This temporal logic is also known as "regular alternation-free mu-calculus". A description of the regular alternation-free mu-calculus can be found in the articles [MS03] and [Mat06], which also describe the verification methods implemented in versions 3.0 and 3.5 of EVALUATOR, respectively.
The regular alternation-free mu-calculus is an extension of the alternation-free fragment of the modal mu-calculus [Koz83, EL86] with action predicates and regular expressions over action sequences. In this setting, action labels are merely handled as character strings.
Note: There exists an extended version of this logic, able to express temporal properties involving data values; see the mcl4 manual page for details. This extended version is supported by evaluator4 but not by evaluator3 .
Regular alternation-free mu-calculus allows direct encodings of "pure" branching-time logics like CTL [CES86] or ACTL [DV90], as well as of regular logics like PDL [FL79] or PDL-delta [Str82]. Moreover, it has an efficient model checking algorithm, with a linear-time complexity in the size of the formula (number of operators) and the size of the LTS model (number of states and transitions). The logic is built from three types of formulas, indicated in the table below.
+--------+-----------------+ | Symbol | Description | +--------+-----------------+ | A | action formula | | R | regular formula | | F | state formula | +--------+-----------------+
The BNF syntax and the informal semantics of these formulas are defined
below. In the grammar, terminal symbols are written between double quotes.
The axiom of the grammar is the
Identifiers are built from letters,
digits, and underscores (beginning with a letter or an underscore). Keywords
must be written in lowercase. Comments are enclosed between '
(*' and '
comments are not allowed. evaluator3 is case-sensitive.
The formulas are interpreted over an LTS <S, A, T, s0>, where: S is the set of states, A is the set of actions (transition labels), T is the transition relation (a subset of S * A * S), and s0 is the initial state. A transition (s1, a, s2) of T, also noted s1-a->s2, indicates that the program from which the LTS has been generated can move from state s1 to state s2 by performing action a.
A ::= string | regexp | "true" | "false" | "not" A | A1 "or" A2 | A1 "and" A2 | A1 "implies" A2 | A1 "equ" A2
A string is a sequence of zero or more characters, enclosed between double
"'), which denotes a label of the LTS. A string may contain any
character but '
\n' (end-of-line). Double quotes are also allowed, if preceded
by a backslash ('
\'). Strings can be concatenated using the binary operator
string ::= "(any char but end-of-line)*" | string1 "#" string2
A transition label of the LTS satisfies a string iff it is identical to the corresponding character string (obtained after concatenation whenever needed).
A regexp is a UNIX regular expression (see the regexp
page for a detailed description of UNIX regular expressions), enclosed
between single quotes ('
''), which denotes a predicate on the labels of the
LTS. Regexp's can be concatenated using the binary operator '
#'. Strings can
be concatenated to regexp's, in which case they are implicitly converted
regexp ::= 'UNIX_regular_expression' | regexp1 "#" regexp2 | string1 "#" regexp2 | regexp1 "#" string2
A label of the LTS satisfies a regexp if it matches the corresponding UNIX_regular_expression (obtained after concatenation whenever needed).
Syntactically, all binary operators on action formulas are left-associative.
not" operator has the highest precedence, followed by "
or", followed by "
implies", followed by "
The boolean operators
have the usual semantics: a label of the LTS always satisfies "
never satisfies "
false"; it satisfies "
not A" iff it does not satisfy A;
it satisfies "A1
or A2" iff it satisfies A1 or it satisfies A2; it satisfies
and A2" iff it satisfies both A1 and A2; it satisfies "A1
iff it does not satisfy A1 or it satisfies A2; it satisfies "A1
iff either it satisfies both A1 and A2, or none of them.
R ::= A | "nil" | R1 "." R2 | R1 "|" R2 | R "?" | R "*" | R "+"
nil" is the empty operator, "
." is the concatenation operator,
|" is the choice operator, "
?" is the option operator, "
*" is the transitive
and reflexive closure operator, and "
+" is the transitive closure operator.
Syntactically, all binary operators on regular formulas are left-associative.
*", and "
+" operators have the highest precedence, followed by
.", followed by "
Note: In early versions of evaluator3
, the "
operator had a higher precedence than "
.". To ensure that "old" MCL version
3 regular formulas are interpreted by the current version of evaluator3
according to their original intended meaning, it is recommended to add
parentheses at appropriate places. For example, an "old" MCL version 3 regular
formula "R1 | R2 . R3" should be rewritten as "(R1 | R2) . R3" to maintain
its original meaning, otherwise the current version of evaluator3
would parse it as "R1 | (R2 . R3)".
A regular formula R denotes a sequence of (consecutive) LTS transitions such that the word obtained by concatenating their labels belongs to the regular language defined by R.
The regular operators
have the following semantics: a sequence of LTS transitions satisfies A
iff it has the form s1-a->s2, where the label a satisfies the formula A; it
nil" iff it is empty (i.e., it contains no transition); it satisfies
. R2" iff it is the concatenation of two sequences satisfying R1 and
R2, respectively; it satisfies "R1
| R2" iff it satisfies R1 or it satisfies
R2; it satisfies "R
?" iff it is either empty, or it satisfies R; it satisfies
*" iff it is the concatenation of zero or more sequences satisfying
R; it satisfies "R
+" iff it is the concatenation of one or more sequences
F ::= "true" | "false" | "not" F | F1 "or" F2 | F1 "and" F2 | F1 "implies" F2 | F1 "equ" F2 | "<" R ">" F | "[" R "]" F | "<" R ">" "@" | "[" R "]" "-|" | X | "mu" X "." F | "nu" X "." F
> F" and "
] F" are the possibility and necessity modal operators,
@" is the infinite looping operator, "
-|" is the saturation operator,
. F" and "
. F" are the minimal and maximal fixed point operators,
and X is a propositional variable.
Syntactically, all binary operators on
state formulas are left-associative. The "
< >", "
[ ]", "
mu", and "
operators have the highest precedence, followed by "
and", followed by "
followed by "
implies", followed by "
equ". The fixed point operators act
as binders for the variables X in a way similar to quantifiers in first-order
logic. In each meaningful "
. F" or "
. F" formula, X is supposed
to have free occurrences inside F. State formulas are assumed to be syntactically
monotonic (i.e., in each fixed point formula "
. F" or "
. F", free
occurrences of X in F may appear only under an even number of negations
and/or left-hand sides of implications) and alternation-free (i.e., without
mutually recursive minimal and maximal fixed point variables).
operators have the usual semantics: a state of the LTS always satisfies
true"; it never satisfies "
false"; it satisfies "
not F" iff it does not
satisfy F; it satisfies "F1
or F2" iff it satisfies F1 or it satisfies
F2; it satisfies "F1
and F2" iff it satisfies both F1 and F2; it satisfies
implies F2" iff it does not satisfy F1 or it satisfies F2; it satisfies
equ F2" iff either it satisfies both F1 and F2, or none of them.
modal operators have the following semantics: a state of the LTS satisfies
> F" iff there is (at least) one transition sequence starting at the
state, satisfying R, and leading to a state satisfying F; it satisfies
] F" iff all transition sequences starting at the state and satisfying
R are leading to states satisfying F.
The infinite looping and saturation
operators have the following semantics: a state of the LTS satisfies "
@" iff there is a transition sequence starting at the state and consisting
of an infinite concatenation of sequences satisfying R; it satisfies "
-|" iff all transition sequences starting at the state and consisting
of a concatenation of sequences satisfying R are finite.
The fixed point
operators have the following semantics: a state satisfies "
. F" iff
it belongs to the minimal solution of the fixed point equation X = F (X),
and it satisfies "
. F" iff it belongs to the maximal solution of the
same equation, where the propositional variable X denotes a set of LTS
states. Intuitively, minimal (resp. maximal) fixed point operators allow
to characterize finite (resp. infinite) tree-like patterns in the LTS.
An LTS satisfies a state formula F iff its initial state s0 satisfies F.
implies", and "
equ" can be expressed in terms of "
or", and "
not" in the usual way. The diamond and box modalities are dual:
[ R ] F = not < R > not FThe same holds for minimal and maximal fixed point operators:
nu X . F = not mu X . not F (not X)where F
)denotes the syntactic substitution of X by
notX in F. The saturation operator is the negation of the infinite looping operator:
[ R ] -| = not < R > @The modalities containing regular formulas can be translated in terms of boolean operators, fixed point operators, and modalities containing only action formulas, by recursively applying the identities below:
< nil > F = < false* > F < R1 . R2 > F = < R1 > < R2 > F < R1 | R2 > F = < R1 > F or < R2 > F < R? > F = < nil | R > F < R* > F = mu X . (F or < R > X) < R+ > F = < R . R* > Fwhere X is a "fresh" propositional variable (the corresponding identities for box modalities are obtained by duality). The infinite looping operator is equivalent to the maximal fixed point formula below:
< R > @ = nu X . < R > Xwhere X is a "fresh" propositional variable.
@" were not allowed to contain "
*" or "
+" operators in their regular formulas R. The current version of evaluator3 accepts regular formulas with "
*" or "
+" in infinite looping operators, which are now able to characterize complex cycles in the LTS (e.g., generalized Buchi accepting cycles). An example of formula accepted by the current version of evaluator3 but not expressible in alternation-free mu-calculus is the following:
< true* . "A" > @This formula is equivalent (by applying the identities above) to a fixed point formula of alternation depth 2:
nu X . mu Y . (< "A" > X or < true > Y)Although the mu-calculus fragment of alternation depth 2 has in general a quadratic-time model checking complexity in the size of the LTS, the alternation depth 2 formulas resulting from the translation of infinite looping operators "
@" containing "
*" or "
+" operators in their regular formulas R have a linear-time model checking complexity in the size of the LTS [MT08].
)" for the infinite looping operator. This syntax is now obsolete, but still accepted by evaluator3 for backward compatibility. It is recommended to use the new syntax "
@", which is closer to the syntax of possibility modalities and reflects more intuitively the existence of an infinite sequence, terminated by a loop ("@") in a finite state LTS.
A fixed point formula "
. F" or
. F" is unguarded [Koz83] if F contains at least one free occurrence
of X which is not preceded (not necessarily immediately) by a modality.
The evaluation of an unguarded formula on an LTS may yield a BES with cyclic
dependencies between variables even if the LTS is acyclic.
A state formula containing regular modalities with nested star operators may yield after translation an unguarded mu-calculus formula. For example, in the following formula:
< A1** . A2 > true = mu X1 . (< A2 > true or mu X2 . (X1 or < A1 > X2)the free occurrence of X1 is not preceded by any modality, and hence the formula is unguarded. Unguarded occurrences of propositional variables can always be eliminated from a mu-calculus formula, at the price of an increase in size [Koz83,Mat02].
[ true* . "OPEN !1" . (not "CLOSE !1")* . "OPEN !2" ] false
which states that every time process 1 enters its critical section (action "OPEN !1"), it is impossible that process 2 also enters its critical section (action "OPEN !2") before process 1 has left its critical section (action "CLOSE !1").
Other typical safety properties are the invariants, expressing that every state of the LTS satisfies some "good" property. For example, deadlock freedom can be expressed by the formula below:
[ true* ] < true > true
stating that every state has at least one successor. Alternately, this formula may be expressed directly using a fixed point operator:
nu X . (< true > true and [ true ] X)
but less concisely than by using a regular formula.
Potentiality assertions can be directly expressed using diamond modalities containing regular formulas. For instance, the following formula:
< true* . "GET !0" > true
states that there exists a sequence leading to a "GET !0" action after performing zero or more transitions. Regular formulas allow to express succinctly complex potentiality assertions, such as the formula below:
< true* . "SEND" . (true* . "ERROR")* . true* . "RECV" > true
stating that there exists a sequence leading (after zero or more transitions) to a "SEND" action, possibly followed by a sequence of "ERROR" actions (possibly separated by other actions) and leading (after zero or more transitions) to a "RECV" action.
Inevitability assertions can be expressed using fixed point operators. For instance, the following formula:
mu X . (< true > true and [ not "START" ] X)
states that all transition sequences starting at the current state lead to "START" actions after a finite number of steps.
[ true* . "SEND" . (not "RECV")* ] < (not "RECV")* . "RECV" > true
Intuitively, the formula above considers the sequences following the "SEND" action by "skipping" the cycles of the LTS that do not contain "RECV" actions: it states that from every state of such a cycle, there is still a finite sequence leading to a "RECV" action.
< true* . 'SEND !1.*' and not 'SEND !1.*!2' > true
states the potential reachability of an action having the gate SEND and the value of the first offer equal to 1, possibly followed by other offers with values different from 2. Moreover, action formulas combined with modalities allow to express invariants over actions (i.e., action formulas that must be satisfied by all transition labels of the LTS). For instance, the following formula:
[ true* . not ('RECV !.* !.*' and 'RECV !\(.*\) !\1') ] false
states that all message receptions (actions "RECV !source !dest") have
different source and destination fields. The UNIX regular expression construct
\( \)' enables to match a portion of a string and to re-use it later in the
"macro" M "(" P1"," ..."," Pn ")" "=" <text> "end_macro"
The above construct defines a macro M having the parameters P1, ..., Pn and the body <text>, which is a string of alpha-numeric characters (normally) containing occurrences of the parameters P1, ..., Pn. For example, the following macro-definition:
macro EU_A (F1, A, F2) = mu X . ((F2) or ((F1) and < A > X)) end_macro
encodes the "Exists Until" operator of ACTL, which states that there exists a sequence of transitions leading to a state satisfying F2 such that all intermediate states satisfy F1 and all intermediate labels satisfy A.
The calls of a macro M have the following form:
M "(" <text1>"," ..."," <textn> ")"
where the arguments <text1>, ..., <textn> are strings. The result of the call is the body <text> of the macro M in which all occurrences of the parameters Pi have been syntactically substituted with the arguments <texti>, for all i between 1 and n. For example, the following call:
EU_A (true, not "SEND", < "RECV" > true)
expands into the formula below:
mu X . ((< "RECV" > true) or ((true) and < not "SEND" > X))
A macro is visible from the point of its definition until the end of the program. The macros may be overloaded: several macros with the same name, but different arities, may be defined in the same scope.
Various macro-definitions (typically encoding the operators of some particular temporal logic) can be grouped into files called libraries. These files may be included in the source program using the following command:
"library" <file0.mcl>"," ..."," <filen.mcl> "end_library"
At the compilation of the program, the above construct is syntactically replaced with the contents of the files <file0.mcl>, ..., <filen.mcl>, placed one after the other in this order. For example, the following command:
library actl.mcl end_library
is syntactically replaced with the content of the file actl.mcl, which implements the ACTL operators.
The included files are searched first in the current directory, then in the directory referenced by $CADP/src/xtl. Multiple inclusions of the same file are silently discarded.
Additional information is available from the CADP Web page located at http://cadp.inria.fr
Directives for installation are given in files $CADP/INSTALLATION_*.
Recent changes and improvements to this software are reported and commented in file $CADP/HISTORY.Radu.Mateescu@inria.fr