COSE 215: Theory of Computation. Undecidability (2) Hakjoo Oh 2016 Spring
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1 COSE 215: Theory of Computation Undecidability (2) Hakjoo Oh 2016 Spring
2 Contents Real Examples of undecidable problems Halting problem Program verification Properties of CFGs Post Correspondence Problem (PCP) How to deal with undecidable problems?
3 Halting Problem
4 Program Verification
5 Program Verification
6 Properties of CFGs Is a given CFG ambiguous? For CFGs G 1 and G 2,isL(G 1 ) \ L(G 2 )=;? Is L(G 1 )=L(G 2 )? Is L(G 1 )=L(R) for some regular expression R? Is L(G 1 )=T for some alphabet T?
7 Post Correspondence Problem Is there a list of dominos (repetitions permitted) such that reading off top yields the string obtained by reading off bottom? Possible: { b a ca abc,,, } ca ab a c Impossible: { abc ab, ca a, acc ba }
8 Programming Technologies Program Analysis Program Synthesis
9 27 int i; 28 for (i = 0; i <= 12; i++) txtbuf[i] = 0; junk = (char *)malloc(sizeof(char) * BUFLEN); 31 binbuf = (char *)malloc(sizeof(char)); if ((junk!= NULL) && (binbuf!= NULL)) { 34 isc_buffer_init(binbuf, junk, BUFLEN); 35 dns_name_init(dns_name, NULL); 36 dns_name_setbuffer(dns_name, binbuf); 37 result = dns_name_fromtext(dns_name, txtbuf, NULL, 0, NULL); 38 free(junk); 39 free(binbuf); Current Technology for Safe SW Manual, ad-hoc, postmortem: code review, testing, simulation, debugging, etc
10 Program Analysis Technology Technology for Software MRI prog.c prog.c
11 Aims to detect memory errors in C programs e.g., buffer-overrun, memory leak, null-dereference, etc Features (vs. testing) Full automation Find bugs early All bugs found
12 16 static char *curfinal = "HDACB FE"; keysym = read_from_input (); curfinal: buffer of size if ((((KeySym)(keysym) >= 0xFF91) keysym: && ((KeySym)(keysym) any integer <= 0xFF94))) 21 { 22 unparseputc((char)(keysym-0xff91 +'P'), pty); 23 key = 1; 24 } 25 else if (keysym >= 0) 26 { 27 if (keysym < 16) keysym: [0,15] 28 { 29 if (read_from_input()) 30 { 31 if (keysym >= 10) return; 32 curfinal[keysym] = 1; safe 33 } 34 else 35 { 36 curfinal[keysym] = 2; buffer-overrun 37 } 38 } 39 if (keysym < 10) 40 { 41 unparseputc(curfinal[keysym], pty); 42 } safe Sparrow automatically pinpoints the buffer-overrun bug curfinal:[10,10] keysym: [10,15] 43 }
13 Static Program Analysis Predict SW behavior statically and automatically static: before execution, before sell / embed automatic: sw is analyzed by sw ( static analyzers ) Applications bug-finding. e.g., find runtime failures of programs security. e.g., is this app malicious or benign? verification. e.g., does the program meet its specification? compiler optimization, e.g., automatic parallelization
14 14
15 How Program Analysis Works sound unsound vs.
16 How Program Analysis Works imprecise precise vs.
17 Program Synthesis Technology Currently, programs are written solely by programmers programming is often repetitive, tedious, and errorprone end-users are not capable of fully leveraging computational devices In the future, most programs will be written by programs
18 Program Synthesis Technology
19 ex) Regular Expressions
20 Synthesizing Regular Expressions ((0 + 1)(0 + 1)(0 + 1))* 0, 1 3
21 Synthesizing Regular Expressions * ( + )+ ( )+ ( * ) ( + ) 0+( )
22 Performance 3 ((0 + 1)(0 + 1)(0 + 1))* (1 (0 + 1)* 0)* (0 + 1(0 + 1))* ((0 + 1)* 01)* (1 + 01*0)* (0 + 1)(0 + 1)0(0 + 1)* ((0 + 1) + (0 + 1)1)* n 3 m 00*00(11)* (0 + 1)* 0101 (0 + 1)* 1.273
23 an help autoand structured uding problem eneration, and These tasks a wide variety ains including programming, d geometry. For s the output of technique for roof problems ple problem.. We have deworld applicahere are many ch IP can and ear future. Any t of high-level sts is a strong xample, the If service ( s end users to d programs usis an excellent on of IP. IFTTT gers (such as ) with actions message) over s Facebook. In e used to learn An Excel 2013 feature that automates repetitive string transformations using examples. Once the user performs one instance of the desired transformation (row 2, col. B) and proceeds to transforming another instance (row 3, col. B), Flash Fill learns a program Concatenate(ToLower(Substring(v,WordToken,1)), " ", ToLower(SubString(v,WordToken,2))), which extracts the first two words in input string v (col. A), converts them to lowercase, and concatenates them separated by a space character, to automate the repetitive task. FlashFill in Microsoft Excel
24 Synthesizing Programs [] 7! [] [[1]] 7! [[]] [[1, 3, 5], [5, 3, 2]] 7! [[3, 5], [5, 3]] [[8, 4, 7, 2], [4, 6, 2, 9], [3, 4, 1, 0]] 7! [[8, 4, 7] [4, 6, 9], [3, 4, 1]] 2.2 Transforming trees dropmins x = map f x where f y = filter g y where g z = foldl h False y where h t w = t (w < z)
25 Summary Many real problems are unsolvable by algorithms They can be approximately but usefully solved
26 Announcement Last class: 6/7 (Tuesday) Final exam: 6/9 (Thursday) in class
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