Non-Obfuscated Yet Unprovable Programs John Case Michael Ralston Computer and Information Sciences Department University of Delaware Newark, DE 19716 USA Email: { case , mralston } @udel.edu Revision of Talk at Asian Logic Conference 2011 Wellington, NZ
For Your Speed Reading Pleasure & Quick Impression ( .. ⌣ ) 1 Introduction Background Mathematical Preliminaries I Mathematical Preliminaries II Mathematical Preliminaries III Mathematical Preliminaries IV 2 Results Main Result Proof of Main Result Next Results 3 References J. Case & M. Ralston (CIS Dept, UD) Non-Obfuscated Unprovable Programs ALC 2011, Wellington 2 / 11
Introduction Background Background The International Obfuscated C Code Contest was a programming contest for the most creatively obfuscated C code, held annually between 1984 and 1996, and thereafter in 1998, 2000, 2001, 2004, and 2006. In many cases, the winning programmer did something simple in such an obscure but succinct way that it was hard for other (human) programmers to see how his/her code actually worked. By contrast, the interest herein is in programs which are, in a sense, easily seen to be correct, but which cannot be proved correct in pre-assigned, computably axiomatized, powerful, true theories T . For any deterministic, multi-tape TM program p , there will be an easily seen equivalent such program q almost (i.e., within small linear factors) as fast and succinct as p , but this equivalence will not be provable in T . My orig. motive: [Put80] says ≈ the short-fast (s-f) programs p for prop. calc. well-formedness are similar/intrinsic, but my s-f corr. q isn’t. Next frames: complexity-bounded computability. J. Case & M. Ralston (CIS Dept, UD) Non-Obfuscated Unprovable Programs ALC 2011, Wellington 3 / 11
Introduction Background Background The International Obfuscated C Code Contest was a programming contest for the most creatively obfuscated C code, held annually between 1984 and 1996, and thereafter in 1998, 2000, 2001, 2004, and 2006. In many cases, the winning programmer did something simple in such an obscure but succinct way that it was hard for other (human) programmers to see how his/her code actually worked. By contrast, the interest herein is in programs which are, in a sense, easily seen to be correct, but which cannot be proved correct in pre-assigned, computably axiomatized, powerful, true theories T . For any deterministic, multi-tape TM program p , there will be an easily seen equivalent such program q almost (i.e., within small linear factors) as fast and succinct as p , but this equivalence will not be provable in T . My orig. motive: [Put80] says ≈ the short-fast (s-f) programs p for prop. calc. well-formedness are similar/intrinsic, but my s-f corr. q isn’t. Next frames: complexity-bounded computability. J. Case & M. Ralston (CIS Dept, UD) Non-Obfuscated Unprovable Programs ALC 2011, Wellington 3 / 11
Introduction Background Background The International Obfuscated C Code Contest was a programming contest for the most creatively obfuscated C code, held annually between 1984 and 1996, and thereafter in 1998, 2000, 2001, 2004, and 2006. In many cases, the winning programmer did something simple in such an obscure but succinct way that it was hard for other (human) programmers to see how his/her code actually worked. By contrast, the interest herein is in programs which are, in a sense, easily seen to be correct, but which cannot be proved correct in pre-assigned, computably axiomatized, powerful, true theories T . For any deterministic, multi-tape TM program p , there will be an easily seen equivalent such program q almost (i.e., within small linear factors) as fast and succinct as p , but this equivalence will not be provable in T . My orig. motive: [Put80] says ≈ the short-fast (s-f) programs p for prop. calc. well-formedness are similar/intrinsic, but my s-f corr. q isn’t. Next frames: complexity-bounded computability. J. Case & M. Ralston (CIS Dept, UD) Non-Obfuscated Unprovable Programs ALC 2011, Wellington 3 / 11
Introduction Background Background The International Obfuscated C Code Contest was a programming contest for the most creatively obfuscated C code, held annually between 1984 and 1996, and thereafter in 1998, 2000, 2001, 2004, and 2006. In many cases, the winning programmer did something simple in such an obscure but succinct way that it was hard for other (human) programmers to see how his/her code actually worked. By contrast, the interest herein is in programs which are, in a sense, easily seen to be correct, but which cannot be proved correct in pre-assigned, computably axiomatized, powerful, true theories T . For any deterministic, multi-tape TM program p , there will be an easily seen equivalent such program q almost (i.e., within small linear factors) as fast and succinct as p , but this equivalence will not be provable in T . My orig. motive: [Put80] says ≈ the short-fast (s-f) programs p for prop. calc. well-formedness are similar/intrinsic, but my s-f corr. q isn’t. Next frames: complexity-bounded computability. J. Case & M. Ralston (CIS Dept, UD) Non-Obfuscated Unprovable Programs ALC 2011, Wellington 3 / 11
Introduction Background Background The International Obfuscated C Code Contest was a programming contest for the most creatively obfuscated C code, held annually between 1984 and 1996, and thereafter in 1998, 2000, 2001, 2004, and 2006. In many cases, the winning programmer did something simple in such an obscure but succinct way that it was hard for other (human) programmers to see how his/her code actually worked. By contrast, the interest herein is in programs which are, in a sense, easily seen to be correct, but which cannot be proved correct in pre-assigned, computably axiomatized, powerful, true theories T . For any deterministic, multi-tape TM program p , there will be an easily seen equivalent such program q almost (i.e., within small linear factors) as fast and succinct as p , but this equivalence will not be provable in T . My orig. motive: [Put80] says ≈ the short-fast (s-f) programs p for prop. calc. well-formedness are similar/intrinsic, but my s-f corr. q isn’t. Next frames: complexity-bounded computability. J. Case & M. Ralston (CIS Dept, UD) Non-Obfuscated Unprovable Programs ALC 2011, Wellington 3 / 11
Introduction Background Background The International Obfuscated C Code Contest was a programming contest for the most creatively obfuscated C code, held annually between 1984 and 1996, and thereafter in 1998, 2000, 2001, 2004, and 2006. In many cases, the winning programmer did something simple in such an obscure but succinct way that it was hard for other (human) programmers to see how his/her code actually worked. By contrast, the interest herein is in programs which are, in a sense, easily seen to be correct, but which cannot be proved correct in pre-assigned, computably axiomatized, powerful, true theories T . For any deterministic, multi-tape TM program p , there will be an easily seen equivalent such program q almost (i.e., within small linear factors) as fast and succinct as p , but this equivalence will not be provable in T . My orig. motive: [Put80] says ≈ the short-fast (s-f) programs p for prop. calc. well-formedness are similar/intrinsic, but my s-f corr. q isn’t. Next frames: complexity-bounded computability. J. Case & M. Ralston (CIS Dept, UD) Non-Obfuscated Unprovable Programs ALC 2011, Wellington 3 / 11
Introduction Preliminaries I Mathematical Preliminaries I Let ϕ TM be the efficiently laid out and G¨ odel-numbered acceptable programming system (numbering) from [RC94] and which is based on deterministic multi-tape Turing Machines (with base 2 I/O). Its programs are named by numbers in N def = { 0 , 1 , 2 , . . . } . The numerical naming mentioned above does not feature prime powers and factorization, but, instead, is a linear-time computable and invertible coding. Let Φ TM be the corresponding step-counting Blum Complexity Measure. For p , a numerically named program in ϕ TM , let | p | = the length of p written in binary, which = ( ⌈ log 2 ( p + 1) ⌉ ) + . ( · ) + turns 0 into 1; else, leaves unchanged. J. Case & M. Ralston (CIS Dept, UD) Non-Obfuscated Unprovable Programs ALC 2011, Wellington 4 / 11
Introduction Preliminaries I Mathematical Preliminaries I Let ϕ TM be the efficiently laid out and G¨ odel-numbered acceptable programming system (numbering) from [RC94] and which is based on deterministic multi-tape Turing Machines (with base 2 I/O). Its programs are named by numbers in N def = { 0 , 1 , 2 , . . . } . The numerical naming mentioned above does not feature prime powers and factorization, but, instead, is a linear-time computable and invertible coding. Let Φ TM be the corresponding step-counting Blum Complexity Measure. For p , a numerically named program in ϕ TM , let | p | = the length of p written in binary, which = ( ⌈ log 2 ( p + 1) ⌉ ) + . ( · ) + turns 0 into 1; else, leaves unchanged. J. Case & M. Ralston (CIS Dept, UD) Non-Obfuscated Unprovable Programs ALC 2011, Wellington 4 / 11
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