Can't figure out how to resolve reduce/reduce conflict - grammar

I'm trying to create grammar for GNU MathProg language from glpk package https://www3.nd.edu/~jeff/mathprog/glpk-4.47/doc/gmpl.pdf
Unfortunately grammar I've written so far is ambiguous.
I don't know how to tell bison which branch of parsing tree is correct when some identifier is used. For example:
numericExpression : numericLiteral
| identifier
| numericFunctionReference
| iteratedNumericExpression
| conditionalNumericExpression
| '(' numericExpression ')' %prec PARENTH
| '-' numericExpression %prec UNARY
| '+' numericExpression %prec UNARY
| numericExpression binaryArithmeticOperator numericExpression
;
symbolicExpression : stringLiteral
| symbolicFunctionReference
| identifier
| conditionalSymbolicExpression
| '(' symbolicExpression ')' %prec PARENTH
| symbolicExpression '&' symbolicExpression
;
indexingExpression : '{' indexingEntries '}'
| '{' indexingEntries ':' logicalExpression '}'
;
setExpression : literalSet
| identifier
| aritmeticSet
| indexingExpression
| iteratedSetExpression
| conditionalSetExpression
| '(' setExpression ')' %prec PARENTH
| setExpression setOperator setExpression
;
numericLiteral : INT
| FLT
;
linearExpression : identifier
| iteratedLinearExpression
| conditionalLinearExpression
| '(' linearExpression ')' %prec PARENTH
| '-' linearExpression %prec UNARY
| '+' linearExpression %prec UNARY
| linearExpression '+' linearExpression
| linearExpression '-' linearExpression
| linearExpression '*' numericExpression
| numericExpression '*' linearExpression
| linearExpression '/' numericExpression
;
logicalExpression : numericExpression
| relationalExpression
| iteratedLogicalExpression
| '(' logicalExpression ')' %prec PARENTH
| NOT logicalExpression %prec NEG
| logicalExpression AND logicalExpression
| logicalExpression OR logicalExpression
;
identifier : SYMBOLIC_NAME
| SYMBOLIC_NAME '[' listOfIndices ']'
;
listOfIndices : SYMBOLIC_NAME
| listOfIndices ',' SYMBOLIC_NAME
;
Identifier is simply name of 'variable'. Variable has a specific type (parameter, set, decision variable) and might be indexed. In code programmer has to declare variable type in statements like eg.
param p1;
param p2{1, 2} >=0;
set s1;
set s2{i in 1..5};
var v1 >=0;
var v2{S1,S2};
But when bison sees identifier doesn't know which rule should use and i'm getting reduce/reduce conflicts like
113 numericExpression: identifier .
123 symbolicExpression: identifier .
'&' reduce using rule 123 (symbolicExpression)
ELSE reduce using rule 113 (numericExpression)
ELSE [reduce using rule 123 (symbolicExpression)]
INTEGER reduce using rule 113 (numericExpression)
INTEGER [reduce using rule 123 (symbolicExpression)]
BINARY reduce using rule 113 (numericExpression)
BINARY [reduce using rule 123 (symbolicExpression)]
ASIGN reduce using rule 113 (numericExpression)
ASIGN [reduce using rule 123 (symbolicExpression)]
',' reduce using rule 113 (numericExpression)
',' [reduce using rule 123 (symbolicExpression)]
'>' reduce using rule 113 (numericExpression)
'>' [reduce using rule 123 (symbolicExpression)]
'}' reduce using rule 113 (numericExpression)
'}' [reduce using rule 123 (symbolicExpression)]
113 numericExpression: identifier .
123 symbolicExpression: identifier .
130 setExpression: identifier .
UNION reduce using rule 130 (setExpression)
DIFF reduce using rule 130 (setExpression)
SYMDIFF reduce using rule 130 (setExpression)
ELSE reduce using rule 113 (numericExpression)
ELSE [reduce using rule 123 (symbolicExpression)]
ELSE [reduce using rule 130 (setExpression)]
WITHIN reduce using rule 130 (setExpression)
IN reduce using rule 113 (numericExpression)
IN [reduce using rule 123 (symbolicExpression)]
I have also other problems but this one is blocker for me

Basically the problem is that identifier appears in multiple rules:
numericExpression : identifier
symbolicExpression : identifier
setExpression: identifier
which may apply in the same context. One way to resolve this is to introduce different token types for set names and scalar (parameters and variables) names:
symbolicExpression : SCALAR_NAME
setExpression: SET_NAME
This will resolve conflict with set names. I don't think that you need this rule
numericExpression : identifier
because there is an automatic conversion from strings to numbers in AMPL and therefore MathProg, which is a subset of AMPL, so symbolicExpression should be allowed in the context of numericExpression.
Note that the grammar is not context-free, so you'll need to pull additional information like the name category above from the symbol table to resolve problems like this.
My flex is a bit rusty but I think you can do something like that in an identifier rule:
return is_setname(...) ? TOK_SET_NAME : TOK_SCALAR_NAME;
where is_setname is a function to check whether the current identifier is a set. You'll need to define such function and get the necessary information from a symbol table.

Related

Jq strange behavior of pipe involving variable on the left hand side

I came across a weird behavior of jq involving a variable on the left hand side of a pipe.
For your information, this question was inspired by the jq manual: under Scoping (https://stedolan.github.io/jq/manual/#Advancedfeatures) where it mentions an example filter ... | .*3 as $times_three | [. + $times_three] | .... I believe the correct version is ... | (.*3) as $times_three | [. + $times_three] | ....
First (https://jqplay.org/s/ffMPsqmsmt)
filter:
. * 3 as $times_three | .
input:
3
output:
9
Second (https://jqplay.org/s/yOFcjRAMLL)
filter:
. * 4 as $times_four | .
input:
3
output:
9
What is happening here?
But (https://jqplay.org/s/IKrTNZjKI8)
filter:
(. * 3) as $times_three | .
input:
3
output:
3
And (https://jqplay.org/s/8zoq2-HN1G)
filter:
(. * 4) as $times_four | .
input:
3
output:
3
So if parenthesis (.*3) or (.*4) is used when the variable is declared then filter behaves predictably.
But if parenthesis is not used .*3 or .*4 then strangely the output is 9 for both.
Can you explain?
Contrary to what the examples in the Scoping section assume, . * 4 as $times_four | . is equivalent to . * ( 4 as $times_four | . ) and therefore squares its input.
You might expect
. * 4 as $times_four | .
to be equivalent to
( . * 4 ) as $times_four | .
And as you point out, some example even suggest this is the case. However, the first snippet is actually equivalent to the following:
. * ( 4 as $times_four | . )
And since … as $x produces its context[1], that's the same as
. * ( . | . )
or
. * .
jq's operator precedence is inconsistent and/or quirky.
"def" | "abc" + "def" | length means"def" | ( "abc" + "def" ) | length, but"def" | "abc" + "def" as $x | length means"def" | "abc" + ( "def" as $x | length ).
This behaviour suggests that that as isn't a binary operator of the form X as $Y as one might expect, but a ternary operator of the form X as $Y | Z.
And, in fact, this is how it's documented:
Variable / Symbolic Binding Operator: ... as $identifier | ...
This leads to surprises, especially since it binds a lot more tightly than expected. And it looks like whomever authored the examples in the Scoping section fell into the trap.
It might produce it multiple times e.g. .[] as $x.
Indeed, there seems to be a mistake in the manual. In section Scoping it is contrasting the (faulty) examples
... | .*3 as $times_three | [. + $times_three] | ... # faulty!
and
... | (.*3 as $times_three | [. + $times_three]) | ... # faulty!
While the overall statement stays valid, both examples are missing additional parentheses around .*3. Thus, it should actually read
... | (.*3) as $times_three | [. + $times_three] | ...
and
... | ((.*3) as $times_three | [. + $times_three]) | ...
respectively.
From the manual under section Variable / Symbolic Binding Operator:
The expression exp as $x | ... means: for each value of expression exp, run the rest of the pipeline with the entire original input, and with $x set to that value. Thus as functions as something of a foreach loop.
This means that a variable assignment takes the one expression left of as and assigns its evaluation to the defined variable right of as (and this happens as many times as exp produces an output). But, as everything in jq is a filter, the assignment itself also is, and as such it needs to have an output itself. If you look closely, the full title of that section
Variable / Symbolic Binding Operator: ... as $identifier | ...
also features a pipe symbol next to it, which indicates that it belongs to the assignment's structure. Try just running . as $x. You will get an error because the | ... part is missing. Thus, to simply keep the input context as is (apart from maybe duplicating it as many times as the expression left of as produced an output), a complete assignment would rather look like … as $x | ., or, if the input context is what you wanted to capture in the variable, . as $x | .
That said, let's clarify what happens with your examples by putting explicit parentheses around the assignments:
3 | . * 3 as $times_three | .
3 | . * (3 as $times_three | .)
3 | . * . # with $times_three set to 3
3 * 3 # with $times_three set to 3
9 # with $times_three set to 3
3 | . * 4 as $times_four | .
3 | . * (4 as $times_four | .)
3 | . * . # with $times_four set to 4
3 * 3 # with $times_four set to 4
9 # with $times_four set to 4
3 | (. * 3) as $times_three | .
3 | ((. * 3) as $times_three | .)
3 | ((3 * 3) as $times_three | .)
3 | (9 as $times_three | .)
3 | . # with $times_three set to 9
3 # with $times_three set to 9
3 | (. * 4) as $times_four | .
3 | ((. * 4) as $times_four | .)
3 | ((3 * 4) as $times_four | .)
3 | (12 as $times_four | .)
3 | . # with $times_four set to 12
3 # with $times_four set to 12

Issue creating grammar definition with antlr

I am currently following the Frederico Tomasetti Antlr tutorial, however I getting the following error when trying to generate my antlr grammar definition.
Chat.g4:52:26: syntax error: ']' came as a complete surprise to me
Chat.g4:52:25 syntax error: mismatched input ')' expecting SEMI while matching a lexer rule
Can anyone see where I've gone wrong?
My g4 file:
1 grammar Chat;
2
3 /*
4 * Parser Rules
5 */
6
7 chat : line+ EOF ;
8
9 line : name command message NEWLINE;
10
11 message : (emoticon | link | color | mention | WORD | WHITESPACE)+ ;
12
13 name : WORD WHITESPACE;
14
15 command : (SAYS | SHOUTS) ':' WHITESPACE ;
16
17 emoticon : ':' '-'? ')'
18 | ':' '-'? '('
19 ;
20
21 link : '[' TEXT ']' '(' TEXT ')' ;
22
23 color : '/' WORD '/' message '/';
24
25 mention : '#' WORD ;
26
27 /*
28 * Lexer Rules
29 */
30
31 fragment A : ('A'|'a') ;
32 fragment S : ('S'|'s') ;
33 fragment Y : ('Y'|'y') ;
34 fragment H : ('H'|'h') ;
35 fragment O : ('O'|'o') ;
36 fragment U : ('U'|'u') ;
37 fragment T : ('T'|'t') ;
38
39 fragment LOWERCASE : [a-z] ;
40 fragment UPPERCASE : [A-Z] ;
41
42 SAYS : S A Y S ;
43
44 SHOUTS : S H O U T S;
45
46 WORD : (LOWERCASE | UPPERCASE | '_')+ ;
47
48 WHITESPACE : (' ' | '\t') ;
49
50 NEWLINE : ('\r'? '\n' | '\r')+ ;
51
52 TEXT : ~[])]+ ;
Any help would be appreciated.
TEXT : ~[])]+ ;
You can't use ] unescaped in a character class - not even in the beginning. You'll need to precede it with a backslash: ~[\])]+.

Grammar for given language

I am given the language {w ∈ {a,b}∗| |w|a = |w|b + 1}. and am asked to find a grammar.
I have come up with the following:
S->aSb | bSa | aAa | bBb | a
A->bS
B->?
and was wondering if this was correct, or if not why?
It's not correct, because it cannot generate the valid sentence:
baaab
which has one more a than b. It should be obvious that this sentence cannot be generated because every sentence generated by your language has different start and end characters.
Edit The edited question is also not correct because the productions:
S -> ... | aAa | a | ...
A -> bS
is equivalent to (by substituting the RHS of A for its use in S):
S -> ... | abSa | a | ...
which will match as follows:
S -> abSa -> abaa

Left-Linear and Right-Linear Grammars

I need help with constructing a left-linear and right-linear grammar for the languages below?
a) (0+1)*00(0+1)*
b) 0*(1(0+1))*
c) (((01+10)*11)*00)*
For a) I have the following:
Left-linear
S --> B00 | S11
B --> B0|B1|011
Right-linear
S --> 00B | 11S
B --> 0B|1B|0|1
Is this correct? I need help with b & c.
Constructing an equivalent Regular Grammar from a Regular Expression
First, I start with some simple rules to construct Regular Grammar(RG) from Regular Expression(RE).
I am writing rules for Right Linear Grammar (leaving as an exercise to write similar rules for Left Linear Grammar)
NOTE: Capital letters are used for variables, and small for terminals in grammar. NULL symbol is ^. Term 'any number' means zero or more times that is * star closure.
[BASIC IDEA]
SINGLE TERMINAL: If the RE is simply e (e being any terminal), we can write G, with only one production rule S --> e (where S is the start symbol), is an equivalent RG.
UNION OPERATION: If the RE is of the form e + f, where both e and f are terminals, we can write G, with two production rules S --> e | f, is an equivalent RG.
CONCATENATION: If the RE is of the form ef, where both e and f are terminals, we can write G, with two production rules S --> eA, A --> f, is an equivalent RG.
STAR CLOSURE: If the RE is of the form e*, where e is a terminal and * Kleene star closure operation, we can write two production rules in G, S --> eS | ^, is an equivalent RG.
PLUS CLOSURE: If the RE is of the form e+, where e is a terminal and + Kleene plus closure operation, we can write two production rules in G, S --> eS | e, is an equivalent RG.
STAR CLOSURE ON UNION: If the RE is of the form (e + f)*, where both e and f are terminals, we can write three production rules in G, S --> eS | fS | ^, is an equivalent RG.
PLUS CLOSURE ON UNION: If the RE is of the form (e + f)+, where both e and f are terminals, we can write four production rules in G, S --> eS | fS | e | f, is an equivalent RG.
STAR CLOSURE ON CONCATENATION: If the RE is of the form (ef)*, where both e and f are terminals, we can write three production rules in G, S --> eA | ^, A --> fS, is an equivalent RG.
PLUS CLOSURE ON CONCATENATION: If the RE is of the form (ef)+, where both e and f are terminals, we can write three production rules in G, S --> eA, A --> fS | f, is an equivalent RG.
Be sure that you understands all above rules, here is the summary table:
+-------------------------------+--------------------+------------------------+
| TYPE | REGULAR-EXPRESSION | RIGHT-LINEAR-GRAMMAR |
+-------------------------------+--------------------+------------------------+
| SINGLE TERMINAL | e | S --> e |
| UNION OPERATION | e + f | S --> e | f |
| CONCATENATION | ef | S --> eA, A --> f |
| STAR CLOSURE | e* | S --> eS | ^ |
| PLUS CLOSURE | e+ | S --> eS | e |
| STAR CLOSURE ON UNION | (e + f)* | S --> eS | fS | ^ |
| PLUS CLOSURE ON UNION | (e + f)+ | S --> eS | fS | e | f |
| STAR CLOSURE ON CONCATENATION | (ef)* | S --> eA | ^, A --> fS |
| PLUS CLOSURE ON CONCATENATION | (ef)+ | S --> eA, A --> fS | f |
+-------------------------------+--------------------+------------------------+
note: symbol e and f are terminals, ^ is NULL symbol, and S is the start variable
[ANSWER]
Now, we can come to you problem.
a) (0+1)*00(0+1)*
Language description: All the strings consist of 0s and 1s, containing at-least one pair of 00.
Right Linear Grammar:
S --> 0S | 1S | 00A
A --> 0A | 1A | ^
String can start with any string of 0s and 1s thats why included rules s --> 0S | 1S and Because at-least one pair of 00 ,there is no null symbol. S --> 00A is included because 0, 1 can be after 00. The symbol A takes care of the 0's and 1's after the 00.
Left Linear Grammar:
S --> S0 | S1 | A00
A --> A0 | A1 | ^
b) 0*(1(0+1))*
Language description: Any number of 0, followed any number of 10 and 11.
{ because 1(0 + 1) = 10 + 11 }
Right Linear Grammar:
S --> 0S | A | ^
A --> 1B
B --> 0A | 1A | 0 | 1
String starts with any number of 0 so rule S --> 0S | ^ are included, then rule for generating 10 and 11 for any number of times using A --> 1B and B --> 0A | 1A | 0 | 1.
Other alternative right linear grammar can be
S --> 0S | A | ^
A --> 10A | 11A | 10 | 11
Left Linear Grammar:
S --> A | ^
A --> A10 | A11 | B
B --> B0 | 0
An alternative form can be
S --> S10 | S11 | B | ^
B --> B0 | 0
c) (((01+10)*11)*00)*
Language description: First is language contains null(^) string because there a * (star) on outside of every thing present inside (). Also if a string in language is not null that defiantly ends with 00. One can simply think this regular expression in the form of ( ( (A)* B )* C )* , where (A)* is (01 + 10)* that is any number of repeat of 01 and 10.
If there is a instance of A in string there would be a B defiantly because (A)*B and B is 11.
Some example strings { ^, 00, 0000, 000000, 1100, 111100, 1100111100, 011100, 101100, 01110000, 01101100, 0101011010101100, 101001110001101100 ....}
Left Linear Grammar:
S --> A00 | ^
A --> B11 | S
B --> B01 | B10 | A
S --> A00 | ^ because any string is either null, or if it's not null it ends with a 00. When the string ends with 00, the variable A matches the pattern ((01 + 10)* + 11)*. Again this pattern can either be null or must end with 11. If its null, then A matches it with S again i.e the string ends with pattern like (00)*. If the pattern is not null, B matches with (01 + 10)*. When B matches all it can, A starts matching the string again. This closes the out-most * in ((01 + 10)* + 11)*.
Right Linear Grammar:
S --> A | 00S | ^
A --> 01A | 10A | 11S
Second part of you question:
For a) I have the following:
Left-linear
S --> B00 | S11
B --> B0|B1|011
Right-linear
S --> 00B | 11S
B --> 0B|1B|0|1
(answer)
You solution are wrong for following reasons,
Left-linear grammar is wrong Because string 0010 not possible to generate.
Right-linear grammar is wrong Because string 1000 is not possible to generate. Although both are in language generated by regular expression of question (a).
EDIT
Adding DFA's for each regular expression. so that one can find it helpful.
a) (0+1)*00(0+1)*
b) 0*(1(0+1))*
c) (((01+10)*11)*00)*
Drawing DFA for this regular expression is trick and complex.
For this I wanted to add DFA's
To simplify the task, we should think the kind formation of RE
to me the RE (((01+10)*11)*00)* looks like (a*b)*
(((01+10)*11)* 00 )*
( a* b )*
Actually in above expression a it self in the form of (a*b)*
that is ((01+10)*11)*
RE (a*b)* is equals to (a + b)*b + ^. The DFA for (ab) is as belows:
DFA for ((01+10)*11)* is:
DFA for (((01+10)*11)* 00 )* is:
Try to find similarity in construction of above three DFA. don't move ahead till you don't understand first one
Rules to convert regular expressions to left or right linear regular grammar

Grammar: difference between a top down and bottom up? (Example)

This is a follow up question from Grammar: difference between a top down and bottom up?
I understand from that question that:
the grammar itself isn't top-down or bottom-up, the parser is
there are grammars that can be parsed by one but not the other
(thanks Jerry Coffin
So for this grammar (all possible mathematical formulas):
E -> E T E
E -> (E)
E -> D
T -> + | - | * | /
D -> 0
D -> L G
G -> G G
G -> 0 | L
L -> 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
Would this be readable by a top down and bottom up parser?
Could you say that this is a top down grammar or a bottom up grammar (or neither)?
I am asking because I have a homework question that asks:
"Write top-down and bottom-up grammars for the language consisting of all ..." (different question)
I am not sure if this can be correct since it appears that there is no such thing as a top-down and bottom-up grammar. Could anyone clarify?
That grammar is stupid, since it unites lexing and parsing as one. But ok, it's an academic example.
The thing with bottoms-up and top-down is that is has special corner cases that are difficult to implement with you normal 1 look ahead. I probably think that you should check if it has any problems and change the grammar.
To understand you grammar I wrote a proper EBNF
expr:
expr op expr |
'(' expr ')' |
number;
op:
'+' |
'-' |
'*' |
'/';
number:
'0' |
digit digits;
digits:
'0' |
digit |
digits digits;
digit:
'1' |
'2' |
'3' |
'4' |
'5' |
'6' |
'7' |
'8' |
'9';
I especially don't like the rule digits: digits digits. It is unclear where the first digits starts and the second ends. I would implement the rule as
digits:
'0' |
digit |
digits digit;
An other problem is number: '0' | digit digits; This conflicts with digits: '0' and digits: digit;. As a matter of fact that is duplicated. I would change the rules to (removing digits):
number:
'0' |
digit |
digit zero_digits;
zero_digits:
zero_digit |
zero_digits zero_digit;
zero_digit:
'0' |
digit;
This makes the grammar LR1 (left recursive with one look ahead) and context free. This is what you would normally give to a parser generator such as bison. And since bison is bottoms up, this is a valid input for a bottoms-up parser.
For a top-down approach, at least for recursive decent, left recursive is a bit of a problem. You can use roll back, if you like but for these you want a RR1 (right recursive one look ahead) grammar. To do that swap the recursions:
zero_digits:
zero_digit |
zero_digit zero_digits;
I am not sure if that answers you question. I think the question is badly formulated and misleading; and I write parsers for a living...