Abstract family of acceptors
An abstract family of acceptors (AFA) is a grouping of generalized acceptors. Informally, an acceptor is a device with a finite state control, a finite number of input symbols, and an internal store with a read and write function. Each acceptor has a start state and a set of accepting states. The device reads a sequence of symbols, transitioning from state to state for each input symbol. If the device ends in an accepting state, the device is said to accept the sequence of symbols. A family of acceptors is a set of acceptors with the same type of internal store. The study of AFA is part of AFL (abstract families of languages) theory.
Seymour Ginsburg, "Algebraic and automata theoretic properties of formal languages", North-Holland, 1975, ISBN 0 7204 2506 9.]
An "AFA Schema" is an ordered 4-tuple , where
# and are nonempty abstract sets.
# is the "write" function: (N.B. * is the
# is the "read" function, a mapping from into the finite subsets of , such that and is in if and only if . (N.B. is the empty word).
# For each in , there is an element in satisfying for all such that is in .
# For each "u" in "I", there exists a finite set ⊆ , such that if ⊆ , is in , and , then is in .
Abstract family of acceptors
An "abstract family of acceptors (AFA)" is an ordered pair such that:
# is an ordered 6-tuple (, , , , , ), where
## (, , , ) is an AFA schema; and
## and are infinite abstract sets
# is the family of all acceptors = (, , , , ), where
## and are finite subsets of , and respectively, ⊆ , and is in ; and
## (called the "transition" function) is a mapping from into the finite subsets of such that the set | ≠ ø for some and is finite.
For a given acceptor, let be the relation on defined by: For in , if there exists a and such that is in , is in and . Let denote the
transitive closureof .
Let be an AFA and = (, , , , ) be in . Define to be the set . For each subset of , let .
Define to be the set . For each subset of , let .
An AFA schema defines a store or memory with read and write function. The symbols in are called "storage symbols" and the symbols in are called "instructions". The write function returns a new storage state given the current storage state and an instruction. The read function returns the current state of memory. Condition (3) insures the empty storage configuration is distinct from other configurations. Condition (4) requires there be an identity instruction that allows the state of memory to remain unchanged while the acceptor changes state or advances the input. Condition (5) assures that the set of storage symbols for any given acceptor is finite.
Abstract family of acceptors
An AFA is the set of all acceptors over a given pair of state and input alphabets which have the same storage mechanism defined by a given AFA schema. The relation defines one step in the operation of an acceptor. is the set of words accepted by acceptor by having the acceptor enter an accepting state. is the set of words accepted by acceptor by having the acceptor simultaneously enter an accepting state and having an empty storage.
The abstract acceptors defined by AFA are generalizations of other types of acceptors (e.g.
finite state automata, pushdown automata, etc.). They have a finite state control like other automata, but their internal storage may vary widely from the stacks and tapes used in classical automata.
Results from AFL theory
The main result from AFL theory is that a family of languages is a full AFL if and only if for some AFA . Equally important is the result that is a full semi-AFL if and only if for some AFA .
Seymour Ginsburgof the University of Southern Californiaand Sheila Greibachof Harvard Universityfirst presented their AFL theory paper at the IEEEEighth Annual Symposium on Switching and Automata Theory in 1967. [ [http://www.worldcat.org/oclc/2891921 IEEE conference record of 1967 Eighth Annual Symposium on Switching and Automata Theory] : papers presented at the Eighth Annual Symposium, University of Texas, October 18-20, 1967.]
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