- 1 Summary of B Syntax
- 1.1 Logical predicates:
- 1.2 Equality:
- 1.3 Booleans:
- 1.4 Sets:
- 1.5 Numbers:
- 1.6 Relations:
- 1.7 Functions:
- 1.8 Sequences:
- 1.9 Records:
- 1.10 Strings:
- 1.11 Trees:
- 1.12 LET and IF-THEN-ELSE
- 1.13 Statements (aka Substitutions):
- 1.14 Machine header:
- 1.15 Machine sections:
- 1.16 Machine inclusion:
- 1.17 Definitions:
- 1.18 Comments and Pragmas
- 1.19 Differences with AtelierB/B4Free
- 1.20 Other notes

- 2 Feedback

P & Q conjunction P or Q disjunction P => Q implication P <=> Q equivalence not P negation !(x).(P=>Q) universal quantification #(x).(P&Q) existential quantification

Above, `P` and `Q` stand for predicates. Inside the universal quantification, `P` must give a value type to the quantified variable.
Note: you can also introduce multiple variables inside a universal or existential quantification, e.g., `!(x,y).(P => Q)`.

E = F equality E /= F disequality

TRUE FALSE BOOL set of boolean values ({TRUE,FALSE}) bool(P) convert predicate into BOOL value

Warning: `TRUE` and `FALSE` are values and *not* predicates in B and cannot be combined using logical connectives.
To combine two boolean values `x` and `y` using conjunction you have to write `x=TRUE & y=TRUE`.
To convert a predicate such as `z>0` into a boolean value you have to use `bool(z>0)`.

{} empty set {E} singleton set {E,F} set enumeration {x|P} comprehension set POW(S) power set POW1(S) set of non-empty subsets FIN(S) set of all finite subsets FIN1(S) set of all non-empty finite subsets card(S) cardinality S*T cartesian product S\/T set union S/\T set intersection S-T set difference E:S element of E/:S not element of S<:T subset of S/<:T not subset of S<<:T strict subset of S/<<:T not strict subset of union(S) generalised union over sets of sets inter(S) generalised intersection over sets of sets UNION(z).(P|E) generalised union with predicate INTER(z).(P|E) generalised intersection with predicate

INTEGER set of integers NATURAL set of natural numbers NATURAL1 set of non-zero natural numbers INT set of implementable integers (MININT..MAXINT) NAT set of implementable natural numbers NAT1 set of non-zero implementable natural numbers n..m set of numbers from n to m MININT the minimum implementable integer MAXINT the maximum implementable integer m>n greater than m<n less than m>=n greater than or equal m<=n less than or equal max(S) maximum of a set of numbers min(S) minimum of a set of numbers m+n addition m-n difference m*n multiplication m/n division m**n power m mod n remainder of division PI(z).(P|E) Set product SIGMA(z).(P|E) Set summation succ(n) successor (n+1) pred(n) predecessor (n-1)

S<->T relation E|->F maplet dom(r) domain of relation ran(r) range of relation id(S) identity relation S<|r domain restriction S<<|r domain subtraction r|>S range restriction r|>>S range subtraction r~ inverse of relation r[S] relational image r1<+r2 relational overriding (r2 overrides r1) r1><r2 direct product {x,(y,z) | x,y:r1 & x,z:r2} (r1;r2) relational composition {x,y| x|->z:r1 & z|->y:r2} (r1||r2) parallel product {((x,v),(y,w)) | x,y:r1 & v,w:r2} prj1(S,T) projection function (usage prj1(Dom,Ran)(Pair)) prj2(S,T) projection function (usage prj2(Dom,Ran)(Pair)) closure1(r) transitive closure closure(r) reflexive & transitive closure (non-standard version: closure({}) = {}; see iterate(r,0) below) iterate(r,n) iteration of r with n>=0 (Note: iterate(r,0) = id(s) where s = dom(r)\/ran(r)) fnc(r) translate relation A<->B into function A+->POW(B) rel(r) translate relation A<->POW(B) into relation A<->B

S+->T partial function S-->T total function S+->>T partial surjection S-->>T total surjection S>+>T partial injection S>->T total injection S>+>>T partial bijection S>->>T total bijection %x.(P|E) lambda abstraction f(E) function application f(E1,...,En) is now supported (as well as f(E1|->E2))

<> or [] empty sequence [E] singleton sequence [E,F] constructed sequence seq(S) set of sequences over Sequence seq1(S) set of non-empty sequences over S iseq(S) set of injective sequences iseq1(S) set of non-empty injective sequences perm(S) set of bijective sequences (permutations) size(s) size of sequence s^t concatenation E->s prepend element s<-E append element rev(s) reverse of sequence first(s) first element last(s) last element front(s) front of sequence (all but last element) tail(s) tail of sequence (all but first element) conc(S) concatenation of sequence of sequences s/|\n take first n elements of sequence s\|/n drop first n elements from sequence

struct(ID:S,...,ID:S) set of records with given fields and field types rec(ID:E,...,ID:E) construct a record with given field names and values E'ID get value of field with name ID

"astring" a specific string value STRING the set of all strings Note: for the moment enumeration of strings is limited (if a variable of type STRING is not given a value by the machine, then ProB assumes STRING = { "STR1", "STR2" })

Atelier-B does not support any operations on strings, apart from equality and disequality. However, the ProB external function library contains several operators on strings.

Nodes in the tree are denoted by index sequences (branches), e.g, n=[1,2,1] Each node in the tree is labelled with an element from a domain S A tree is a function mapping of branches to elements of the domain S.

tree(S) set of trees over domain S btree(S) set of binary trees over domain S top(t) top of a tree const(E,s) construct a tree from info E and sequence of subtrees s rank(t,n) rank of the node at end of branch n in the tree t father(t,n) father of the node denoted by branch n in the tree t son(t,n,i) the ith son of the node denoted by branch n in tree t sons(t) the sequence of sons of the root of the tree t subtree(t,n) arity(t,n) bin(E) construct a binary tree with a single node E bin(tl,E,tr) construct a binary tree with root info E and subtrees tl,tr left(t) the left (first) son of the root of the binary tree t right(t) the right (last) son of the root of the binary tree t sizet(t) the size of the tree (number of nodes) prefix(t) the nodes of the tree t in prefix order postfix(t) the nodes of the tree t in prefix order mirror, infix are recognised by the parser but not yet supported by ProB itself

ProB allows the following for predicates and expressions:

IF P THEN E1 ELSE E2 END conditional for expressions or predicates E1,E2 LET x1,... BE x1=E1 & ... IN E END

Note: the expressions E1,... defining x1,... are not allowed to use x1,...

skip no operation x := E assignment f(x) := E functional override x :: S choice from set x : (P) choice by predicate P (constraining x) x <-- OP(x) call operation and assign return value G||H parallel substitution** G;H sequential composition** ANY x,... WHERE P THEN G END non deterministic choice LET x,... BE x=E & ... IN G END VAR x,... IN G END generate local variables PRE P THEN G END ASSERT P THEN G END CHOICE G OR H END IF P THEN G END IF P THEN G ELSE H END IF P1 THEN G1 ELSIF P2 THEN G2 ... END IF P1 THEN G1 ELSIF P2 THEN G2 ... ELSE Gn END SELECT P THEN G WHEN ... WHEN Q THEN H END SELECT P THEN G WHEN ... WHEN Q THEN H ELSE I END CASE E OF EITHER m THEN G OR n THEN H ... END END CASE E OF EITHER m THEN G OR n THEN H ... ELSE I END END WHEN P THEN G END is a synonym for SELECT P THEN G END **: cannot be used at the top-level of an operation, but needs to be wrapped inside a BEGIN END or another statement (to avoid problems with the operators ; and ||).

MACHINE or REFINEMENT or IMPLEMENTATION Note: machine parameters can either be SETS (if identifier is all upper-case) or scalars (i.e., integer, boolean or SET element; if identifier is not all upper-case; typing must be provided be CONSTRAINTS) You can also use MODEL or SYSTEM as a synonym for MACHINE, as well as EVENTS as a synonym for OPERATIONS.

CONSTRAINTS P (logical predicate) SETS S;T={e1,e2,...};... CONSTANTS x,y,... CONCRETE_CONSTANTS cx,cy,... PROPERTIES P (logical predicate) DEFINITIONS m(x,...) == BODY;.... VARIABLES x,y,... CONCRETE_VARIABLES cv,cw,... INVARIANT P (logical predicate) ASSERTIONS P;...;P (list of logical predicates separated by ;) INITIALISATION OPERATIONS

USES list of machines INCLUDES list of machines SEES list of machines EXTENDS list of machines PROMOTES list of operations REFINES machine CSP_CONTROLLER controller will use controller.csp to guide machine (currently disabled in 1.3) Note: Refinement machines should express the operation preconditions in terms of their own variables.

NAME1 == Expression ; NAME2(ID,...,ID) == E2 ... Note: we now support definitions with arguments. There are a few Definitions which can be used to influence the animator: GOAL == P to define a custom Goal predicate for Model Checking (the Goal is also set by using "Advanced Find...") SCOPE == P to limit the search space to "interesting" nodes scope_SETNAME == n..n to define custom cardinality for set SETNAME scope_SETNAME == n equivalent to 1..n SET_PREF_MININT == n SET_PREF_MAXINT == n SET_PREF_MAX_INITIALISATIONS == n max. number of intialisations computed SET_PREF_MAX_OPERATIONS == n max. number of enablings per operation computed SET_PREF_SYMBOLIC == TRUE/FALSE ASSERT_LTL... == "LTL Formula" using X,F,G,U,R LTL operators + Y,O,H,S Past-LTL operators + atomic propositions: e(OpName), [OpName], {BPredicate} ANIMATION_FUNCTION == e a function (INT*INT) +-> INT or an INT ANIMATION_FUNCTION_DEFAULT == e a function (INT*INT) +-> INT or an INT instead of any INT above you can also use BOOL or any SET ANIMATION_IMGn == "PATH to .gif" a path to a gif file ANIMATION_STRn == "sometext" a string without spaces

B supports two styles of comments: /* ... */ block comments // ... line comments

ProB recognises several pragma comments of the form /*@ PRAGMA VALUE */ The whitespace between @ and PRAGMA is optional. /*@label LBL */ associates a label LBL with the following predicate /*@desc DESC */ associates a description DESC with the preceding predicate /*@file PATH */ associates a file for machines in SEES,...

Basically, ProB tries to be compatible with Atelier B and conforms to the semantics of Abrial's B-Book and of Atelier B's reference manual (http://www.atelierb.eu/php/documents-en.php#manuel-reference). Here are the main differences with Atelier B: - tuples without parentheses are not supported; write (a,b,c) instead of a,b,c - relational composition has to be wrapped into parentheses; write (f;g) - parallel product also has to be wrapped into parentheses; write (f||g) - trees are not yet fully supported - the VALUES clause is only partially supported - definitions have to be syntactically correct and be either an expression, predicate or substitution; the arguments to definitions have to be expressions; definitions which are predicates or substitutions must be declared before first use - definitions are local to a machine - for ProB the order of fields in a record is not relevant (internally the fields are sorted), Atelier-B reports a type error if the order of the name of the fields changes - well-definedness: for disjunctions and implications ProB uses the L-system of well-definedness (i.e., for P => Q, P should be well-defined and if P is true then Q should also be well-defined) - ProB allows WHILE loops and sequential composition in abstract machines - ProB now allows the IF-THEN-ELSE and LET for expressions and predicates (e.g., IF x<0 THEN -x ELSE x END or LET x BE x=f(y) IN x+x END) - ProB's type inference is much stronger than Atelier-B's, much less typing predicates are required - ProB allows identifiers consisting of a single character - ProB allows multi-line strings and supports UTF-8 characters in strings - ProB allows a she-bang line in machine files starting with #! (If you discover more differences, please let us know!) See also our Wiki for documentation: http://www.stups.hhu.de/ProB/index.php5/Current_Limitations http://www.stups.hhu.de/ProB/index.php5/Using_ProB_with_Atelier_B Also note that there are various differences between BToolkit and AtelierB/ProB: - AtelierB/ProB do not allow true as predicate; e.g., PRE true THEN ... END is not allowed (use BEGIN ... END instead) - AtelierB/ProB do not allow a machine parameter to be used in the PROPERTIES - AtelierB/ProB require a scalar machine parameter to be typed in the CONSTRAINTS clause - In AtelierB/ProB the BOOL type is pre-defined and cannot be redefined

ProB is best at treating universally quantified formulas of the form !x.(x:SET => RHS), or !(x,y).(x|->y:SET =>RHS), !(x,y,z).(x|->y|->z:SET =>RHS), ...; otherwise the treatment of !(x1,...,xn).(LHS => RHS) may delay until all values treated by LHS are known. Similarly, expressions of the form SIGMA(x).(x:SET|Expr) and PI(x).(x:SET|Expr) lead to better constraint propagation. The construction S:FIN(S) is recognised by ProB as equivalent to the Event-B finite(S) operator. ProB assumes that machines and STRING values are encoded using UTF-8.

Please help us to improve this documentation by providing feedback in our bug tracker, asking questions in our prob-users group or sending an email to Michael Leuschel.