This Appendix gives an example of a grammar using some
of the advanced features discussed in Section 10.
The desk calculator example in Appendix A is
modified to provide a desk calculator that
does floating point interval arithmetic.
The calculator understands floating point
constants, the arithmetic operations +, -, *, /,
unary -, and = (assignment), and has 26 floating
point variables, ``a'' through ``z''.
Moreover, it also understands
*intervals ,*
written

( x , y )where

This example explores a number of interesting features
of Yacc and C.
Intervals are represented by a structure, consisting of the
left and right endpoint values, stored as
*double 's.*
This structure is given a type name, INTERVAL, by using
*typedef .*
The Yacc value stack can also contain floating point scalars, and
integers (used to index into the arrays holding the variable values).
Notice that this entire strategy depends strongly on being able to
assign structures and unions in C.
In fact, many of the actions call functions that return structures
as well.

It is also worth noting the use of YYERROR to handle error conditions: division by an interval containing 0, and an interval presented in the wrong order. In effect, the error recovery mechanism of Yacc is used to throw away the rest of the offending line.

In addition to the mixing of types on the value stack, this grammar also demonstrates an interesting use of syntax to keep track of the type (e.g. scalar or interval) of intermediate expressions. Note that a scalar can be automatically promoted to an interval if the context demands an interval value. This causes a large number of conflicts when the grammar is run through Yacc: 18 Shift/Reduce and 26 Reduce/Reduce. The problem can be seen by looking at the two input lines:

2.5 + ( 3.5 - 4. )and

2.5 + ( 3.5 , 4. )Notice that the 2.5 is to be used in an interval valued expression in the second example, but this fact is not known until the ``,'' is read; by this time, 2.5 is finished, and the parser cannot go back and change its mind. More generally, it might be necessary to look ahead an arbitrary number of tokens to decide whether to convert a scalar to an interval. This problem is evaded by having two rules for each binary interval valued operator: one when the left operand is a scalar, and one when the left operand is an interval. In the second case, the right operand must be an interval, so the conversion will be applied automatically. Despite this evasion, there are still many cases where the conversion may be applied or not, leading to the above conflicts. They are resolved by listing the rules that yield scalars first in the specification file; in this way, the conflicts will be resolved in the direction of keeping scalar valued expressions scalar valued until they are forced to become intervals.

This way of handling multiple types is very instructive, but not very general. If there were many kinds of expression types, instead of just two, the number of rules needed would increase dramatically, and the conflicts even more dramatically. Thus, while this example is instructive, it is better practice in a more normal programming language environment to keep the type information as part of the value, and not as part of the grammar.

Finally, a word about the lexical analysis.
The only unusual feature is the treatment of floating point constants.
The C library routine
*atof*
is used to do the actual conversion from a character string
to a double precision value.
If the lexical analyzer detects an error,
it responds by returning a token that
is illegal in the grammar, provoking a syntax error
in the parser, and thence error recovery.

%{ # include <stdio.h> # include <ctype.h> typedef struct interval { double lo, hi; } INTERVAL; INTERVAL vmul(), vdiv(); double atof(); double dreg[ 26 ]; INTERVAL vreg[ 26 ]; %} %start lines %union { int ival; double dval; INTERVAL vval; } %token <ival> DREG VREG /* indices into dreg, vreg arrays */ %token <dval> CONST /* floating point constant */ %type <dval> dexp /* expression */ %type <vval> vexp /* interval expression */ /* precedence information about the operators */ %left '+' '-' %left '*' '/' %left UMINUS /* precedence for unary minus */ %% lines : /* empty */ | lines line ; line : dexp '\n' { printf( "%15.8f\n", $1 ); } | vexp '\n' { printf( "(%15.8f , %15.8f )\n", $1.lo, $1.hi ); } | DREG '=' dexp '\n' { dreg[$1] = $3; } | VREG '=' vexp '\n' { vreg[$1] = $3; } | error '\n' { yyerrok; } ; dexp : CONST | DREG { $$ = dreg[$1]; } | dexp '+' dexp { $$ = $1 + $3; } | dexp '-' dexp { $$ = $1 - $3; } | dexp '*' dexp { $$ = $1 * $3; } | dexp '/' dexp { $$ = $1 / $3; } | '-' dexp %prec UMINUS { $$ = - $2; } | '(' dexp ')' { $$ = $2; } ; vexp : dexp { $$.hi = $$.lo = $1; } | '(' dexp ',' dexp ')' { $$.lo = $2; $$.hi = $4; if( $$.lo > $$.hi ){ printf( "interval out of order\n" ); YYERROR; } } | VREG { $$ = vreg[$1]; } | vexp '+' vexp { $$.hi = $1.hi + $3.hi; $$.lo = $1.lo + $3.lo; } | dexp '+' vexp { $$.hi = $1 + $3.hi; $$.lo = $1 + $3.lo; } | vexp '-' vexp { $$.hi = $1.hi - $3.lo; $$.lo = $1.lo - $3.hi; } | dexp '-' vexp { $$.hi = $1 - $3.lo; $$.lo = $1 - $3.hi; } | vexp '*' vexp { $$ = vmul( $1.lo, $1.hi, $3 ); } | dexp '*' vexp { $$ = vmul( $1, $1, $3 ); } | vexp '/' vexp { if( dcheck( $3 ) ) YYERROR; $$ = vdiv( $1.lo, $1.hi, $3 ); } | dexp '/' vexp { if( dcheck( $3 ) ) YYERROR; $$ = vdiv( $1, $1, $3 ); } | '-' vexp %prec UMINUS { $$.hi = -$2.lo; $$.lo = -$2.hi; } | '(' vexp ')' { $$ = $2; } ; %% # define BSZ 50 /* buffer size for floating point numbers */ /* lexical analysis */ yylex(){ register c; while( (c=getchar()) == ' ' ){ /* skip over blanks */ } if( isupper( c ) ){ yylval.ival = c - 'A'; return( VREG ); } if( islower( c ) ){ yylval.ival = c - 'a'; return( DREG ); } if( isdigit( c ) || c=='.' ){ /* gobble up digits, points, exponents */ char buf[BSZ+1], *cp = buf; int dot = 0, exp = 0; for( ; (cp-buf)<BSZ ; ++cp,c=getchar() ){ *cp = c; if( isdigit( c ) ) continue; if( c == '.' ){ if( dot++ || exp ) return( '.' ); /* will cause syntax error */ continue; } if( c == 'e' ){ if( exp++ ) return( 'e' ); /* will cause syntax error */ continue; } /* end of number */ break; } *cp = '\0'; if( (cp-buf) >= BSZ ) printf( "constant too long: truncated\n" ); else ungetc( c, stdin ); /* push back last char read */ yylval.dval = atof( buf ); return( CONST ); } return( c ); } INTERVAL hilo( a, b, c, d ) double a, b, c, d; { /* returns the smallest interval containing a, b, c, and d */ /* used by *, / routines */ INTERVAL v; if( a>b ) { v.hi = a; v.lo = b; } else { v.hi = b; v.lo = a; } if( c>d ) { if( c>v.hi ) v.hi = c; if( d<v.lo ) v.lo = d; } else { if( d>v.hi ) v.hi = d; if( c<v.lo ) v.lo = c; } return( v ); } INTERVAL vmul( a, b, v ) double a, b; INTERVAL v; { return( hilo( a*v.hi, a*v.lo, b*v.hi, b*v.lo ) ); } dcheck( v ) INTERVAL v; { if( v.hi >= 0. && v.lo <= 0. ){ printf( "divisor interval contains 0.\n" ); return( 1 ); } return( 0 ); } INTERVAL vdiv( a, b, v ) double a, b; INTERVAL v; { return( hilo( a/v.hi, a/v.lo, b/v.hi, b/v.lo ) ); }

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