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A common critique that was directed at Parsons’ conception of theory was that it was much too abstract. According to Robert K. Merton, for example, sociological theory should focus on the middle range and stay at that level (Merton, 1949, 1968). Attempts to formulate a general theory, at the highest and most abstract level of sociology, were in
- Nondeterministic Machines
- Representing Machines as Logo Lists
- Text Editors: a Use for Acceptors
- Regular Expressions
- Regular Expressions and Finite-State Machines
- How to Translate
- Making the Machine Deterministic
- Eliminating Redundant States
- Counting and Finite-State Machines
- Turing Machines
- Turing’s Thesis
- The Halting Theorem
- Proving the Halting Theorem in Logo
- true ? print haltp "piglatin "mxyzptlk
Here is rule 6: “To be accepted a string must begin with A and end with C .” Strings accepted by this rule include AC (the shortest possible), ABC , AACC , ACAC , ABCABC , and so on. Between the initial A and the final C an accepted string can have any combination of A s, B s, and C s. It’s natural to think of the string as having three parts: a fi...
The game program uses finite-state machines to represent the rules by which it accepts or rejects strings. (The machines must be deterministic for the program to work.) Logo programs can’t read circles and arrows, so a machine is represented as a list. What information is actually contained in an FSM diagram? The diagram shows that there are a cert...
It may seem to you that accepting or rejecting strings isn’t much like what you usually do with computers. You may wonder how this mathematical model is related to real computer programming. There are two answers to this question. One is that it’s possible to design finite-state machines that have more versatile outputs than simply yes or no. I’ll ...
The notation I’m about to describe allows an acceptance rule, like the rules in the game program or the rules for ed searches, to be represented in Logo. The representation of such a rule is called a regular expression. I’m going to tell you some rules for what a regular expression can look like. Don’t be confused: Any particular regular expression...
I’ve hinted at something that I haven’t actually made explicit: Regular expressions are equivalent to finite-state machines. In other words, if you can express a rule as a regular expression, you can design a finite-state machine that carries out the rule. If you can’t write a regular expression for the rule, you can’t design a finite-state machine...
The general claim that regular expressions are equivalent in power to finite-state machines is called Kleene’s Theorem, named after the mathematician Stephen C. Kleene, its discoverer. You can find a proof of this theorem in any textbook on automata theory. I’m not going to give a proof here, but I’ll indicate how the translation is done in my prog...
In the first volume of this series we explored the techniques of depth-first and breadth-first tree traversal. Given a tree structure, these algorithms allow us to “visit” every node of the tree once. A finite state machine can be viewed as a structure almost like a tree. The machine’s start state corresponds to the root node; the states that can b...
The machines produced by determine are runnable, but often ugly; they contain many more states than necessary. Procedure optimize eliminates many redundancies and also combines arrows with the same head and tail but with different letters. First it goes through the machine’s arrow list, creating a list for each state representing the exits from tha...
Earlier we saw that you can’t write a regular expression for a rule that requires balanced parentheses. Since regular expressions are equivalent to finite-state machines, you won’t be surprised to learn that finite-state machines can’t count. Actually, they can count up to a point; it’s just that each finite-state machine can only count up to a fix...
One way we might explore infinite machines is to imagine that they’re represented by state diagrams, like those of finite-state machines, but with an infinite number of states. For example, here is a picture of an infinite-capacity parenthesis counter:
Turing invented his abstract machine because he was trying to formalize the idea of an effective procedure : What does it mean to specify a technique for solving some problem well enough that we can be sure it will really work? As an analogy, think about directions for driving from here to somewhere. Someone can hand you a piece of paper with a lis...
I’m not going to get into specific examples of Turing machine programming here. That would take too much space for a single chapter; if you’re interested you should pursue the topic in a book on automata theory. But I want to give one example of the theoretical value of Turing machines. You’ve undoubtedly had the experience of writing a Logo progra...
What makes it possible to raise the question of whether a Turing machine can decide whether another Turing machine would halt for a given input tape is the fact that one Turing machine’s “program” can be represented as data for another Turing machine. This is also true of Logo procedures. In particular, the higher-order procedures like map and filt...
false Remember that haltp itself must always stop, even in the case where piglatin wouldn’t stop.) Now consider this Logo procedure: to try :proc if haltp :proc :proc [loop] end to loop loop end Since haltp works, we’re assuming, on any Logo procedure with one input, it must work on try in particular. What happens if we say ? try "try Does this sto...
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Jan 1, 2014 · Abstract. We introduce computational models, such as sequential machines and automata, using the category theory. In particular, we introduce a generalized theorem which states the existence of the most efficient finite state automaton, called the minimal realization. First, we introduce set theoretical elementary models using sets and functions.
- Yoshihiro Mizoguchi
- ym@imi.kyushu-u.ac.jp
- 2014
What is Automata Theory? Study of abstract computing devices, or “machines”. Automaton = an abstract computing device. Note: A “device” need not even be a physical hardware! A fundamental question in computer science: Find out what different models of machines can do and cannot do. The theory of computation.
What is Automata Theory? Study of abstract computing devices, or “machines” Automaton = an abstract computing device Note: A “device” need not even be a physical hardware! A fundamental question in computer science: Find out what different models of machines can do and cannot do The theory of computation
Theory of Computation: A Historical Perspective. 1930s. 1940-1950s. Alan Turing studies Turing machines. Decidability. Halting problem. “Finite automata” machines studied. Noam Chomsky proposes the “Chomsky Hierarchy” for formal languages Cook introduces “intractable” problems or “NP-Hard” problems Modern computer science ...
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What is abstraction in sociology?
e range of opinion on the matter.There are five basic approaches in sociological theory for generating theoretical statements and formats: (1) meta-theoretical schemes, (2) ana-lytical schemes, (3) discursive schemes, (4) propositional. chemes, and (5) modeling schemes. Figure 1.2 summarizes the relations among these schemes.
