Regular Languages and Automata

The domain of languages and automata encompasses three concrete models, finite state automata, regular expressions and linear temporal logic. They are described separately below.

Regular Expressions (RegEx)

Regular expressions are discussed in Section 2 of the reader 'Languages'.

Example

regular expression re = a + b* . d

Syntax RegEx

The syntax of a regular expression always starts with regular expression <name> =. After that, the actual regular expression is written. Regular expressions are constructed from the following primitives and operators (see the reader):

  • the empty set, which is denoted by \o;
  • the empty word, which is denoted by \e;
  • a symbol from the alphabet is either written by a single upper or lower case letter, such as a or Z, or an arbitrary sequence of characters, placed between single or double quotes, e.g., '.' or "'";
  • the union of two or more expressions is written by placing the + symbol between them;
  • the concatenation of two or more expressions is written by placing the . symbol between them;
  • the Kleene closure of a regular expression is written with the * symbol behind it.
  • for ω-regular expressions, the indefinite repetition of a regular expression is written with the ** symbols behind it.
  • parentheses can be used to group expressions, e.g., (a+b)**; without parentheses, the order of priority of operators is: Kleene star. ω repetition, concatenation, union.

where clauses can be used to introduce short hand to avoid a lot of repetition of similar subexpressions.

regular expression number = @Digit . @Digit* . '.' . @Digit . @Digit*
where Digit = '0' + '1' +'2' +'3' +'4' +'5' +'6' +'7' +'8' +'9'

It is used as in the following example. where is added after the expression, followed by a list of definitions of the form <name> = <regular expression>. @ followed by the name of a definitions refers to the regular expression of the definition. The regular expressions can in turn use references to their definitions, but definitions cannot be cyclic.

Operations RegEx

In the workbench, regular expressions support the following operations.

  • Convert to FSA, which converts the regular expression to a non-deterministic finite state automaton that accepts the same language.
  • Convert ω Regular Expression to NBA - converts an ω-regular expression to a non-deterministic Büchi automaton (NBA) that accepts the same language;

Finite State Automata (FSA)

Finite State Automata on finite words are discussed in Section 2 of the reader 'Languages'. Finite State Automata on infinite words, $\omega$-automata or Büchi automata are discussed in Section 4 of that reader.

Example FSA

finite state automaton Model {
        S [initial] -- a ----> T [final]
        T -- a;b --> S
}

Syntax FSA

Finite State Automata are a graph-like structure and the discussion about syntax of graphs applies. The syntax of a finite state automaton always starts with finite state automaton <name> {. It ends with a closing curly bracket }. The core of the description of the FSA is in the form of transitions S -- a --> T. Transitions are labelled with symbols from the alphabet of the FSA (a in the example). A transition without a symbol represents an $\epsilon$ transition. A transition can be labelled with multiple symbols separated by commas, for example S -- a, b, # --> T. This is a short-hand notation for a transition for each of the symbols. In this case, # can be used for an $\epsilon$ transition.

Symbol names are combinations of letters, numbers and the underscore character _ that start with a letter. Additionally, arbitrary symbol names can be written by enclosing the name with single or double quotation marks.

State names follow the same requirements, but have the additional possibility of using tuple-like (such as (S,T)) or set-like (such as {S,t}) notation. These can also be mixed arbitrarily, recursively, so that ({A,B},{X,Y}) is also a valid state name. It is important to note that in the set-like notation, state names are still compared literally {A,B} is not the same as {B,A}

States can have different annotations to denote if the state is an initial state or a final (accepting) state. For example, S [initial; final] denotes that S is both an initial state and a final state. Annotations are separated by semicolon ;. final can be abbreviated to f. initial can be abbreviated to i and the enclosing square brackets are optional, so S f;i is also valid. Annotations do not need to be repeated every time a state is mentioned, but can be added anywhere.

Generalized acceptance conditions can de specified by adding a label after the final label, e.g., S final[A], T final[A,B] defines two acceptance sets A = {S,T} and B = {T}. If the model has additional final states that are not labelled with a named acceptance set, then those states together also for a separate acceptance set.

Occasionally there is a need to add states to an FSA that have no adjacent transitions. In that case, unconnected states can be added by adding the keyword states at the end of the description followed by a list of states (possibly including annotations), such as in the following model.

finite state automaton A {
    S --> T
    states U V W f;i
}

Operations FSA

The following operations can be performed on FSA models.

  • Convert to DFA - converts a non-deterministic automaton (NFA) to a deterministic automaton (DFA);
  • Eliminate $\epsilon$ - eliminates all $\epsilon$ transitions from an NFA;
  • Make Complete - make the automaton complete, by adding transitions and a state where needed to make sure that every states has an outgoing transition for every symbol of the alphabet; non-deterministic automata (NFA) are first converted to a DFA.
  • Check for Emptiness - analyze if the language defined by the automaton, interpreted as an automaton on finite words is empty or not;
<!-- - *Synchronous Product* - compute the synchronous product between two automata on finite words. Automata are **allowed** to take transitions without synchronization on symbols that are not present in the alphabet of the other automaton; -->
  • Synchronous Product - compute the synchronous product between two automata on finite words. Automata are not allowed to take transitions without synchronization on symbols that are not present in the alphabet of the other automaton;
<!-- - *Synchronous Product NBA* - compute the synchronous product between two automata on infinite words. Automata are **allowed** to take transitions without synchronization on symbols that are not present in the alphabet of the other automaton; -->
  • Complement - compute the complement of the automaton on finite words;
  • Check Word Acceptance - analyze if a given word is accepted by the automaton on finite words;
  • Check Determinism - check if the automaton is deterministic;
  • Determine Alphabet - determine the alphabet of the automaton;
  • Check Language Inclusion - check if the language of one automaton on finite words is included in the language of another automaton;
  • Minimize - minimize an automaton. Tries to find a smaller automaton that accepts the same language. Note that it does not necessarily give the smallest possible automaton that accepts the same language;
  • Relabel States - replace the state labels of an automaton with labels of the form S followed by a number;
  • Convert to Regular Expression - converts the automaton on finite words into a regular expression that defines the same language.

Operations NBA

The following operations can be performed on FSA models that represent Büchi automata.

  • Check for Emptiness NBA - analyze if the language defined by the automaton, interpreted as an automaton on infinite words is empty or not;
  • Synchronous Product NBA - compute the synchronous product between two automata on infinite words. Automata are not allowed to take transitions without synchronization on symbols that are not present in the alphabet of the other automaton;
  • Minimize - minimize an automaton. Tries to find a smaller automaton that accepts the same language. Note that it does not necessarily give the smallest possible automaton that accepts the same language;

Linear Temporal Logic (LTL)

Linear Temporal Logic is discussed in Section 5 of the reader 'Languages'.

Example LTL

ltl formula phi =  G a => X ((not a) U b)

Syntax LTL

The syntax of LTL formulas always starts with ltl formula <name> =. After that, the actual formula is written. LTL formulas are constructed from the following primitives and operators (see the reader):

  • the literals, true and false;
  • atomic propositions, e.g., p;
  • conjunctions, formulas joined by the keyword and, e.g., p and q and r;
  • disjunctions, formulas joined by the keyword or, e.g., p or q and r;
  • the until operator U, e.g., p U q;
  • the release operator R, e.g., p R q;
  • the next operator X, e.g., X p;
  • the eventually operator F, e.g., F p;
  • the always operator G, e.g., G p;
  • the negation operator not, e.g., not X q;
  • the implication =>, e.g., p => X not p.
  • parentheses can be used to group expressions, e.g., p and (q or r); without parentheses, the order of priority of operators is: negation, always, eventually, next, implication (which is right-associative), release, until, conjunction, disjunction.

The formula is optionally followed by a definition of the alphabet considered for the language of the formula, for example:

alphabet {a, b, c}

If no alphabet is explicitly specified, the alphabet is assumed to consist of all symbols mentioned in the definition

Finally, an optional where clauses can be used to introduce the atomic propositions. It uses the syntax

where 
    p = {a, b}
    q = {c}

The example above defines the àtomic proposition p to refer to the set of symbols including a and b. If an atomic proposition p in a formula is not defined in the where clause, it is assumed to be defined as p = {p}.

Operations LTL

In the workbench, LTL formulas support a single operation, convert to FSA, which converts the formula to a non-deterministic finite state automaton that accepts the same language.