Mechanized Operational Semantics
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1 Mechanized Operational Semantics J Strother Moore Department of Computer Sciences University of Texas at Austin Marktoberdorf Summer School 2008 (Lecture 5: Boyer-Moore Fast String Searching) 1
2 The Problem One of the classic problems in computing is string searching : find the first occurrence of one character string ( the pattern ) in another ( the text ). Generally, the text is very large (e.g., gigabytes) but the patterns are relatively small. 2
3 Examples Find the word comedy in this NY Times article: Fred Armisen s office at Saturday Night Live is deceptively small, barely big enough to fit a desk, a couch, and an ipod. The glorified closet, the subject of a running joke on the comedy show, now in its 31st season, can simultaneously house a wisecracking... 3
4 AAAAAAAAAAAAACAAAGACAGGGGCAACAAAGTGAGACCCTAAAAAAAAAAAAACCCCA AAACGGAGAACTTGGAATCCTGTGTCCAAAAAAAAAAGCAGGAAGAGAGCGTGTAGAAAC TGAAGCTGAAGTGGAAAAAAAAAAGTCGCCAGCACCTACTGTGGAGACCAGAAAGGAAAA AAAAAATTGGCAGTCTCGTAGCATACCAAAACTAGGCTTGAAAAAAAAAACACACAAAAA AACACAGGCTACCCAGTATTTTATCGTCCAAAAAAAAAGAGGGAAGAAGGACATTTATAT TTGCCTTCTGCCAAAAAAAAAAGTACCTCCCGCCTAGAAGAGAGTTTAGAAATCACCAAA AAAAAATAGAGAGTCCCAAAATGTTCGGAATACTCAGAAAAAAAAATCTTAGTCAGTGCT CACTCAGAGGGACCGGGTATTTAAAAAAAACCTAGACCAGATGCAGCAGGTACAAATTAA TCAATCCCAAAAAAAAGACCTTCTACCCTTCCAAAAAATGATAGTTGTCTGCAATCCAAA AAAAAGACTCTCCGGAAGGTGGACATGCAGAACCTACCAAAAAAAAAGAGAAGAAAGAAT TGCCGGGCAAAAAGTTCCACGTAAAAAAAAAAGGAAATGGGAATGGAGTGTTGTTCTCCT TCCTACCTAGTTTTGAAAAAAAAGGATGGATGTGGGTCACCTGCTCACGTTCTCCAAAAA AAAGTGGGTGCTCTCTCACAATATTCTTAGAGGTGGCAAAAAAAATAAAGTTGATGGAAA CAGTACTGTGTGGGCCAAACAAAAAAAAAATGGCACCACCTTTTCATTGGCTGAAAAAAA AATTCAACTGAAAAACACAAGTCATACCTTCCTGTTTTATTTGCAAAAAAAATTTTTCAA ACCCCACGGCAACAAACGACAGTATCAAAAAAACAACTTCATTTGACATTCTGCTATATT AATGCTCTATGTGGAAAAAAAAACCATCAAGTTGTGCCTTTTTTCAAAGAAATCCATGCA AAAAAAAGACCCATGAAATAATTTTCTGGATCATCCATACAGAACCAAAAAAAAGAGGTG 4
5 C O M E D Y J O K E O N T H E C O M E D Y 5
6 C O M E D Y J O K E O N T H E C O M E D Y 6
7 C J J O M E D Y O K E O N T H E C O M E D Y 7
8 C O M E D Y J O K E O N T H E C O M E D Y 8
9 C O O M E D Y J O K E O N T H E C O M E D Y 9
10 C O M E D Y J O K E O N T H E C O M E D Y 10
11 C K O M E D Y J O K E O N T H E C O M E D Y 11
12 C O M E D Y J O K E O N T H E C O M E D Y 12
13 C O M E D Y J O K E O N T H E C O M E D Y 13
14 C O M E D Y J O K E O N T H E C O M E D Y 14
15 J O K E O N T H E C O M E D Y C O M E D Y C O M E D Y 15
16 C O M E D Y J O K E O N T H E C O M E D Y 16
17 C O M E D Y J O K E O N T H E C O M E D Y 17
18 C O M E D Y O J O K E O N T H E C O M E D Y 18
19 C O M E D Y O J O K E O N T H E C O M E D Y 19
20 C O M E D Y J O K E O N T H E C O M E D Y 20
21 C O M E D Y H J O K E O N T H E C O M E D Y 21
22 H C O M E D Y J O K E O N T H E C O M E D Y 22
23 C O M E D Y J O K E O N T H E C O M E D Y 23
24 C O M E D Y E J O K E O N T H E C O M E D Y 24
25 C O M E D Y E J O K E O N T H E C O M E D Y 25
26 C O M E D Y J O K E O N T H E C O M E D Y 26
27 C O M E D Y C O M E D Y J O K E O N T H E C O M E D Y Key Property: The longer the pattern, the faster the search! 27
28 Pre-Computing the Skip Distance pat: COMEDY txt: xxxxxoxxxxxxxxxxx... A 6 F 6 K 6 P 6 U 6 <space> 6 B 6 G 6 L 6 Q 6 V 6 C 5 H 6 M 3 R 6 W 6 D 1 I 6 N 6 S 6 X 6 E 2 J 6 O 4 T 6 Y 0 Z 6 28
29 This is a 1-dimensional array, skip[c], as big as the alphabet. 29
30 C O M E D Y J O K E O N T H E C O M E D Y skip[c]: A 6 F 6 K 6 P 6 U 6 <space> 6 B 6 G 6 L 6 Q 6 V 6 C 5 H 6 M 3 R 6 W 6 D 1 I 6 N 6 S 6 X 6 E 2 J 6 O 4 T 6 Y 0 Z 6 30
31 C O M E D Y J O K E O N T H E C O M E D Y skip[c]: A 6 F 6 K 6 P 6 U 6 <space> 6 B 6 G 6 L 6 Q 6 V 6 C 5 H 6 M 3 R 6 W 6 D 1 I 6 N 6 S 6 X 6 E 2 J 6 O 4 T 6 Y 0 Z 6 31
32 C O M E D Y O J O K E O N T H E C O M E D Y skip[c]: A 6 F 6 K 6 P 6 U 6 <space> 6 B 6 G 6 L 6 Q 6 V 6 C 5 H 6 M 3 R 6 W 6 D 1 I 6 N 6 S 6 X 6 E 2 J 6 O 4 T 6 Y 0 Z 6 32
33 C O M E D Y O J O K E O N T H E C O M E D Y skip[c]: A 6 F 6 K 6 P 6 U 6 <space> 6 B 6 G 6 L 6 Q 6 V 6 C 5 H 6 M 3 R 6 W 6 D 1 I 6 N 6 S 6 X 6 E 2 J 6 O 4 T 6 Y 0 Z 6 33
34 C O M E D Y J O K E O N T H E C O M E D Y skip[c]: A 6 F 6 K 6 P 6 U 6 <space> 6 B 6 G 6 L 6 Q 6 V 6 C 5 H 6 M 3 R 6 W 6 D 1 I 6 N 6 S 6 X 6 E 2 J 6 O 4 T 6 Y 0 Z 6 34
35 C O M E D Y H J O K E O N T H E C O M E D Y skip[c]: A 6 F 6 K 6 P 6 U 6 <space> 6 B 6 G 6 L 6 Q 6 V 6 C 5 H 6 M 3 R 6 W 6 D 1 I 6 N 6 S 6 X 6 E 2 J 6 O 4 T 6 Y 0 Z 6 35
36 H C O M E D Y J O K E O N T H E C O M E D Y skip[c]: A 6 F 6 K 6 P 6 U 6 <space> 6 B 6 G 6 L 6 Q 6 V 6 C 5 H 6 M 3 R 6 W 6 D 1 I 6 N 6 S 6 X 6 E 2 J 6 O 4 T 6 Y 0 Z 6 36
37 C O M E D Y J O K E O N T H E C O M E D Y skip[c]: A 6 F 6 K 6 P 6 U 6 <space> 6 B 6 G 6 L 6 Q 6 V 6 C 5 H 6 M 3 R 6 W 6 D 1 I 6 N 6 S 6 X 6 E 2 J 6 O 4 T 6 Y 0 Z 6 37
38 C O M E D Y E J O K E O N T H E C O M E D Y skip[c]: A 6 F 6 K 6 P 6 U 6 <space> 6 B 6 G 6 L 6 Q 6 V 6 C 5 H 6 M 3 R 6 W 6 D 1 I 6 N 6 S 6 X 6 E 2 J 6 O 4 T 6 Y 0 Z 6 38
39 C O M E D Y E J O K E O N T H E C O M E D Y skip[c]: A 6 F 6 K 6 P 6 U 6 <space> 6 B 6 G 6 L 6 Q 6 V 6 C 5 H 6 M 3 R 6 W 6 D 1 I 6 N 6 S 6 X 6 E 2 J 6 O 4 T 6 Y 0 Z 6 39
40 C O M E D Y J O K E O N T H E C O M E D Y skip[c]: A 6 F 6 K 6 P 6 U 6 <space> 6 B 6 G 6 L 6 Q 6 V 6 C 5 H 6 M 3 R 6 W 6 D 1 I 6 N 6 S 6 X 6 E 2 J 6 O 4 T 6 Y 0 Z 6 40
41 C O M E D Y C O M E D Y J O K E O N T H E C O M E D Y skip[c]: A 6 F 6 K 6 P 6 U 6 <space> 6 B 6 G 6 L 6 Q 6 V 6 C 5 H 6 M 3 R 6 W 6 D 1 I 6 N 6 S 6 X 6 E 2 J 6 O 4 T 6 Y 0 Z 6 41
42 But Wait! There s More! pat: NONPARTIPULAR txt:
43 But Wait! There s More! pat: NONPARTIPULAR txt: R
44 But Wait! There s More! pat: NONPARTIPULAR txt: A
45 But Wait! There s More! pat: NONPARTIPULAR txt: P
46 But Wait! There s More! pat: NONPARTIPULAR txt: P Slide 2 to match the discovered character. 46
47 But Wait! There s More! pat: NONPARTIPULAR txt: P Slide 2 to match the discovered character. 47
48 But Wait! There s More! pat: NONPARTIPULAR txt: P??
49 But Wait! There s More! pat: NONPARTIPULAR txt: PAR
50 But Wait! There s More! pat: NONPARTIPULAR txt:
51 But Wait! There s More! pat: NONPARTIPULAR txt: R
52 But Wait! There s More! pat: NONPARTIPULAR txt: AR
53 But Wait! There s More! pat: NONPARTIPULAR txt: PAR
54 But Wait! There s More! pat: NONPARTIPULAR txt: PAR
55 But Wait! There s More! pat: NONPARTIPULAR txt: PAR Slide 7 to match the discovered substring! 55
56 But Wait! There s More! pat: NONPARTIPULAR txt: PAR Slide 7 to match the discovered substring! There are only α pat combinations, where α is the alphabet size. We can still pre-compute the skip distance. 56
57 The Delta Array delta[c,j] is an array of size α pat that gives the skip distance when a mismatch occurs after comparing c from txt to pat[j]. 57
58 The Algorithm fast(pat, txt) If pat = "" then If txt = "" then return Not-Found; else return 0; end; end; 58
59 preprocess pat to produce delta; j := pat 1; i := j; 59
60 while (0 j i < txt ) do If pat[j] = txt[i] then i := i 1; j := j 1; else i := i + delta[txt[i], j]; j := pat 1; end; 60
61 If (j < 0) then return i + 1; else return Not-Found; end; end; 61
62 Performance How does the algorithm perform? In our test: txt: English text of length 177,985. pat: 100 randomly chosen patterns of length 5 30, chosen from another English text and filtered so they do not occur in the search text. 62
63 63
64 Naive algorithm would be a line at 180,000 reads. 64
65 65
66 Goal Prove the correctness of an M1 program for the Boyer-Moore fast string searching algorithm. We will not code the preprocessing in M1. We will write code for the Boyer-Moore algorithm that assumes that the contents of a certain local contains a 2-dimensional delta array. We will initialize the array variable with ACL2 code, not M1 code. 66
67 We will proceed as previously advised: Step 1: prove that the code implements the algorithm Step 2: prove that the algorithm implements the spec We ll do Step 2 first. It s always the hardest. 67
68 Demo 1 68
69 The Obviously Correct Algorithm (defun correct-loop (pat txt i) (cond ((>= i (length txt)) nil) ((matchp pat 0 txt i) i) (t (correct-loop pat txt (+ 1 i))))) (defun correct (pat txt) (correct-loop pat txt 0)) (I omit type-like tests here.) 69
70 The Fast Algorithm (defun fast-loop (pat j txt i) (declare :measure (measure pat j txt i) :well-founded-relation l<)) (cond... ((equal (char pat j) (char txt i)) (fast-loop pat (- j 1) txt (- i 1))) (t (fast-loop pat (- (length pat) 1) txt (+ i (delta (char txt i) j pat)))))) 70
71 (defun fast (pat txt) (if (equal pat "") (if (equal txt "") nil 0) (fast-loop pat (- (length pat) 1) txt (- (length pat) 1)))) 71
72 Step 2: Fast Algorithm is Correct (defthm fast-is-correct (implies (and (stringp pat) (stringp txt)) (equal (fast pat txt) (correct pat txt)))) 72
73 Decomposition (a) correct-loop can skip ahead if there are no matches in the region skipped (b) there are no matches in the region skipped by the delta computation. 73
74 Summary of Step 2 A total of 9 definitions and lemmas are proved to establish (defthm fast-is-correct (implies (and (stringp pat) (stringp txt)) (equal (fast pat txt) (correct pat txt)))) (On top of a library of useful utilities having nothing to do with this problem.) 74
75 Step 1 (defconst *m1-boyer-moore-program* ; Allocation of locals ; pat 0 ; j 1 ; txt 2 ; i 3 ; pmax 4 = (length pat) ; tmax 5 = (length txt) ; array 6 = (preprocess pat) ; c 7 = temp - last char read from txt ( (load 0) ; 0 (load pat) (push "") ; 1 (push "") 75
76 (ifane 5) ; 2 (ifane loop) (load 2) ; 3 (load txt) (push "") ; 4 (push "") (ifane 40) ; 5 (ifane win) (goto 43) ; 6 (goto lose) ; loop: (load 1) ; 7 (load j) (iflt 37) ; 8 (iflt win)) (load 5) ; 9 (load tmax) (load 3) ; 10 (load i) (sub) ; 11 (sub) (ifle 37) ; 12 (ifle lose) (load 0) ; 13 (load pat) (load 1) ; 14 (load j) (aload) ; 15 (aload) (load 2) ; 16 (load txt) (load 3) ; 17 (load i) (aload) ; 18 (aload) (store 7) ; 19 (store c) 76
77 (load 7) ; 20 (load c) (sub) ; 21 (sub) (ifne 10) ; 22 (ifne skip) (load 1) ; 23 (load j) (push 1) ; 24 (push 1) (sub) ; 25 (sub) (store 1) ; 26 (store j) (load 3) ; 27 (load i) (push 1) ; 28 (push 1) (sub) ; 29 (sub) (store 3) ; 30 (store i) (goto -24) ; 31 (goto loop) ; skip: (load 3) ; 32 (load i) (load 6) ; 33 (load array) (load 7) ; 34 (load c) (aload) ; 35 (aload) (load 1) ; 36 (load j) (aload) ; 37 (aload) 77
78 (add) ; 38 (add) (store 3) ; 39 (store i) (load 4) ; 40 (load pmax) (push 1) ; 41 (push 1) (sub) ; 42 (sub) (store 1) ; 43 (store j) (goto -37) ; 44 (goto loop) ; win: (load 3) ; 45 (load i) (push 1) ; 46 (push 1) (add) ; 47 (add) (return) ; 48 (return) ; lose: (push nil) ; 49 (push nil) (return) ) ; 50 (return)) ) 78
79 The Schedule How do we define the schedule for such a complicated piece of code? 79
80 The Schedule (defun m1-boyer-moore-loop-sched (pat j txt i) (cond ((< j 0) (repeat 0 6)) ((<= (length txt) i) (repeat 0 8)) ((equal (char-code (char pat j)) (char-code (char txt i))) (append (repeat 0 25) (m1-boyer-moore-loop-sched pat (- j 1) txt (- i 1)))) (t (append (repeat 0 29) (m1-boyer-moore-loop-sched pat (- (length pat) 1) txt (+ i (delta (char txt i) j pat)))))) 80
81 The Schedule (defun m1-boyer-moore-loop-sched (pat j txt i) (cond ((< j 0) (repeat 0 6)) ((<= (length txt) i) (repeat 0 8)) ((equal (char-code (char pat j)) (char-code (char txt i))) (append (repeat 0 25) (m1-boyer-moore-loop-sched pat (- j 1) txt (- i 1)))) (t (append (repeat 0 29) (m1-boyer-moore-loop-sched pat (- (length pat) 1) txt (+ i (delta (char txt i) j pat)))))) 81
82 (defun m1-boyer-moore-sched (pat txt) (if (equal pat "") (if (equal txt "") (repeat 0 9) (repeat 0 10)) (append (repeat 0 3) (m1-boyer-moore-loop-sched pat (- (length pat) 1) txt (- (length pat) 1))))) 82
83 The Schedule Defining the schedule is trivial if you have verified the algorithm. They have identical recursive structure and justification. 83
84 (defthm m1-boyer-moore-is-fast (implies (and (stringp pat) (stringp txt)) (equal (top (stack (run (m1-boyer-moore-sched pat txt) (make-state 0 (list pat (- (length pat) 1) txt (- (length pat) 1) (length pat) (length txt) (preprocess pat) 0) nil *m1-boyer-moore-program*)))) (fast pat txt)))) 84
85 (defthm m1-boyer-moore-halts (implies (and (stringp pat) (stringp txt)) (haltedp (run (m1-boyer-moore-sched pat txt) (make-state 0 (list pat (- (length pat) 1) txt (- (length pat) 1) (length pat) (length txt) (preprocess pat) 0) nil *m1-boyer-moore-program*))))) 85
86 Main Theorem Given the two steps: Step 1: The code computes the same thing as the function fast Step 2: The function fast computes the same as correct It is trivial to show 86
87 (defthm m1-boyer-moore-is-correct (implies (and (stringp pat) (stringp txt)) (equal (top (stack (run (m1-boyer-moore-sched pat txt) (make-state 0 (list pat (- (length pat) 1) txt (- (length pat) 1) (length pat) (length txt) (preprocess pat) 0) nil *m1-boyer-moore-program*)))) (correct pat txt)))) 87
88 Conclusion Mechanized operational (interpretive) semantics are entirely within a logical framework and so permit logical analysis of programs by traditional formal proofs, without introduction of meta-logical transformers (VCGs) are generally executable are easily related to implementations allow derivation of language properties 88
89 may allow derivation of intensional properties (e.g., how many steps a program takes to terminate) allow verification of system hierarchies (multiple layers of abstraction can be formalized and related within the proof system) 89
90 Thank You 90
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