Summary of B Syntax


Summary of B Syntax

Logical predicates:

 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).

Equality:

 E = F      equality
 E /= F     disequality

Booleans:

 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).

Sets:

 {}          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

Numbers:

 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)
 0xH         hexadecimal literal, where H is a sequence of letters in [0-9A-Fa-f]

Relations:

 S<->T     relation
 S<<->T    total relation
 S<->>T    surjective relation
 S<<->>T   total surjective 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

Functions:

  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))

Sequences:

  <> 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
 

Records:

  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

Strings:

  "astring"     a specific (single-line) string value
  '''astring''' an alternate way of writing (multi-line) strings, no need to escape "
  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. ProB also allows multi-line strings. As of version 1.7.0, ProB will support the following escape sequences within strings:

\n   newline (ASCII character 13)
\r   carriage return (ASCII 10)
\t  tab (ASCII 9)
\"   the double quote symbol "
\'   the single quote symbol '
\\   the backslash symbol

Within single-line string literals, you do not need to escape '. Within multi-line string literals, you do not need to escape " and you can use tabs and newlines. ProB assumes that all B machines and strings use the UTF-8 encoding.

The library LibraryStrings.def in stdlib contains additional useful external functions (like TO_STRING, STRING_SPLIT, FORMAT_TO_STRING, INT_TO_HEX_STRING, ...). Some of the sequence operators work also on strings:

  size(s)     the length of a string s
  rev(s)      the reverse a string s
  s ^ t       the concatenation of two strings
  conc(ss)    the concatenation of a sequence of strings

You can turn this support off using the STRING_AS_SEQUENCE preference.

Reals:

 REAL        set of reals
 FLOAT       set of floating point numbers
 i.f         real literal, where i and f are sequences of digits
 real(n)     convert an integer n into a real number
 floor(r)    convert a real r to an integer
 ceiling(r)  convert a real r to an integer

Standard arithmetic operators can be applied to reals: +, - , *, /, SIGMA, PI. Exponentiation of a real with an integer is also allowed. The comparison predicates =, /=, <, >, <=, >= also all work. Support for reals and floats is experimental. The definition in Atelier-B is also not stable yet. Currently ProB supports floating point numbers only. Warning: properties such as associativity and commutativity of arithmetic operators thus does not hold. The library LibraryReals.def in stdlib contains additional useful external functions (like RSIN, RCOS, RLOG, RSQRT, RPOW, ...). You can turn off support for REALS using the preference ALLOW_REALS.

Trees:

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

LET and IF-THEN-ELSE

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,...

Statements (aka Substitutions):

  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 header:

  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.
  ProB also supports the ref keyword of Atelier-B for event refinement.

Machine sections:

  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

Machine inclusion:

  USES list of machines
  INCLUDES list of machines
  SEES list of machines
  EXTENDS list of machines
  PROMOTES list of operations
  REFINES machine

Note: Refinement machines should express the operation preconditions in terms of their own variables.

Definitions:

  NAME1 == Expression;          Definition without arguments
  NAME2(ID,...,ID) == E2;       Definition with arguments
 "FILE.def";                   Include definitions from file 

There are a few Definitions which can be used to influence the animator:

There are a few specific definitions which can be used to influence ProB:
  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
  SET_PREF_TIME_OUT == n             time out for operation computation in ms
  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}

The following definitions allow providing a custom state visualization:

  ANIMATION_FUNCTIONn == 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
                    as a result you can also use STRING values,
                    or even other values which are pretty printed
  ANIMATION_IMGn == "PATH to .gif"   a path to a gif file
  ANIMATION_STRn == "sometext"       a string without spaces;
                                     the result integer n will be rendered as a string
  ANIMATION_STR_JUSTIFY_LEFT == TRUE computes the longest string in the outputs and pads
                                     the other strings accordingly
  SET_PREF_TK_CUSTOM_STATE_VIEW_PADDING == n          additional padding between images in pixels
  SET_PREF_TK_CUSTOM_STATE_VIEW_STRING_PADDING == n   additional padding between text in pixels

The following definitions allow providing a custom state graph:

  CUSTOM_GRAPH_NODESn == e    define a set of nodes to be shown,
                              nodes can also be pairs (node,colour) or triples (node,shape,colour)
                              colours are strings of valid Dot/Tk colors (e.g., "maroon" or "red")
                              shapes are strings of valid Dot shapes (e.g., "rect" or "hexagon")
  CUSTOM_GRAPH_EDGESn == e    define a relation to be shown
                              edges can either be pairs (node1,node2) or triples (node1,label,node2)
                              where label is either a Dot/Tk color or a string or value representing
                              the label to be used for the edges

In both cases e can also be a record which defines default dot attributes like color, shape, style and description, e.g., rec(color:"red", style:"dotted", edges:Expression).

Comments and Pragmas

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.

  /*@symbolic */      put before comprehension set or lambda to instruct ProB
                      to keep it symbolic and not try to compute it explicitly
  /*@label LBL */     associates a label LBL with the following predicate
                      (LBL must be identifier or a string "....")
  /*@desc DESC */     associates a description DESC with the preceding predicate or
                      introduced identifier (in VARIABLES, CONSTANTS,... section)
                      There are two special descriptions
                      /*@desc memo*/ to be put after identifiers in the ABSTRACT_CONSTANTS section
                                     indicating that these functions should be memoized
                      /*@desc prob-ignore */ to be put after predicates (e.g., in PROPERTIES) which
                                             should be ignored by ProB
                                             when the preference USE_IGNORE_PRAGMAS is TRUE
  /*@file PATH */     associates a file for machines in SEES, INCLUDES, ...
                      put pragma after a seen or included machine
  /*@package NAME */  at start of machine, machine file should be in folder NAME/...
                      NAME can be qualified N1.N2...Nk, in which case the machine
                      file should be in N1/N2/.../Nk
  /*@import-package NAME */  adds ../NAME to search paths for SEES,...
                      NAME can also be qualified N1.N2...Nk, use after package pragma
  /*@generated */     can be put at the top of a machine file; indicates the machine
                      is generated from some other source and should not be edited

File Extensions

   .mch   for abstract machine files
   .ref   for refinement machines
   .imp   for implementation machines
   .def   for DEFINITIONS files
   .rmch  for Rules machines for data validation

Differences with AtelierB/B4Free

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. 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 to use the Event-B relation operators <<->, <->>, <<->>
  - ProB allows multi-line strings and supports UTF-8 characters in strings,
    and ProB allows string literals written using three apostrophes ('''string''')
  - 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:

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

Other notes

 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.

Feedback

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.