Summary of B Syntax: Difference between revisions

Latest revision as of 13:07, 13 June 2024

Summary of B Syntax

Below we describe the "classical" B syntax as supported by ProB. You may also wish to consult

The B summary by Ken Robinson (File:B-summary.pdf)
The Atelier-B reference manual (b-language-reference-manual.pdf)

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
 btrue        truth (this is a predicate)
 bfalse       falsity (this is a predicate)

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     truth value (this is an expression)
 FALSE    falsity value (this is an expression)
 BOOL     set of boolean values ({TRUE,FALSE})
 bool(P)  convert predicate into BOOL value

Warning: TRUE and FALSE are expression 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
 {(x).P|E}       Event-B style comprehension set (brackets needed)
 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 or 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

Integers

 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 (all pairs (x,(y,z)) with x,y:r1 and x,z:r2)
 (r1;r2)       relational composition {x,y| x|->z:r1 & z|->y:r2}
 (r1||r2)      parallel product (all pairs ((x,v),(y,w)) with x,y:r1 and v,w:r2)
 prj1(S,T)     projection function (usage prj1(Dom,Ran)(Pair))
 prj2(S,T)     projection function (usage prj2(Dom,Ran)(Pair))
               prj1(Pair) and prj2(Pair) are also allowed
 fnc(r)        translate relation A<->B into function A+->POW(B)
 rel(r)        translate relation A<->POW(B) into relation A<->B
 closure1(r)   transitive closure
 closure(r)    reflexive & transitive closure
               (equal to id(TYPEOF_r) \/ closure1(r))
 iterate(r,n)  iteration of r with n>=0
               (Note: iterate(r,0)=id(s) where s=TYPEOF_r)

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 also supported (as well as f(E1|->E2...|->En))

Sequences

 [] or <>  empty sequence
 [E]       singleton sequence
 [E,F]     constructed sequence
 seq(S)    set of sequences over S
 seq1(S)   set of non-empty sequences over S
 iseq(S)   set of injective sequences over S
 iseq1(S)  set of non-empty injective sequences over S
 perm(S)   set of bijective sequences (permutations) over S
 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

Identifiers

 ID    must start with letter (ASCII or Unicode), can then contain
       letters (ASCII or Unicode), digits and underscore (_) and
       can end with Unicode subscripts followed by Unicode primes
 M.ID  composed identifier for identifier coming from included machine M
 `ID`  an identifier in backquotes can contain almost any character (except newline)

Strings

 "astring"      a specific (single-line) string value
 '''astring'''  an alternate way of writing (multi-line) strings, no need to escape "
 ```tstring```  template strings, where ${Expr} parts are evaluated and converted to string,
                you can provide options separated by commas in square brackets like $[2f]{Expr}.
                Valid options are: Nf (for floats/reals), Nd (for integer), Np (padding),
                ascii (can be abbreviated to a), unicode (can be abbreviated to u).
 STRING         the set of all strings

Atelier-B does not support any operations on strings, apart from equality and disequality. In ProB, however, some of the sequence operators work also on strings:

 size(s)   the length of a string s
 rev(s)    the reverse of 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. The library LibraryStrings.def in stdlib contains additional useful external functions (like TO_STRING, STRING_SPLIT, FORMAT_TO_STRING, INT_TO_HEX_STRING, ...).

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.

Reals

 REAL        set of reals
 FLOAT       set of floating point numbers
 i.f         real literal in decimal notation, where i and f are natural numbers
 i.fEg       real literal in scientific notation, where i,f are natural numbers and g is an integer
 real(n)     convert an integer n into a real number
 floor(r)    convert a real r into an integer
 ceiling(r)  convert a real r into an integer

One can also use a lowercase e for literals in scientific notation (e.g. 1.0e-10). 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 do 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. The REAL_SOLVER preference how constraints are solved.

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, expressions and substitutions:

 IF P THEN E1 END                    conditional branching
 IF P THEN E1 ELSIF E2 END           we also allow multiple ELSIF branches
 IF P THEN E1 ELSE E2 END            but you always need an ELSE branch for expressions and predicates
 IF P THEN E1 ELSIF E2 ELSE E3 END
 LET x1,... BE x1=E1 & ... IN E END  introduce local variables

Note: the expression Ei defining xi is allowed to use x1,...,x(i-1) for predicates/expressions. By setting the preference ALLOW_COMPLEX_LETS to TRUE, this is also allowed for substitutions.

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; previous value of x is x$0)
 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
 WHILE P1 DO G INVARIANT P2 VARIANT E 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 confusion with the operators ; and || on relations).

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,...};...
 FREETYPES           x=x1,x2(arg2),...;...
 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      S                     (substitution)
 OPERATIONS          O;...                 (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 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
 MAX_OPERATIONS_OPNAME == n         max. number of enablings for the operation OPNAME
 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}
 HEURISTIC_FUNCTION == n            in directed model-checking mode nodes with smalles value will be processed first

The following definitions allow providing a custom state visualization (n can be empty or a number):

 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 (n can be empty or a number):

 CUSTOM_GRAPH_NODESn == e  define a set of nodes to be shown,
                           nodes can also be pairs (Node,Colour), triples (Node,Shape,Colour) or
                           records or sets of records like
                           rec(color:Colour, shape:Shape, style:Style, label:Label, value:Node, ...)
                           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"), and
                           Styles are valid Dot shape styles (e.g., "rounded" or "solid" or "dashed")
 CUSTOM_GRAPH_EDGESn == e  define a relation to be shown as a graph
                           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.:

 CUSTOM_GRAPH_NODES == rec(color:"blue", shape:"rect", style:"filled", nodes:e);
 CUSTOM_GRAPH_EDGES == rec(color:"red", style:"dotted", dir:"none", penwidth:2, edges:e)

Alternatively, the complete graph can be put into one definition using CUSTOM_GRAPH. You have to define a single CUSTOM_GRAPH definition of a record with global graph attributes

  (like rankdir or layout) and optionally with edges and nodes attributes (replacing
   CUSTOM_GRAPH_EDGES and CUSTOM_GRAPH_NODES respectively), e.g.:

    CUSTOM_GRAPH == rec(layout:"circo", nodes:mynodes, edges:myedges)

You can now also use a single CUSTOM_GRAPH definition of a record with global graph attributes (like rankdir or layout) and optionally with edges and nodes attributes (replacing CUSTOM_GRAPH_EDGES and CUSTOM_GRAPH_NODES respectively), e.g.:

 CUSTOM_GRAPH == rec(layout:"circo", nodes:mynodes, edges:myedges)

You can also provide SEQUENCE_CHART_opname definitions for generating UML sequence charts.

These DEFINITIONS affect VisB:

 VISB_JSON_FILE == "PATH to .json"  a path to a default VisB JSON file for visualisation;
                                    if it is "" an empty SVG will be created
 VISB_SVG_OBJECTSn == ...           define a record or set of records for creating new SVG objects
 VISB_SVG_UPDATESn == ...           define a record or set of records containing updates of SVG objects
 VISB_SVG_HOVERSn == ...            define a record or set of records for VisB hover functions
 VISB_SVG_BOX == ...                record with dimensions (height, width) of a default empty SVG
 VISB_SVG_CONTENTS == ...           defines a string to be included into a created empty SVG file

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, lambda, union or composition 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 three special descriptions:
                            /*@desc memo*/          to be put after identifiers in the ABSTRACT_CONSTANTS section
                                                    indicating that these functions should be memoized
                            /*@desc expand*/        to be put after identifiers (in VARIABLES, CONSTANTS,... section)
                                                    indicating that they should be expanded and not kept symbolically
                            /*@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

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

Free Types

More information can be found here.

Free types exist in Z and in the Rodin theory plugin and are supported by ProB. You can also define new free types in classical B by adding a FREETYPES clause with free type definitions separated by semicolon.

Here is a definition of an inductive type IntList for lists of integers constructed using inil and icons:

FREETYPES
  IntList = inil, icons(INTEGER*IntList)

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)
 - not all tree operators are 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 stronger than Atelier-B's, much less typing predicates
   are required
 - You can apply prj1 and prj2 without providing the type arguments, e.g., prj2(prj1(1|->2|->3))
 - ProB accepts operations with parameters but without pre-conditions
 - ProB allows identifiers consisting of a single character and identifiers in single backquotes (`id`)
 - ProB allows to use <> for the empty sequence (but this use is deprecated)
 - ProB allows escape codes (\n, \', \", see above) and supports UTF-8 characters in strings,
   and ProB allows multi-line string literals written using three apostrophes ('''string''')
   as well as template strings using three backquotes (e.g., ```1+2=${1+2}```)
 - ProB allows a she-bang line in machine files starting with #!
 - ProB allows btrue and bfalse as predicates in B machines
 - ProB allows to use the Event-B relation operators <<->, <->>, <<->>
 - ProB allows set comprehensions with an extra expression like {x•x:1..10|x*x}.
 - The FREETYPES section and the external libraries (LibraryStrings.def, ...) do not exist in Atelier-B

Other notes

 ProB now supports the Unicode mathematical symbols, exactly like Atelier-B
 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.

Event-B Syntax

Note that the Event-B syntax in Rodin is slightly different (e.g, no sequences or strings built-in). There is also an Event-B summary by Ken Robinson (File:EventB-summary.pdf). The Event-B syntax is only available for Event-B models in Rodin, ProB2-UI and ProB Jupyter notebooks.

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.