Fuzzing Without Specifications: Learning Structure from Behaviour Part I

1. Fuzzing Without Speci fi cations:
 Learning Structure from Behaviour Part I Rahul Gopinath 1
2. Fuzzing Without Speci fi cations:
 Learning Structure from Behaviour Part I Rahul Gopinath 2
3. Prerequisites https://github.com/vrthra/summer2025#readme • Install Python 3.12 • Install Graphviz • Install Z3 • Install Jupyter • Start Jupyter 3
4. Prerequisites http://localhost:8888/tree • Install Python 3.10 • Install Graphviz • Install Jupyter • Start Jupyter 4 See the README
5. http://localhost:8888/notebooks/RoadMap.ipynb 5
6. http://localhost:8888/notebooks/x0_0_Prerequisites.ipynb 6
7. 7 Finding hidden bugs
8. 8 Fuzzing Crash? Program Trash deck technique: 1950s - Gerald Weinberg
9. 9 $ ./fuzz [;x1-GPZ+wcckc];,N9J+?#6^6\e?]9lu
 2_%'4GX"0VUB[E/r ~fApu6b8<{%siq8Z
 h.6{V,hr?;{Ti.r3PIxMMMv6{xS^+'Hq!
 AxB"YXRS@!Kd6;wtAMefFWM(`|J_<1~o}
 z3K(CCzRH JIIvHz>_*.\>JrlU32~eGP?
 lR=bF3+;y$3lodQ<B89!5"W2fK*vE7v{'
 )KC-i,c{<[~m!]o;{.'}Gj\(X}EtYetrp
 bY@aGZ1{P!AZU7x#4(Rtn!q4nCwqol^y6
 }0|Ko=*JK~;zMKV=9Nai:wxu{J&UV#HaU
 )*BiC<),`+t*gka<W=Z.%T5WGHZpI30D<
 Pq>&]BS6R&j?#tP7iaV}-}`\?[_[Z^LBM
 PG-FKj'\xwuZ1=Q`^`5,$N$Q@[!CuRzJ2
 D|vBy!^zkhdf3C5PAkR?V((-%><hn|3='
 i2Qx]D$qs4O`1@fevnG'2\11Vf3piU37@
 5:dfd45*(7^%5ap\zIyl"'f,$ee,J4Gw:
 cgNKLie3nx9(`efSlg6#[K"@WjhZ}r[Sc
 un&sBCS,T[/3]KAeEnQ7lU)3Pn,0)G/6N
 -wyzj/MTd#A;r*(ds./df3r8Odaf?/<#r Program ✘ Random Fuzzing
11. http://localhost:8888/notebooks/x1_0_GeneratingSamples.ipynb 11
12. http://localhost:8888/notebooks/x1_0_GeneratingSamples.ipynb 12
14. @app.route('/admin') def admin(): username = request.cookies.get("username") if not username: return {"Error": "Specify username in Cookie"} username = urllib.quote(os.path.basename(username)) url = "http://permissions:5000/permissions/{}".format(username) resp = requests.request(method="GET", url=url) # "superadmin\ud888" will be simpli fi ed to "superadmin" ret = ujson.loads(resp.text) if resp.status_code == 200: if "superadmin" in ret["roles"]: return {"OK": "Superadmin Access granted"} else: e = u"Access denied. User has following roles: {}".format(ret["roles"]) return {"Error": e}, 401 else:return {"Error": ret["Error"]}, 500 [ ; x 1 - G P Z + w c c k c ] ; , N 9 J + ? # 6 ^ 6 \ e ? ] 9 l u 2 _ % ' 4 G X " 0 V U B [ E / r ~ f A p u 6 b 8 < { % s i q 8 Z h . 6 { V , h r ? ; {Ti.r3PIxMMMv6{xS^+'Hq!AxB"YXRS@! Kd6;wtAMefFWM(`|J_<1~o}z3K(CCzRH J I I v H z > _ * . \ > J r l U 3 2 ~ e G P ? lR=bF3+;y$3lodQ<B89!5"W2fK*vE7v{')KC- i,c{<[~m!]o;{.'}Gj\(X}EtYetrpbY@aGZ1{P! A Z U 7 x # 4 ( R t n ! q 4 n C w q o l ^ y 6 } 0 | Ko=*JK~;zMKV=9Nai:wxu{J&UV#HaU)*Bi C < ) , ` + t * g k a < W = Z . % T 5 W G H Z p I 3 0 D < P q > & ] B S 6 R & j ? # t P 7 i a V } - } ` \ ? [ _ [ Z ^ L B M P G - FKj'\xwuZ1=Q`^`5,$N$Q@[!CuRzJ2D|vBy! ^ z k h d f 3 C 5 P A k R ? V ( ( - % > < h n | 3='i2Qx]D$qs4O`1@fevnG'2\11Vf3piU37@ 5 : d f d 4 5 * ( 7 ^ % 5 a p \ z I y l " ' f , $ee,J4Gw:cgNKLie3nx9(`efSlg6#[K"@Wjh Z}r[Scun&sBCS,T[/3]KAeEnQ7lU)3Pn,0)G/ 6N-wyzj/MTd#A;r Program 14 https://www.fuzzingbook.org/html/Fuzzer.html Traditional Fuzzing
15. 15 (CACM '90) No longer very effective
16. • Insert Instrumentation • Generate inputs • Collect execution feedback • Branches covered during execution • Slightly Mutate Input and try again Collect inputs obtaining new coverage 16 Feedback Driven Fuzzing 16 https://www.fuzzingbook.org/html/MutationFuzzer.html
17. 17 def triangle(a, b, c): if a == b: if b == c: return Equilateral else: return Isosceles else: if b == c: return Isosceles else: if a == c: return Isosceles else: return Scalene def triangle(a, b, c): __probe_enter() if a == b: __probe_1() if b == c: __probe_2() return Equilateral else: __probe_3() return Isosceles else: __probe_4() if b == c: __probe_5() return Isosceles else: __probe_6() if a == c: __probe_7() return Isosceles else: __probe_8() return Scalene def triangle(a, b, c): if a == b: if b == c: return Equilateral else: return Isosceles else: if b == c: return Isosceles else: if a == c: return Isosceles else: return Scalene • Insert Instrumentation • Generate inputs • Collect execution feedback • Branches covered during execution • Slightly Mutate Input and try again Collect inputs obtaining new coverage 17 https://www.fuzzingbook.org/html/MutationFuzzer.html Feedback Driven Fuzzing
18. 18 Feedback Driven Fuzzing triangle (1,1,1) def triangle(a, b, c): if a == b: if b == c: return Equilateral else: return Isosceles else: if b == c: return Isosceles else: if a == c: return Isosceles else: return Scalene • Insert Instrumentation • Generate inputs • Collect execution feedback • Branches covered during execution • Slightly Mutate Input and try again Collect inputs obtaining new coverage 18 https://www.fuzzingbook.org/html/MutationFuzzer.html
19. 19 triangle (1,1,1) def triangle(a, b, c): if a == b: if b == c: return Equilateral else: return Isosceles else: if b == c: return Isosceles else: if a == c: return Isosceles else: return Scalene • Insert Instrumentation • Generate inputs • Collect execution feedback • Branches covered during execution • Slightly Mutate Input and try again Collect inputs obtaining new coverage 19 https://www.fuzzingbook.org/html/MutationFuzzer.html Feedback Driven Fuzzing
20. triangle (1,1,1) 20 triangle (1,1,2) def triangle(a, b, c): if a == b: if b == c: return Equilateral else: return Isosceles else: if b == c: return Isosceles else: if a == c: return Isosceles else: return Scalene • Insert Instrumentation • Generate inputs • Collect execution feedback • Branches covered during execution • Slightly Mutate Input and try again Collect inputs obtaining new coverage Mutated 20 https://www.fuzzingbook.org/html/MutationFuzzer.html Feedback Driven Fuzzing
21. 21 triangle (1,1,3) def triangle(a, b, c): if a == b: if b == c: return Equilateral else: return Isosceles else: if b == c: return Isosceles else: if a == c: return Isosceles else: return Scalene • Insert Instrumentation • Generate inputs • Collect execution feedback • Branches covered during execution • Slightly Mutate Input and try again Collect inputs obtaining new coverage Mutated 21 https://www.fuzzingbook.org/html/MutationFuzzer.html Feedback Driven Fuzzing
22. Feedback Driven Fuzzing 22 triangle (1,1,2) def triangle(a, b, c): if a == b: if b == c: return Equilateral else: return Isosceles else: if b == c: return Isosceles else: if a == c: return Isosceles else: return Scalene • Insert Instrumentation • Generate inputs • Collect execution feedback • Branches covered during execution • Slightly Mutate Input and try again Collect inputs obtaining new coverage 22 https://www.fuzzingbook.org/html/MutationFuzzer.html triangle (1,1,1) triangle (1,1,3)
23. Feedback Driven Fuzzing • Insert Instrumentation • Generate inputs • Collect execution feedback • Branches covered during execution • Slightly Mutate Input and try again Collect inputs obtaining new coverage 23 triangle (1,1,2) def triangle(a, b, c): if a == b: if b == c: return Equilateral else: return Isosceles else: if b == c: return Isosceles else: if a == c: return Isosceles else: return Scalene triangle (1,1,1) AFL 23
24. 24 Feedback Driven Fuzzing Weakness: static int is_reserved_word_token(const char *s, int len) { const char *reserved[] = { "break", "case", "catch", "continue", "debugger", "default", "delete", "do", "else", "false", "finally", "for", "function", "if", "in", "instanceof", "new", "null", "return", "switch", "this", "throw", "true", "try", "typeof", "var", "void", "while", "with", "let", "undefined", ((void *)0)}; int i; if (!mjs_is_alpha(s[0])) return 0; for (i = 0; reserved[i] != ((void *)0); i++) { if (len == (int)strlen(reserved[i]) && strncmp(s, reserved[i], len) == 0) return i + 1; } return 0; } Tokens if (x > 100) { } coverage: 20% if (x > 100) { } e coverage: 5% if (x > 100) { } el coverage: 5% if (x > 100) { } els coverage: 5% if (x > 100) { } else coverage: 25% No smooth coverage gradient in parsers 24 Note: Constant unrolling (e.g. AFL) does not help in such lexical tokens
25. 25 def json_raw(stm): while True: stm.skipspaces() c = stm.peek() if c == 't': return json_fixed(stm, 'true') elif c == 'f': return json_fixed(stm, 'false') elif c == 'n': return json_fixed(stm, 'null') elif c == '"': return json_string(stm) elif c == '{': return json_dict(stm) elif c == '[': return json_list(stm) elif c in NUMSTART: return json_number(stm) raise JSONError(E_MALF, stm, stm.pos) Weak points: • Need for smooth coverage gradient • Coverage only provides first level guidance 1. {"abc":[]} 2. [{"a":[]}, {"b":[]}, {"c":["ab","c"]}] 25 Feedback Driven Fuzzing
26. 26 def json_raw(stm): while True: stm.skipspaces() c = stm.peek() if c == 't': return json_fixed(stm, 'true') elif c == 'f': return json_fixed(stm, 'false') elif c == 'n': return json_fixed(stm, 'null') elif c == '"': return json_string(stm) elif c == '{': return json_dict(stm) elif c == '[': return json_list(stm) elif c in NUMSTART: return json_number(stm) raise JSONError(E_MALF, stm, stm.pos) Weak points: • Need for smooth coverage gradient • Coverage only provides first level guidance 1. {"abc":[]} 2. [{"a":[]}, {"b":[]}, {"c":["ab","c"]}] 26 Feedback Driven Fuzzing
27. 27 def json_raw(stm): while True: stm.skipspaces() c = stm.peek() if c == 't': return json_fixed(stm, 'true') elif c == 'f': return json_fixed(stm, 'false') elif c == 'n': return json_fixed(stm, 'null') elif c == '"': return json_string(stm) elif c == '{': return json_dict(stm) elif c == '[': return json_list(stm) elif c in NUMSTART: return json_number(stm) raise JSONError(E_MALF, stm, stm.pos) Weak points: • Need for smooth coverage gradient • Coverage only provides first level guidance 1. {"abc":[]} 2. [{"a":[]}, {"b":[]}, {"c":["ab","c"]}] 27 Feedback Driven Fuzzing
29. 29 Fuzzing Parsers
30. 30 1y78 NO ( NO (1+2) YES 21214/91*293 YES &12133 NO
31. 31 Fuzzing Parsers $ ./fuzz [;x1-GPZ+wcckc];,N9J+?#6^6\e?]9lu
 2_%'4GX"0VUB[E/r ~fApu6b8<{%siq8Z
 h.6{V,hr?;{Ti.r3PIxMMMv6{xS^+'Hq!
 AxB"YXRS@!Kd6;wtAMefFWM(`|J_<1~o}
 z3K(CCzRH JIIvHz>_*.\>JrlU32~eGP?
 lR=bF3+;y$3lodQ<B89!5"W2fK*vE7v{'
 )KC-i,c{<[~m!]o;{.'}Gj\(X}EtYetrp
 bY@aGZ1{P!AZU7x#4(Rtn!q4nCwqol^y6
 }0|Ko=*JK~;zMKV=9Nai:wxu{J&UV#HaU
 )*BiC<),`+t*gka<W=Z.%T5WGHZpI30D<
 Pq>&]BS6R&j?#tP7iaV}-}`\?[_[Z^LBM
 PG-FKj'\xwuZ1=Q`^`5,$N$Q@[!CuRzJ2
 D|vBy!^zkhdf3C5PAkR?V((-%><hn|3='
 i2Qx]D$qs4O`1@fevnG'2\11Vf3piU37@
 5:dfd45*(7^%5ap\zIyl"'f,$ee,J4Gw:
 cgNKLie3nx9(`efSlg6#[K"@WjhZ}r[Sc
 un&sBCS,T[/3]KAeEnQ7lU)3Pn,0)G/6N
 -wyzj/MTd#A;r*(ds./df3r8Odaf?/<#r Parser Syntax Error Interpreter #
32. 32 Leveraging Instrumentation Based Feedback def parse_num(s,i): n = '' while s[i:] and is_digit(s[i]): n += s[i] i = i +1 return i,n def parse_paren(s, i): assert s[i] == '(' i, v = parse_expr(s, i+1) if s[i:] == '': raise Exception(s, i) assert s[i] == ')' return i+1, v def parse_expr(s, i = 0): expr = [] is_op = True while s[i:]: c = s[i] if c in string.digits: if not is_op: raise Exception(s,i) i,num = parse_num(s,i) expr.append(num) is_op = False elif c in ['+', '-', '*', '/']: if is_op: raise Exception(s,i) expr.append(c) is_op = True i = i + 1 elif c == '(': if not is_op: raise Exception(s,i) i, cexpr = parse_paren(s, i) expr.append(cexpr) is_op = False elif c == ')': break else: raise Exception(s,i) if is_op: raise Exception(s,i) return i, expr Parser Syntax Error Interpreter #
33. http://localhost:8888/notebooks/x1_1_TrackingAccess.ipynb 33
34. 34 Tracker for Access
35. 35 AST Rewriting
36. 36 Find The Last Compared Index
37. 37 Speci fi cation Free Generator
38. 38 Speci fi cation Free Generators A ( 2 - B 9 ) 4 ) A ∉ (,+,-,1,2,3,4,5,6,7,8,9,0 B ∉ +,-,1,2,3,4,5,6,7,8,9,0,) ) ∉ +,-,1,2,3,4,5,6,7,8,9,0 (2-94)
40. 40 Limitation: Lack of control
41. 41 Constraining the Input Domain with Grammars
42. 42 Grammar
43. 43 Formal Languages Formal Language Descriptions 3. Regular Context Free Recursively Enumerable (Chomsky,1956) Easy to produce and parse Argument Stack Return Stack
44. 44 Grammar <start> := <expr> <expr> := <expr> '+' <expr> | <expr> '-' <expr> | <expr> '/' <expr> | <expr> '*' <expr> | '(' <expr> ')' | <number> <number> := <integer> | <integer> '.' <integer> <integer>:= <digit> <integer> | <digit> <digit> := [0-9] Arithmetic expression grammar De f inition for <expr> <expr> key
45. 45 <start> := <expr> <expr> := <expr> '+' <expr> | <expr> '-' <expr> | <expr> '/' <expr> | <expr> '*' <expr> | '(' <expr> ')' | <number> <number> := <integer> | <integer> '.' <integer> <integer>:= <digit> <integer> | <digit> <digit> := [0-9] Grammar Arithmetic expression grammar Expansion Rule Terminal Symbol Nonterminal Symbol
46. http://localhost:8888/notebooks/x0_1_Grammars.ipynb 46
47. 47 Grammars For Parsing (8 / 3) * 49 <start> := <expr> <expr> := <expr> '+' <expr> | <expr> '-' <expr> | <expr> '/' <expr> | <expr> '*' <expr> | '(' <expr> ')' | <number> <number> := <integer> | <integer> '.' <integer> <integer>:= <digit> <integer> | <digit> <digit> := [0-9]
48. http://localhost:8888/notebooks/x0_3_Parser.ipynb 48
49. 49 Parsing is Surprisingly Integral to Fuzzing 1. Parsing is necessary for decomposing and recomposing inputs without affecting the validity of inputs constructed (8 / 3) * 49
50. 50 Parsing is Surprisingly Integral to Fuzzing 1. Parsing is necessary for decomposing and recomposing inputs without affecting the validity of inputs constructed (8 / 3) * 49
51. 51 Parsing is Surprisingly Integral to Fuzzing 1. Parsing is necessary for decomposing and recomposing inputs without affecting the validity of inputs constructed (2 + 1) * 49 1 2 + (8 / 3) * 49
52. 52 Parsing is Surprisingly Integral to Fuzzing 2. Use parsers to mine the input distribution, and generate uncommon inputs
53. 53 Parsing is Surprisingly Integral to Fuzzing 2. Use parsers to extract the complexity of an input or a set of inputs 18439249 (8/3)*49 https://rahul.gopinath.org/post/2024/03/23/k-paths-for-context-free-grammars/
55. 55 Grammars 8.2 - 27 - -9 / +((+9 * --2 + --+-+- ((-1 * +(8 - 5 - 6)) * (-(a-+(((+(4) )))) - ++4) / +(-+---((5.6 - --(3 * -1.8 * +(6 * +-(((-(-6) * ---+6)) / +--(+-+-7 * (-0 * (+(((((2)) + 8 - 3 - ++9.0 + ---(--+7 / (1 / +++6.37) + (1) / 482) / +++-+0)))) + 8.2 - 27 - -9 / +((+9 * --2 + --+-+-((-1 * + (8 - 5 - 6)) * (-(a-+(((+(4))))) - + +4) / +(-+---((5.6 - --(3 * -1.8 * + (6 * +-(((-(-6) * ---+6)) / +--(+-+- 7 * (-0 * (+(((((2)) + 8 - 3 - ++9.0 + ---(--+7 / (1 / +++6.37) + (1) / 482) / +++-+0)))) * -+5 + 7.513)))) - (+1 / ++((-84)))))))) * ++5 / +-(- -2 - -++-9.0)))) / 5 * --++090 + * - +5 + 7.513)))) - (+1 / ++((-84)))))) )) * 8.2 - 27 - -9 / +((+9 * --2 + - -+-+-((-1 * +(8 - 5 - 6)) * (-(a-+(( (+(4))))) - ++4) / +(-+---((5.6 - -- (3 * -1.8 * +(6 * +-(((-(-6) * ---+6 )) / +--(+-+-7 * (-0 * (+(((((2)) + 8 - 3 - ++9.0 + ---(--+7 / (1 / +++6 .37) + (1) / 482) / +++-+0)))) * -+5 + 7.513)))) - (+1 / ++((-84)))))))) * ++5 / +-(--2 - -++-9.0)))) / 5 * --++090 ++5 / +-(--2 - -++-9.0)))) / 5 * --++090 <start> := <expr> <expr> := <expr> '+' <expr> | <expr> '-' <expr> | <expr> '/' <expr> | <expr> '*' <expr> | '(' <expr> ')' | <number> <number> := <integer> | <integer> '.' <integer> <integer>:= <digit> <integer> | <digit> <digit> := [0-9] For Fuzzing (Hanford 1970) (Purdom 1972)
56. 56 Grammars As effective producers Interpreter Parser ✘ ✔ 8.2 - 27 - -9 / +((+9 * --2 + --+-+- ((-1 * +(8 - 5 - 6)) * (-(a-+(((+(4) )))) - ++4) / +(-+---((5.6 - --(3 * -1.8 * +(6 * +-(((-(-6) * ---+6)) / +--(+-+-7 * (-0 * (+(((((2)) + 8 - 3 - ++9.0 + ---(--+7 / (1 / +++6.37) + (1) / 482) / +++-+0)))) + 8.2 - 27 - -9 / +((+9 * --2 + --+-+-((-1 * + (8 - 5 - 6)) * (-(a-+(((+(4))))) - + +4) / +(-+---((5.6 - --(3 * -1.8 * + (6 * +-(((-(-6) * ---+6)) / +--(+-+- 7 * (-0 * (+(((((2)) + 8 - 3 - ++9.0 + ---(--+7 / (1 / +++6.37) + (1) / 482) / +++-+0)))) * -+5 + 7.513)))) - (+1 / ++((-84)))))))) * ++5 / +-(- -2 - -++-9.0)))) / 5 * --++090 + * - +5 + 7.513)))) - (+1 / ++((-84)))))) )) * 8.2 - 27 - -9 / +((+9 * --2 + - -+-+-((-1 * +(8 - 5 - 6)) * (-(a-+(( (+(4))))) - ++4) / +(-+---((5.6 - -- (3 * -1.8 * +(6 * +-(((-(-6) * ---+6 )) / +--(+-+-7 * (-0 * (+(((((2)) + 8 - 3 - ++9.0 + ---(--+7 / (1 / +++6 .37) + (1) / 482) / +++-+0)))) * -+5 + 7.513)))) - (+1 / ++((-84)))))))) * ++5 / +-(--2 - -++-9.0)))) / 5 * --++090 ++5 / +-(--2 - -++-9.0)))) / 5 * --++090
57. 57 Grammars <start> := <expr> <expr> := <expr> '+' <expr> | <expr> '-' <expr> | <expr> '/' <expr> | <expr> '*' <expr> | '(' <expr> ')' | <number> <number> := <integer> | <integer> '.' <integer> <integer>:= <digit> <integer> | <digit> <digit> := [0-9] As efficient producers def start(): expr() def expr(): match (random() % 6): case 0: expr(); print('+'); expr() case 1: expr(); print('-'); expr() case 2: expr(); print('/'); expr() case 3: expr(); print('*'); expr() case 4: print('('); expr(); print(')') case 5: number() def number(): match (random() % 2): case 0: integer() case 1: integer(); print('.'); integer() def integer(): match (random() % 2): case 0: digit(); integer() case 1: digit() def digit(): match (random() % 10): case 0: print('0') case 1: print('1') case 2: print('2') case 3: print('3') case 4: print('4') case 5: print('5') case 6: print('6') case 7: print('7') Compiled Grammar (F1)
58. http://localhost:8888/notebooks/x0_2_GrammarFuzzer.ipynb 58
59. 59 Where to Get the Input Grammar From?
60. 60 https://www.fuzzingbook.org/html/GrammarMiner.html AUTOGRAM Where to Get the Grammar From? Hand-written parsers already encode the grammar
61. http://localhost:8888/notebooks/x2_0_MiningGrammar.ipynb 61
62. 62 MIMID Where to Get the Grammar From? Hand-written parsers already encode the grammar
63. 63 Where to Get the Grammar From? 1. Extract the input string accesses 2. Attach control fl ow information Hand-written parsers already encode the grammar Each control- fl ow structure gets wrapped in a context-manager -If conditionals -Loops -Subroutines
64. 64 How to Extract This Grammar? • Inputs + control fl ow -> Dynamic Control Dependence Trees • DCD Trees -> Parse Tree
65. 65 Control Dependence Graph Statement B is control dependent on A if A determines whether B executes. def parse_csv(s,i): while s[i:]: if is_digit(s[i]): n,j = num(s[i:]) i = i+j else: comma(s[i]) i += 1 CDG for parse_csv while: determines whether if: executes
66. 66 def parse_csv(s,i): while s[i:]: if is_digit(s[i]): n,j = num(s[i:]) i = i+j else: comma(s[i]) i += 1 CDG for parse_csv Dynamic Control Dependence Tree Each statement execution is represented as a separate node DCD Tree for call parse_csv()
67. 67 def parse_csv(s,i): while s[i:]: if is_digit(s[i]): n,j = num(s[i:]) i = i+j else: comma(s[i]) i += 1 '1' '2' ',' DCD Tree ~ Parse Tree •No tracking beyond input bu ff er •Characters are attached to nodes where they are accessed last "12," "12,"
68. 68 •Characters are attached to nodes where they are accessed last
70. 70 def is_digit(i): return i in '0123456789' def parse_num(s,i): n = '' while s[i:] and is_digit(s[i]): n += s[i] i = i +1 return i,n def parse_paren(s, i): assert s[i] == '(' i, v = parse_expr(s, i+1) if s[i:] == '': raise Ex(s, i) assert s[i] == ')' return i+1, v def parse_expr(s, i = 0): expr, is_op = [], True while s[i:]: c = s[i] if isdigit(c): if not is_op: raise Ex(s,i) i,num = parse_num(s,i) expr.append(num) is_op = False elif c in ['+', '-', '*', '/']: if is_op: raise Ex(s,i) expr.append(c) is_op, i = True, i + 1 elif c == '(': if not is_op: raise Ex(s,i) i, cexpr = parse_paren(s, i) expr.append(cexpr) is_op = False elif c == ')': break else: raise Ex(s,i) if is_op: raise Ex(s,i) return i, expr 9+3/4 Parse tree for parse_expr('9+3/4')
72. 72 def is_digit(i): return i in '0123456789' def parse_num(s,i): n = '' while s[i:] and is_digit(s[i]): n += s[i] i = i +1 return i,n def parse_paren(s, i): assert s[i] == '(' i, v = parse_expr(s, i+1) if s[i:] == '': raise Ex(s, i) assert s[i] == ')' return i+1, v def parse_expr(s, i = 0): expr, is_op = [], True while s[i:]: c = s[i] if isdigit(c): if not is_op: raise Ex(s,i) i,num = parse_num(s,i) expr.append(num) is_op = False elif c in ['+', '-', '*', '/']: if is_op: raise Ex(s,i) expr.append(c) is_op, i = True, i + 1 elif c == '(': if not is_op: raise Ex(s,i) i, cexpr = parse_paren(s, i) expr.append(cexpr) is_op = False elif c == ')': break else: raise Ex(s,i) if is_op: raise Ex(s,i) return i, expr 9+3/4 Identifying Compatible Nodes Which nodes correspond to the same nonterminal
73. 73 (9 + 1) * 3 3 * (9 + 1)
74. 74 9 + 1 3 * (9 + 1)
75. 75 3 (9 + 1) * 3 * (9 + 1)
76. 76 3*(1) 1
77. 77 3*(1) 1 <parse_expr> := <while 1:1> <while 1:0> <while 1:1> <while 1:1> <parse_expr> :=
78. <parse_expr> := <while 1:1> <while 1:0> <while 1:1> | <while 1:1> <while 1:0> <while 1:1> <while 1:0> <while 1:1> <while 1:0> <while 1:1> | <while 1:1> <while 1:0> <while 1:1> <while 1:0> <while 1:1> | <while 1:1> <while 1:1> := <if 1:1> <if 1:1> := <parse_num> | <parse_paren> <parse_num> := <is_digit> <is_digit> := '3' | '1' <parse_paren>:= '(' <parse_expr> ')' <while 1:0> := <if 1:0> <if 1:0> := '*' 78
79. <parse_expr> := <while_s> <while_s> := <while_1:1> <while_1:0> <while_s> | <while_1:1> <parse_expr> := <while 1:1> <while 1:0> <while 1:1> | <while 1:1> <while 1:0> <while 1:1> <while 1:0> <while 1:1> <while 1:0> <while 1:1> | <while 1:1> <while 1:0> <while 1:1> <while 1:0> <while 1:1> | <while 1:1> <while 1:1> := <if 1:1> <if 1:1> := <parse_num> | <parse_paren> <parse_num> := <is_digit> <is_digit> := '3' | '1' <parse_paren>:= '(' <parse_expr> ')' <while 1:0> := <if 1:0> <if 1:0> := '*' 79 Generalizing Loops with Regular Inference
81. 81 def is_digit(i): return i in '0123456789' def parse_num(s,i): n = '' while s[i:] and is_digit(s[i]): n += s[i] i = i +1 return i,n def parse_paren(s, i): assert s[i] == '(' i, v = parse_expr(s, i+1) if s[i:] == '': raise Ex(s, i) assert s[i] == ')' return i+1, v def parse_expr(s, i = 0): expr, is_op = [], True while s[i:]: c = s[i] if isdigit(c): if not is_op: raise Ex(s,i) i,num = parse_num(s,i) expr.append(num) is_op = False elif c in ['+', '-', '*', '/']: if is_op: raise Ex(s,i) expr.append(c) is_op, i = True, i + 1 elif c == '(': if not is_op: raise Ex(s,i) i, cexpr = parse_paren(s, i) expr.append(cexpr) is_op = False elif c == ')': break else: raise Ex(s,i) if is_op: raise Ex(s,i) return i, expr <START> := <parse_expr.0-0-c> <parse_expr.0-0-c> := <parse_expr.0-1-s><parse_expr.0> | <parse_expr.0> <parse_expr.0-1-s> := <parse_expr.0><parse_expr.0-2> | <parse_expr.0><parse_expr.0-2><parse_expr.0-1-s> <parse_expr.0> := '(' <parse_expr.0-0-c> ')' | <parse_num.0-1-s> <parse_expr.0-2> := '*' | '+' | '-' | '/' <parse_num.0-1-s> := <is_digit.0-0-c> | <is_digit.0-0-c><parse_num.0-1-s> <is_digit.0-0-c> : [0-9] calc.py Recovered Arithmetic Grammar
82. 82 8.2 - 27 - -9 / +((+9 * --2 + --+-+- ((-1 * +(8 - 5 - 6)) * (-(a-+(((+(4) )))) - ++4) / +(-+---((5.6 - --(3 * -1.8 * +(6 * +-(((-(-6) * ---+6)) / +--(+-+-7 * (-0 * (+(((((2)) + 8 - 3 - ++9.0 + ---(--+7 / (1 / +++6.37) + (1) / 482) / +++-+0)))) + 8.2 - 27 - -9 / +((+9 * --2 + --+-+-((-1 * + (8 - 5 - 6)) * (-(a-+(((+(4))))) - + +4) / +(-+---((5.6 - --(3 * -1.8 * + (6 * +-(((-(-6) * ---+6)) / +--(+-+- 7 * (-0 * (+(((((2)) + 8 - 3 - ++9.0 + ---(--+7 / (1 / +++6.37) + (1) / 482) / +++-+0)))) * -+5 + 7.513)))) - (+1 / ++((-84)))))))) * ++5 / +-(- -2 - -++-9.0)))) / 5 * --++090 + * - +5 + 7.513)))) - (+1 / ++((-84)))))) )) * 8.2 - 27 - -9 / +((+9 * --2 + - -+-+-((-1 * +(8 - 5 - 6)) * (-(a-+(( (+(4))))) - ++4) / +(-+---((5.6 - -- (3 * -1.8 * +(6 * +-(((-(-6) * ---+6 )) / +--(+-+-7 * (-0 * (+(((((2)) + 8 - 3 - ++9.0 + ---(--+7 / (1 / +++6 .37) + (1) / 482) / +++-+0)))) * -+5 + 7.513)))) - (+1 / ++((-84)))))))) * ++5 / +-(--2 - -++-9.0)))) / 5 * --++090 ++5 / +-(--2 - -++-9.0)))) / 5 * --++090 <START> := <parse_expr.0-0-c> <parse_expr.0-0-c> := <parse_expr.0-1-s><parse_expr.0> | <parse_expr.0> <parse_expr.0-1-s> := <parse_expr.0><parse_expr.0-2> | <parse_expr.0><parse_expr.0-2><parse_expr.0-1-s> <parse_expr.0> := '(' <parse_expr.0-0-c> ')' | <parse_num.0-1-s> <parse_expr.0-2> := '*' | '+' | '-' | '/' <parse_num.0-1-s> := <is_digit.0-0-c> | <is_digit.0-0-c><parse_num.0-1-s> <is_digit.0-0-c> : [0-9]
83. 83 <START> ::= <json_raw> <json_raw> ::= '"' <json_string'> | '[' <json_list'> | '{' <json_dict'> | <json_number'> | 'true' | 'false' | 'null' <json_number'> ::= <json_number>+ | <json_number>+ 'e' <json_number>+ <json_number> ::= '+' | '-' | '.' | [0-9] | 'E' | 'e' <json_string'> ::= <json_string>* '"' <json_list'> ::= ']' | <json_raw> (','<json_raw>)* ']' | ( ',' <json_raw>)+ (',' <json_raw>)* ']' <json_dict'> ::= '}' | ( '"' <json_string'> ':' <json_raw> ',' )* '"'<json_string'> ':' <json_raw> '}' <json_string> ::= ' ' | '!' | '#' | '$' | '%' | '&' | ''' | '*' | '+' | '-' | ',' | '.' | '/' | ':' | ';' | '<' | '=' | '>' | '?' | '@' | '[' | ']' | '^' | '_', ''',| '{' | '|' | '}' | '~' | '[A-Za-z0-9]' | '\' <decode_escape> <decode_escape> ::= '"' | '/' | 'b' | 'f' | 'n' | 'r' | 't' stm.next() if expect_key: raise JSONError(E_DKEY, stm, stm.pos) if c == '}': return result expect_key = 1 continue # parse out a key/value pair elif c == '"': key = _from_json_string(stm) stm.skipspaces() c = stm.next() if c != ':': raise JSONError(E_COLON, stm, stm.pos) stm.skipspaces() val = _from_json_raw(stm) result[key] = val expect_key = 0 continue raise JSONError(E_MALF, stm, stm.pos) def _from_json_raw(stm): while True: stm.skipspaces() c = stm.peek() if c == '"': return _from_json_string(stm) elif c == '{': return _from_json_dict(stm) elif c == '[': return _from_json_list(stm) elif c == 't': return _from_json_fixed(stm, 'true', True, E_BOOL) elif c == 'f': return _from_json_fixed(stm, 'false', False, E_BOOL) elif c == 'n': return _from_json_fixed(stm, 'null', None, E_NULL) elif c in NUMSTART: return _from_json_number(stm) raise JSONError(E_MALF, stm, stm.pos) def from_json(data): stm = JSONStream(data) return _from_json_raw(stm) microjson.py Recovered JSON grammar