[comment {-*- tcl -*- doctools manpage}]
[manpage_begin grammar::fa n 0.4]
[keywords automaton]
[keywords {finite automaton}]
[keywords grammar]
[keywords parsing]
[keywords {regular expression}]
[keywords {regular grammar}]
[keywords {regular languages}]
[keywords state]
[keywords transducer]
[copyright {2004-2009 Andreas Kupries }]
[moddesc {Finite automaton operations and usage}]
[titledesc {Create and manipulate finite automatons}]
[category {Grammars and finite automata}]
[require Tcl 8.4]
[require snit 1.3]
[require struct::list]
[require struct::set]
[require grammar::fa::op [opt 0.2]]
[require grammar::fa [opt 0.4]]
[description]
[para]
This package provides a container class for
[term {finite automatons}] (Short: FA).
It allows the incremental definition of the automaton, its
manipulation and querying of the definition.
While the package provides complex operations on the automaton
(via package [package grammar::fa::op]), it does not have the
ability to execute a definition for a stream of symbols.
Use the packages
[package grammar::fa::dacceptor] and
[package grammar::fa::dexec] for that.
Another package related to this is [package grammar::fa::compiler]. It
turns a FA into an executor class which has the definition of the FA
hardwired into it. The output of this package is configurable to suit
a large number of different implementation languages and paradigms.
[para]
For more information about what a finite automaton is see section
[sectref {FINITE AUTOMATONS}].
[section API]
The package exports the API described here.
[list_begin definitions]
[call [cmd ::grammar::fa] [arg faName] [opt "[const =]|[const :=]|[const <--]|[const as]|[const deserialize] [arg src]|[const fromRegex] [arg re] [opt [arg over]]"]]
Creates a new finite automaton with an associated global Tcl command
whose name is [arg faName]. This command may be used to invoke various
operations on the automaton. It has the following general form:
[list_begin definitions]
[call [cmd faName] [arg option] [opt [arg "arg arg ..."]]]
[arg Option] and the [arg arg]s determine the exact behavior of the
command. See section [sectref {FA METHODS}] for more explanations. The
new automaton will be empty if no [arg src] is specified. Otherwise
it will contain a copy of the definition contained in the [arg src].
The [arg src] has to be a FA object reference for all operators except
[const deserialize] and [const fromRegex]. The [const deserialize]
operator requires [arg src] to be the serialization of a FA instead,
and [const fromRegex] takes a regular expression in the form a of a
syntax tree. See [cmd ::grammar::fa::op::fromRegex] for more detail on
that.
[list_end]
[list_end]
[section {FA METHODS}]
[para]
All automatons provide the following methods for their manipulation:
[list_begin definitions]
[call [arg faName] [method destroy]]
Destroys the automaton, including its storage space and associated
command.
[call [arg faName] [method clear]]
Clears out the definition of the automaton contained in [arg faName],
but does [emph not] destroy the object.
[call [arg faName] [method =] [arg srcFA]]
Assigns the contents of the automaton contained
in [arg srcFA] to [arg faName], overwriting any
existing definition.
This is the assignment operator for automatons. It copies the
automaton contained in the FA object [arg srcFA] over the automaton
definition in [arg faName]. The old contents of [arg faName] are
deleted by this operation.
[para]
This operation is in effect equivalent to
[para]
[example_begin]
[arg faName] [method deserialize] [lb][arg srcFA] [method serialize][rb]
[example_end]
[call [arg faName] [method -->] [arg dstFA]]
This is the reverse assignment operator for automatons. It copies the
automation contained in the object [arg faName] over the automaton
definition in the object [arg dstFA].
The old contents of [arg dstFA] are deleted by this operation.
[para]
This operation is in effect equivalent to
[para]
[example_begin]
[arg dstFA] [method deserialize] [lb][arg faName] [method serialize][rb]
[example_end]
[call [arg faName] [method serialize]]
This method serializes the automaton stored in [arg faName]. In other
words it returns a tcl [emph value] completely describing that
automaton.
This allows, for example, the transfer of automatons over arbitrary
channels, persistence, etc.
This method is also the basis for both the copy constructor and the
assignment operator.
[para]
The result of this method has to be semantically identical over all
implementations of the [package grammar::fa] interface. This is what
will enable us to copy automatons between different implementations of
the same interface.
[para]
The result is a list of three elements with the following structure:
[list_begin enumerated]
[enum]
The constant string [const grammar::fa].
[enum]
A list containing the names of all known input symbols. The order of
elements in this list is not relevant.
[enum]
The last item in the list is a dictionary, however the order of the
keys is important as well. The keys are the states of the serialized
FA, and their order is the order in which to create the states when
deserializing. This is relevant to preserve the order relationship
between states.
[para]
The value of each dictionary entry is a list of three elements
describing the state in more detail.
[list_begin enumerated]
[enum]
A boolean flag. If its value is [const true] then the state is a
start state, otherwise it is not.
[enum]
A boolean flag. If its value is [const true] then the state is a
final state, otherwise it is not.
[enum]
The last element is a dictionary describing the transitions for the
state. The keys are symbols (or the empty string), and the values are
sets of successor states.
[list_end]
[list_end]
[para]
Assuming the following FA (which describes the life of a truck driver
in a very simple way :)
[para]
[example {
Drive -- yellow --> Brake -- red --> (Stop) -- red/yellow --> Attention -- green --> Drive
(...) is the start state.
}]
[para]
a possible serialization is
[para]
[example {
grammar::fa \\
{yellow red green red/yellow} \\
{Drive {0 0 {yellow Brake}} \\
Brake {0 0 {red Stop}} \\
Stop {1 0 {red/yellow Attention}} \\
Attention {0 0 {green Drive}}}
}]
[para]
A possible one, because I did not care about creation order here
[call [arg faName] [method deserialize] [arg serialization]]
This is the complement to [method serialize]. It replaces the
automaton definition in [arg faName] with the automaton described by
the [arg serialization] value. The old contents of [arg faName] are
deleted by this operation.
[call [arg faName] [method states]]
Returns the set of all states known to [arg faName].
[call [arg faName] [method state] [method add] [arg s1] [opt "[arg s2] ..."]]
Adds the states [arg s1], [arg s2], et cetera to the FA definition in
[arg faName]. The operation will fail any of the new states is already
declared.
[call [arg faName] [method state] [method delete] [arg s1] [opt "[arg s2] ..."]]
Deletes the state [arg s1], [arg s2], et cetera, and all associated
information from the FA definition in [arg faName]. The latter means
that the information about in- or outbound transitions is deleted as
well. If the deleted state was a start or final state then this
information is invalidated as well. The operation will fail if the
state [arg s] is not known to the FA.
[call [arg faName] [method state] [method exists] [arg s]]
A predicate. It tests whether the state [arg s] is known to the FA in
[arg faName].
The result is a boolean value. It will be set to [const true] if the
state [arg s] is known, and [const false] otherwise.
[call [arg faName] [method state] [method rename] [arg s] [arg snew]]
Renames the state [arg s] to [arg snew]. Fails if [arg s] is not a
known state. Also fails if [arg snew] is already known as a state.
[call [arg faName] [method startstates]]
Returns the set of states which are marked as [term start] states,
also known as [term initial] states.
See [sectref {FINITE AUTOMATONS}] for explanations what this means.
[call [arg faName] [method start] [method add] [arg s1] [opt "[arg s2] ..."]]
Mark the states [arg s1], [arg s2], et cetera in the FA [arg faName]
as [term start] (aka [term initial]).
[call [arg faName] [method start] [method remove] [arg s1] [opt "[arg s2] ..."]]
Mark the states [arg s1], [arg s2], et cetera in the FA [arg faName]
as [term {not start}] (aka [term {not accepting}]).
[call [arg faName] [method start?] [arg s]]
A predicate. It tests if the state [arg s] in the FA [arg faName] is
[term start] or not.
The result is a boolean value. It will be set to [const true] if the
state [arg s] is [term start], and [const false] otherwise.
[call [arg faName] [method start?set] [arg stateset]]
A predicate. It tests if the set of states [arg stateset] contains at
least one start state. They operation will fail if the set contains an
element which is not a known state.
The result is a boolean value. It will be set to [const true] if a
start state is present in [arg stateset], and [const false] otherwise.
[call [arg faName] [method finalstates]]
Returns the set of states which are marked as [term final] states,
also known as [term accepting] states.
See [sectref {FINITE AUTOMATONS}] for explanations what this means.
[call [arg faName] [method final] [method add] [arg s1] [opt "[arg s2] ..."]]
Mark the states [arg s1], [arg s2], et cetera in the FA [arg faName]
as [term final] (aka [term accepting]).
[call [arg faName] [method final] [method remove] [arg s1] [opt "[arg s2] ..."]]
Mark the states [arg s1], [arg s2], et cetera in the FA [arg faName]
as [term {not final}] (aka [term {not accepting}]).
[call [arg faName] [method final?] [arg s]]
A predicate. It tests if the state [arg s] in the FA [arg faName] is
[term final] or not.
The result is a boolean value. It will be set to [const true] if the
state [arg s] is [term final], and [const false] otherwise.
[call [arg faName] [method final?set] [arg stateset]]
A predicate. It tests if the set of states [arg stateset] contains at
least one final state. They operation will fail if the set contains an
element which is not a known state.
The result is a boolean value. It will be set to [const true] if a
final state is present in [arg stateset], and [const false] otherwise.
[call [arg faName] [method symbols]]
Returns the set of all symbols known to the FA [arg faName].
[call [arg faName] [method symbols@] [arg s] [opt [arg d]]]
Returns the set of all symbols for which the state [arg s] has transitions.
If the empty symbol is present then [arg s] has epsilon transitions. If two
states are specified the result is the set of symbols which have transitions
from [arg s] to [arg t]. This set may be empty if there are no transitions
between the two specified states.
[call [arg faName] [method symbols@set] [arg stateset]]
Returns the set of all symbols for which at least one state in the set
of states [arg stateset] has transitions.
In other words, the union of [lb][arg faName] [method symbols@] [var s][rb]
for all states [var s] in [arg stateset].
If the empty symbol is present then at least one state contained in
[arg stateset] has epsilon transitions.
[call [arg faName] [method symbol] [method add] [arg sym1] [opt "[arg sym2] ..."]]
Adds the symbols [arg sym1], [arg sym2], et cetera to the FA
definition in [arg faName]. The operation will fail any of the symbols
is already declared. The empty string is not allowed as a value for the symbols.
[call [arg faName] [method symbol] [method delete] [arg sym1] [opt "[arg sym2] ..."]]
Deletes the symbols [arg sym1], [arg sym2] et cetera, and all
associated information from the FA definition in [arg faName]. The
latter means that all transitions using the symbols are deleted as
well. The operation will fail if any of the symbols is not known to
the FA.
[call [arg faName] [method symbol] [method rename] [arg sym] [arg newsym]]
Renames the symbol [arg sym] to [arg newsym]. Fails if [arg sym] is
not a known symbol. Also fails if [arg newsym] is already known as a
symbol.
[call [arg faName] [method symbol] [method exists] [arg sym]]
A predicate. It tests whether the symbol [arg sym] is known to the FA
in [arg faName].
The result is a boolean value. It will be set to [const true] if the
symbol [arg sym] is known, and [const false] otherwise.
[call [arg faName] [method next ] [arg s] [arg sym] [opt "[const -->] [arg next]"]]
Define or query transition information.
[para]
If [arg next] is specified, then the method will add a transition from
the state [arg s] to the [term successor] state [arg next] labeled with
the symbol [arg sym] to the FA contained in [arg faName]. The
operation will fail if [arg s], or [arg next] are not known states, or
if [arg sym] is not a known symbol. An exception to the latter is that
[arg sym] is allowed to be the empty string. In that case the new
transition is an [term {epsilon transition}] which will not consume
input when traversed. The operation will also fail if the combination
of ([arg s], [arg sym], and [arg next]) is already present in the FA.
[para]
If [arg next] was not specified, then the method will return
the set of states which can be reached from [arg s] through
a single transition labeled with symbol [arg sym].
[call [arg faName] [method !next] [arg s] [arg sym] [opt "[const -->] [arg next]"]]
Remove one or more transitions from the Fa in [arg faName].
[para]
If [arg next] was specified then the single transition from the state
[arg s] to the state [arg next] labeled with the symbol [arg sym] is
removed from the FA. Otherwise [emph all] transitions originating in
state [arg s] and labeled with the symbol [arg sym] will be removed.
[para]
The operation will fail if [arg s] and/or [arg next] are not known as
states. It will also fail if a non-empty [arg sym] is not known as
symbol. The empty string is acceptable, and allows the removal of
epsilon transitions.
[call [arg faName] [method nextset] [arg stateset] [arg sym]]
Returns the set of states which can be reached by a single transition
originating in a state in the set [arg stateset] and labeled with the
symbol [arg sym].
[para]
In other words, this is the union of
[lb][arg faName] next [var s] [arg symbol][rb]
for all states [var s] in [arg stateset].
[call [arg faName] [method is] [method deterministic]]
A predicate. It tests whether the FA in [arg faName] is a
deterministic FA or not.
The result is a boolean value. It will be set to [const true] if the
FA is deterministic, and [const false] otherwise.
[call [arg faName] [method is] [method complete]]
A predicate. It tests whether the FA in [arg faName] is a complete FA
or not. A FA is complete if it has at least one transition per state
and symbol. This also means that a FA without symbols, or states is
also complete.
The result is a boolean value. It will be set to [const true] if the
FA is deterministic, and [const false] otherwise.
[para]
Note: When a FA has epsilon-transitions transitions over a symbol for
a state S can be indirect, i.e. not attached directly to S, but to a
state in the epsilon-closure of S. The symbols for such indirect
transitions count when computing completeness.
[call [arg faName] [method is] [method useful]]
A predicate. It tests whether the FA in [arg faName] is an useful FA
or not. A FA is useful if all states are [term reachable]
and [term useful].
The result is a boolean value. It will be set to [const true] if the
FA is deterministic, and [const false] otherwise.
[call [arg faName] [method is] [method epsilon-free]]
A predicate. It tests whether the FA in [arg faName] is an
epsilon-free FA or not. A FA is epsilon-free if it has no epsilon
transitions. This definition means that all deterministic FAs are
epsilon-free as well, and epsilon-freeness is a necessary
pre-condition for deterministic'ness.
The result is a boolean value. It will be set to [const true] if the
FA is deterministic, and [const false] otherwise.
[call [arg faName] [method reachable_states]]
Returns the set of states which are reachable from a start state by
one or more transitions.
[call [arg faName] [method unreachable_states]]
Returns the set of states which are not reachable from any start state
by any number of transitions. This is
[para]
[example {
[faName states] - [faName reachable_states]
}]
[call [arg faName] [method reachable] [arg s]]
A predicate. It tests whether the state [arg s] in the FA [arg faName]
can be reached from a start state by one or more transitions.
The result is a boolean value. It will be set to [const true] if the
state can be reached, and [const false] otherwise.
[call [arg faName] [method useful_states]]
Returns the set of states which are able to reach a final state by
one or more transitions.
[call [arg faName] [method unuseful_states]]
Returns the set of states which are not able to reach a final state by
any number of transitions. This is
[para]
[example {
[faName states] - [faName useful_states]
}]
[call [arg faName] [method useful] [arg s]]
A predicate. It tests whether the state [arg s] in the FA [arg faName]
is able to reach a final state by one or more transitions.
The result is a boolean value. It will be set to [const true] if the
state is useful, and [const false] otherwise.
[call [arg faName] [method epsilon_closure] [arg s]]
Returns the set of states which are reachable from the state [arg s]
in the FA [arg faName] by one or more epsilon transitions, i.e
transitions over the empty symbol, transitions which do not consume
input. This is called the [term {epsilon closure}] of [arg s].
[call [arg faName] [method reverse]]
[call [arg faName] [method complete]]
[call [arg faName] [method remove_eps]]
[call [arg faName] [method trim] [opt [arg what]]]
[call [arg faName] [method determinize] [opt [arg mapvar]]]
[call [arg faName] [method minimize] [opt [arg mapvar]]]
[call [arg faName] [method complement]]
[call [arg faName] [method kleene]]
[call [arg faName] [method optional]]
[call [arg faName] [method union] [arg fa] [opt [arg mapvar]]]
[call [arg faName] [method intersect] [arg fa] [opt [arg mapvar]]]
[call [arg faName] [method difference] [arg fa] [opt [arg mapvar]]]
[call [arg faName] [method concatenate] [arg fa] [opt [arg mapvar]]]
[call [arg faName] [method fromRegex] [arg regex] [opt [arg over]]]
These methods provide more complex operations on the FA. Please see
the same-named commands in the package [package grammar::fa::op] for
descriptions of what they do.
[list_end]
[para]
[section EXAMPLES]
[para]
[section {FINITE AUTOMATONS}]
[para]
For the mathematically inclined, a FA is a 5-tuple (S,Sy,St,Fi,T) where
[list_begin itemized]
[item]
S is a set of [term {states}],
[item]
Sy a set of [term {input symbols}],
[item]
St is a subset of S, the set of [term start] states, also known as
[term initial] states.
[item]
Fi is a subset of S, the set of [term final] states, also known as
[term accepting].
[item]
T is a function from S x (Sy + epsilon) to {S}, the [term {transition function}].
Here [const epsilon] denotes the empty input symbol and is distinct
from all symbols in Sy; and {S} is the set of subsets of S. In other
words, T maps a combination of State and Input (which can be empty) to
a set of [term {successor states}].
[list_end]
[para]
In computer theory a FA is most often shown as a graph where the nodes
represent the states, and the edges between the nodes encode the
transition function: For all n in S' = T (s, sy) we have one edge
between the nodes representing s and n resp., labeled with sy. The
start and accepting states are encoded through distinct visual
markers, i.e. they are attributes of the nodes.
[para]
FA's are used to process streams of symbols over Sy.
[para]
A specific FA is said to [term accept] a finite stream sy_1 sy_2
... sy_n if there is a path in the graph of the FA beginning at a
state in St and ending at a state in Fi whose edges have the labels
sy_1, sy_2, etc. to sy_n.
The set of all strings accepted by the FA is the [term language] of
the FA. One important equivalence is that the set of languages which
can be accepted by an FA is the set of [term {regular languages}].
[para]
Another important concept is that of deterministic FAs. A FA is said
to be [term deterministic] if for each string of input symbols there
is exactly one path in the graph of the FA beginning at the start
state and whose edges are labeled with the symbols in the string.
While it might seem that non-deterministic FAs to have more power of
recognition, this is not so. For each non-deterministic FA we can
construct a deterministic FA which accepts the same language (-->
Thompson's subset construction).
[para]
While one of the premier applications of FAs is in [term parsing],
especially in the [term lexer] stage (where symbols == characters),
this is not the only possibility by far.
[para]
Quite a lot of processes can be modeled as a FA, albeit with a
possibly large set of states. For these the notion of accepting states
is often less or not relevant at all. What is needed instead is the
ability to act to state changes in the FA, i.e. to generate some
output in response to the input.
This transforms a FA into a [term {finite transducer}], which has an
additional set OSy of [term {output symbols}] and also an additional
[term {output function}] O which maps from "S x (Sy + epsilon)" to
"(Osy + epsilon)", i.e a combination of state and input, possibly
empty to an output symbol, or nothing.
[para]
For the graph representation this means that edges are additional
labeled with the output symbol to write when this edge is traversed
while matching input. Note that for an application "writing an output
symbol" can also be "executing some code".
[para]
Transducers are not handled by this package. They will get their own
package in the future.
[vset CATEGORY grammar_fa]
[include ../doctools2base/include/feedback.inc]
[manpage_end]