How to solve LR(1) grammar ambiguity between ternary expressions (a ? b : c) and "maybe" expressions (a?)?

Question

I have created a grammar, a stripped-down version of which is reproduced below:

(0) exp1: ternary;
(1) exp1: exp2;
(2) ternary: exp2 "?" exp1 ":" exp1;
(3) exp2: exp2 "+" exp3;
(4) exp2: exp3;
(5) exp3: maybe;
(6) exp3: "1";
(7) maybe: exp3 "?";

I believe this language is unambiguous, and should be LR-parsable. (Please let me know if I'm wrong!)

However, when I attempt to generate an LR(1) parser for this grammar, I get shift/reduce conflicts, because when the parser sees exp3 with lookahead ?, it doesn't know whether to shift or reduce:

Conflicts in state 3:
    Reduction using rule 4: exp2:  exp3 · | "?"
    Shift to state 6

Conflicts in state 9:
    Reduction using rule 3: exp2:  exp2 "+" exp3 · | "?"
    Shift to state 6

Conflicts in state 13:
    Reduction using rule 4: exp2:  exp3 · | "?"
    Shift to state 16

Conflicts in state 20:
    Reduction using rule 4: exp2:  exp3 · | "?"
    Shift to state 23

Conflicts in state 25:
    Reduction using rule 3: exp2:  exp2 "+" exp3 · | "?"
    Shift to state 23

Conflicts in state 28:
    Reduction using rule 3: exp2:  exp2 "+" exp3 · | "?"
    Shift to state 16

Is there a reasonable way for me to make this language LR(1)-parsable (with no conflicts)?

Or are GLR (or maybe LR(2)?) my only realistic options for a language like this?
(Or am I even wrong in believing that the language is unambiguous in the first place?)

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For reference, the ambiguous state machine I generated is the following (where ♦ is EOF):

State 0:
    exp1:  · ternary | {♦} → shift 1
    ternary:  · exp2 "?" exp1 ":" exp1 | {♦} → shift 2
    exp2:  · exp2 "+" exp3 | {"?", "+"} → shift 2
    exp2:  · exp3 | {"?", "+"} → shift 3
    exp3:  · maybe | {"?", "+"} → shift 4
    exp3:  · "1" | {"?", "+"} → shift 5
    maybe:  · exp3 "?" | {"?", "+"} → shift 3

State 1:
    exp1:  ternary · | {♦} → reduce 0

State 2:
    ternary:  exp2 · "?" exp1 ":" exp1 | {♦} → shift 7
    exp2:  exp2 · "+" exp3 | {"?", "+"} → shift 8

State 3:
    exp2:  exp3 · | {"+"} → reduce 4
    exp2:  exp3 · | {"?"} → reduce 4 shift 6
    maybe:  exp3 · "?" | {"?", "+"} → reduce 4 shift 6

State 4:
    exp3:  maybe · | {"?", "+"} → reduce 5

State 5:
    exp3:  "1" · | {"?", "+"} → reduce 6

State 6:
    maybe:  exp3 "?" · | {"?", "+"} → reduce 7

State 7:
    exp1:  · ternary | {":"} → shift 10
    exp1:  · exp2 | {":"} → shift 11
    ternary:  · exp2 "?" exp1 ":" exp1 | {":"} → shift 11
    ternary:  exp2 "?" · exp1 ":" exp1 | {♦} → shift 12
    exp2:  · exp2 "+" exp3 | {"?", ":", "+"} → shift 11
    exp2:  · exp3 | {"?", ":", "+"} → shift 13
    exp3:  · maybe | {"?", ":", "+"} → shift 14
    exp3:  · "1" | {"?", ":", "+"} → shift 15
    maybe:  · exp3 "?" | {"?", ":", "+"} → shift 13

State 8:
    exp2:  exp2 "+" · exp3 | {"?", "+"} → shift 9
    exp3:  · maybe | {"?", "+"} → shift 4
    exp3:  · "1" | {"?", "+"} → shift 5
    maybe:  · exp3 "?" | {"?", "+"} → shift 9

State 9:
    exp2:  exp2 "+" exp3 · | {"+"} → reduce 3
    exp2:  exp2 "+" exp3 · | {"?"} → reduce 3 shift 6
    maybe:  exp3 · "?" | {"?", "+"} → reduce 3 shift 6

State 10:
    exp1:  ternary · | {":"} → reduce 0

State 11:
    exp1:  exp2 · | {":"} → reduce 1
    ternary:  exp2 · "?" exp1 ":" exp1 | {":"} → shift 26
    exp2:  exp2 · "+" exp3 | {"?", ":", "+"} → shift 27

State 12:
    ternary:  exp2 "?" exp1 · ":" exp1 | {♦} → shift 17

State 13:
    exp2:  exp3 · | {":", "+"} → reduce 4
    exp2:  exp3 · | {"?"} → reduce 4 shift 16
    maybe:  exp3 · "?" | {"?", ":", "+"} → reduce 4 shift 16

State 14:
    exp3:  maybe · | {"?", ":", "+"} → reduce 5

State 15:
    exp3:  "1" · | {"?", ":", "+"} → reduce 6

State 16:
    maybe:  exp3 "?" · | {"?", ":", "+"} → reduce 7

State 17:
    exp1:  · ternary | {♦} → shift 1
    exp1:  · exp2 | {♦} → shift 18
    ternary:  · exp2 "?" exp1 ":" exp1 | {♦} → shift 18
    ternary:  exp2 "?" exp1 ":" · exp1 | {♦} → shift 19
    exp2:  · exp2 "+" exp3 | {♦, "?", "+"} → shift 18
    exp2:  · exp3 | {♦, "?", "+"} → shift 20
    exp3:  · maybe | {♦, "?", "+"} → shift 21
    exp3:  · "1" | {♦, "?", "+"} → shift 22
    maybe:  · exp3 "?" | {♦, "?", "+"} → shift 20

State 18:
    exp1:  exp2 · | {♦} → reduce 1
    ternary:  exp2 · "?" exp1 ":" exp1 | {♦} → shift 7
    exp2:  exp2 · "+" exp3 | {♦, "?", "+"} → shift 24

State 19:
    ternary:  exp2 "?" exp1 ":" exp1 · | {♦} → reduce 2

State 20:
    exp2:  exp3 · | {♦, "+"} → reduce 4
    exp2:  exp3 · | {"?"} → reduce 4 shift 23
    maybe:  exp3 · "?" | {♦, "?", "+"} → reduce 4 shift 23

State 21:
    exp3:  maybe · | {♦, "?", "+"} → reduce 5

State 22:
    exp3:  "1" · | {♦, "?", "+"} → reduce 6

State 23:
    maybe:  exp3 "?" · | {♦, "?", "+"} → reduce 7

State 24:
    exp2:  exp2 "+" · exp3 | {♦, "?", "+"} → shift 25
    exp3:  · maybe | {♦, "?", "+"} → shift 21
    exp3:  · "1" | {♦, "?", "+"} → shift 22
    maybe:  · exp3 "?" | {♦, "?", "+"} → shift 25

State 25:
    exp2:  exp2 "+" exp3 · | {♦, "+"} → reduce 3
    exp2:  exp2 "+" exp3 · | {"?"} → reduce 3 shift 23
    maybe:  exp3 · "?" | {♦, "?", "+"} → reduce 3 shift 23

State 26:
    exp1:  · ternary | {":"} → shift 10
    exp1:  · exp2 | {":"} → shift 11
    ternary:  · exp2 "?" exp1 ":" exp1 | {":"} → shift 11
    ternary:  exp2 "?" · exp1 ":" exp1 | {":"} → shift 29
    exp2:  · exp2 "+" exp3 | {"?", ":", "+"} → shift 11
    exp2:  · exp3 | {"?", ":", "+"} → shift 13
    exp3:  · maybe | {"?", ":", "+"} → shift 14
    exp3:  · "1" | {"?", ":", "+"} → shift 15
    maybe:  · exp3 "?" | {"?", ":", "+"} → shift 13

State 27:
    exp2:  exp2 "+" · exp3 | {"?", ":", "+"} → shift 28
    exp3:  · maybe | {"?", ":", "+"} → shift 14
    exp3:  · "1" | {"?", ":", "+"} → shift 15
    maybe:  · exp3 "?" | {"?", ":", "+"} → shift 28

State 28:
    exp2:  exp2 "+" exp3 · | {":", "+"} → reduce 3
    exp2:  exp2 "+" exp3 · | {"?"} → reduce 3 shift 16
    maybe:  exp3 · "?" | {"?", ":", "+"} → reduce 3 shift 16

State 29:
    ternary:  exp2 "?" exp1 · ":" exp1 | {":"} → shift 30

State 30:
    exp1:  · ternary | {":"} → shift 10
    exp1:  · exp2 | {":"} → shift 11
    ternary:  · exp2 "?" exp1 ":" exp1 | {":"} → shift 11
    ternary:  exp2 "?" exp1 ":" · exp1 | {":"} → shift 31
    exp2:  · exp2 "+" exp3 | {"?", ":", "+"} → shift 11
    exp2:  · exp3 | {"?", ":", "+"} → shift 13
    exp3:  · maybe | {"?", ":", "+"} → shift 14
    exp3:  · "1" | {"?", ":", "+"} → shift 15
    maybe:  · exp3 "?" | {"?", ":", "+"} → shift 13

State 31:
    ternary:  exp2 "?" exp1 ":" exp1 · | {":"} → reduce 2

Answer 1

I think this might be a precedence issue. The conflicts you're getting occur when the parser is looking at something like this:

 a + b ? c : d

At the time that the parser has seen a + b ? and is looking at c, it can't decide whether it needs to

1. Reduce b?, so that it will parse an expression of the form a + (b?) and then continue from there, or

2. Reduce a + b, so that it will parse an expression of the form (a + b) ? c : d

I think the challenge here is that in one case, ? has very low precedence (when used as a ternary operator), and in another case it has very high precedence (when used as a unary operator). However, if you did assign the precedences this way, I think that the parser might be able to disambiguate these cases.

Hope this helps!