
 * Camp Merrie Woode, Plainwell, MI
 * Camp O' the Hills, Brooklyn, MI
 * Camp Oak Hills, Harrison, MI
 * Camp Wacousta, Wacousta, MI
 * Barbara Osterman Cabin, Alma, MI
 * Jane Harris Cabin, Mt. Pleasant, MI
 * Program & Training Center, Kalamazoo, MI

Girl Scouts of Michigan Shore to Shore
Headquartered in Grand Rapids, Michigan, the Girl Scouts of Michigan Shore to Shore serves just over 6,000 members in northwestern Michigan (not including the Upper Peninsula). It was formed by the merger of Girl Scouts of Michigan Pine and Dunes Council, Girl Scouts of Michigan Trails, Girl Scouts of Crooked Tree, and Girl Scouts of Mitten Bay.

Camps:
 * Camp Anna Behrens - Greenville, MI
 * Camp Woodlands - Alpena, MI
 * Sherwood Lodge at Camp Anna Behrens - Greenville, MI
 * Betsy Cutler House - Muskegon, MI
 * Grand Haven Little House - Grand Haven, MI
 * Ludington Little House - Ludington, MI

Girl Scouts of Northern Indiana-Michiana
Headquartered in Fort Wayne, Indiana serves girls in Berrien and Cass counties.

Girl Scouts of The Northwestern Great Lakes
Headquartered in Green Bay, Wisconsin, this dividing serves girls in the Upper Peninsula.

 * Camp Birch Trails - in Rock Falls, 8 miles N of Merrill, WI
 * Camp Cuesta - by Bailey's Harbor, West of Traverse City on the across Lake Michigan past South & North Manitou Islands.
 * Camp Del O'Claire - Weston, WI
 * Camp Nawakwa - NW of Cornell, WI
 * Camp Pow-Low - Gwinn, MI near Marquette, MI in waterfall country
 * Camp Sacajawea - Wisconsin Rapids, WI on Lake Tapawingo.
 * Camp Winnecomac - NE of Appleton in Kaukauna, WI surrounded by 1,000 Island Nature Preserve.
 * Appleton Service Center - Appleton, WI
 * Eau Claire Service Center - Eau Claire, WI
 * Green Bay Service Center - Green Bay, WI

Girl Scouts of Southeastern Michigan
Headquartered in Detroit, Michigan, it was formed by a merger on January 1, 2009 of Girl Scouts Fair Winds Council, Girl Scouts of Macomb County - Otsikita Council, Girl Scouts of Metro Detroit, and Girl Scouts - Michigan Waterways Council.

<!-- Efforts are under way to merge and realign councils across Lower Michigan into three new councils since 2006 as a part of across the country effort to reduce the number of councils. Glowing Embers, Huron Valley, Irish Hills, Michigan Capital, Mitten Bay with the addition of Shiawassee County from Fair Winds Council and Branch, St. Joseph, Van Buren Counties from Singing Sands Council will become Girl Scouts Heart of Michigan. Girl Scouts of Southeastern Michigan will be form from Fair Winds, Huron Valley, Macomb County - Otsikita, Metro Detroit, Michigan Waterways Councils with a portion of Monroe coming from Maumee Valley Council. Crooked Tree, Michigan Pine & Dunes and Michigan Trails Councils, and parts of Mitten Bay are merging into Girl Scouts of Michigan Shore to Shore Council. Peninsula Waters Council merged with northern Wisconsin councils: Birch Trails Council (Schofield WI), Fox River Area (Appleton WI), Indian Waters (Eau Claire WI), Lac Baie Council (Green Bay WI), and Girl Scouts of Woodland Council (Wisconsin Rapids WI) into Girl Scouts of the Northwestern Great Lakes, Inc.
 * Camp Hawthorn Hollow - Richmond, MI
 * Camp Innisfree - between Howell and Pinckney, MI - was idled as of September 2014.
 * Playfair Program Center - near Lexington, MI

Post Realignment Councils -->

Scouting museums

 * E. Urner Goodman Scout Museum, Owasippe Scout Reservation, Blue Lake Township, Michigan
 * Thomas D. Trainor Boy Scout Museum
 * Washington Scouting Museum, Washington, Michigan

title: Scouting in Michigan