CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE

Transcription

1 CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE KOJI KOBAYASHI Abstract. This paper describes about relation between circuit compleit and accept inputs structure in Hamming space b using almost all monotone circuit that emulate deterministic Turing machine(dtm). Circuit famil that emulate DTM are almost all monotone circuit famil ecept some NOT-gate which connect input variables (like negation normal form (NNF)). Therefore, we can analze DTM limitation b using this NNF Circuit famil. NNF circuit famil cannot compute sandwich structure effectivel. Sandwich structure is two accept inputs that sandwich reject inputs in Hamming space. So NNF circuit have to use unique AND-gate to identif each different vector of sandwich structure. That is, we can measure problem compleit b counting different vectors. Some dicision problem have caracteristic in sandwich structure. Different vectors of Negate HornSAT problem are at most constant length because we can delete constant part of each negative literal in Horn clauses b using definite clauses. Therefore, number of these different vector is at most polnomial size. The other hand, Negate CNFSAT problem have different vector which number is over polnomial size. 1. Introduction In this paper, we consider the relation between circuit compleit and accept inputs structure in Hamming space b using almost all monotone circuit that emulate deterministic Turing machine(dtm), and confirm Negation HornSAT problem and Negation CNFSAT problem. In computational compleit, we use circuit famil to analze problem compleit, and we find out some result such as Date:

2 CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE 2 P ARIT Y / AC 0 [Ajtai, Furst], CLIQUE / mp monotone circuit famil with polnomial size [Razborov]. The purpose of this paper is to provide new approach to analze problem compleit b corresponding problem input structure in Hamming space and gate in circuit famil which emulate DTM. 2. NNF circuit famil First, we define NNF circuit famil that is almost all monotone circuit. Eplained in book [Sipser] Circuit Compleit section, Circuit famil can emulate DTM onl using NOT-gate in changing input values {0, 1} to {01, 10}. This almost all monotone circuit famil have simple structure like monotone circuit famil. Definition 2.1. We will use the terms; NNF Circuit Famil as circuit famil that have no NOT-gate ecept connecting INPUT-gates directl (like negation normal form). Input variable pair as output pair of INPUT-gate and NOT-gate {01, 10} that correspond to an input variable {0, 1}. Figure 2.1 is eample of a NNF circuit. Theorem 2.2. Let t : N N be a function where t (n) n. If A T IME (t (n)) then NNF circuit famil can emulate DTM that compute A with O ( t 2 (n) ) gate. Proof. This Proof is based on [Sipser] proof. See [Sipser] for detail. NNF circuit famil can emulate DTM b computing ever step s cell values (and head state if head on the cell). Figure 2.2 shows part of a NNF circuit block diagram. Input of this circuit is modified w 1 w n to c 1,1 c 1,n, and finall output result at c out = c t(n),1 cell. This circuit emulate DTM behavior, so c u,v compute cell s

3 CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE 3 Figure 2.1. NNF circuit state of step u from previous step cell c u 1,v and each side cells c u 1,v 1, c u 1,v+1 (because head affect at most side cells in each step). Figure 2.3 shows eample of c u,v sub circuit that transition function is if state is q k and tape value is 0, then move +1 and change state to q m. This circuit shows one of transition configuration which (c u 1,v 1, c u 1,v, c u 1,v+1 ) = (q k 0, q 0, q 0). q means no head on the cell. Each OR-gate w,q in c u,v correspond to ever step s cell condition (cell value w, and head status q if head eist on the c u,v cell), and output 1 if and onl if c u,v cell satisf corresponding condition. Previous step s output in c u 1,v 1, c u 1,v, c u 1,v+1 are connected to net step s AND-gate δ in c u,v with transition wire. Each δ correspond to transition function δ, and each δ output correspond to each transition function s result of c u,v. To simplif, NNF circuit include separate

4 CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE 4 Figure 2.2. NNF circuit block diagram three gates δ, 1, δ,0, δ,+1 according to head eists position c u 1,v 1, c u 1,v, c u 1,v+1, and special transition function δ which correspond to no head transition (keep current tape value). So δ in c u,v output 1 if and onl if previous step s output in c u 1,v 1, c u 1,v, c u 1,v+1 satisf transition function δ condition. Each transition functions affect (or do not affect) net step s condition, so δ output is connected to w,qm in c u,v and decide c u,v condition. Because DTM have constant number of transition functions, NNF can compute each step s cell b using constant number of AND-gates and OR-gates (without NOT-gate). First step s cells are handled in a special wa. Input is {0, 1} and above monotone circuit cannot manage 0 value. So NNF circuit compute {0, 1} {01, 10} b using NOT-gate.

5 CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE 5 Figure 2.3. c u,v circuit Corollar 2.3. NNF circuit famil can compute P problem with polnomial number of gates of input length. Confirm NNF circuit famil behavior. We define some term that decide relation of inputs. Definition 2.4. We will use the term; Neighbor input (pair) as accept inputs pair that no accept inputs eists between these accept input in Hamming space. Boundar input (set) of neighbor input as reject inputs that eist between neighbor inputs in Hamming space.

6 CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE 6 Figure 2.4. First step Different Variables as all difference part of values in neighbor input pair. Different vector as vector and inverse vector pair which start and end point is neighbor input pair in Hamming space. To simplif, we use 1 = 1. Neighbor distance as different vector length. Sandwich structure as connected graph which nodes are accept inputs in Hamming space. Figure 2.5 shows eample of sandwich structure which neighbor input pair is and In this case, and are different variables, and ( ) and ( ) are different vector, neighbor distance is 7. Effective circuit of accept input t as one of minimal sub circuit in NNF circuit that decide circuit output as 1 with accept input t. Effective circuit do not include

7 CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE 7 Figure 2.5. Sandwich structure gates which output 0, or even if gate change output 0 and effective circuit keep output 1. Figure 2.7 shows eample of effective circuit which circuit is 2.1 and input is { 1, 2, 3 } = {1, 1, 0}. Dotted gates do not affect OUTPUT-gate even if the gate negate output, so effective circuit do not include them. Theorem 2.5. All input variable pair of different variables join OR-gate in effective circuit. Proof. Because all input variable pair is {01, 10} and do not include 11 in ever input. NNF circuit is almost monotone circuit, so effective circuit have to to join another accept input at OR-gate to connect OUTPUT-gate. Figure 2.8 shows eample of effective circuit which circuit is 2.1 and input are { 1, 2, 3 } = {1, 1, 0}, {0, 0, 1}. Effective circuit include one of input variable pair,

8 CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE 8 Figure 2.6. Different vector and other side of variable pair do not become 1 in same input. So AND-gate cannot meet another effective circuit. Theorem 2.6. NNF circuit have at least one unique AND-gate which correspond to different vector to differentiate neighbor input and boundar input. Proof. Mentioned above 2.5, all accept input variable pair of different variables join at OR-gate. Because NNF circuit is almost all monotone circuit, there are a) b) case to join effective circuits; a) all different variables meet at AND-gate, and join at OR-gate after meeting AND-gate. (see 2.9) b) some partial different variables meet at AND-gate, and join at OR-gate these AND-gate output, and meet at AND-gate all OR-gate output. (see 2.8)

9 CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE 9 Figure 2.7. Effective circuit Case a), some AND-gate become 1 if and onl if input include one side of different variables. Therefore, trunk of these AND-gate does not become 1 if input AND-gate does not include these different variables. Each pair of different variables correspond to different vector, so the AND-gate correspond to different vector. Case b), because no boundar input become accept input, some OR-gate which join different variables become 0 if input is boundar input. That is, effective circuit become 0 if some of these OR-gate become 0, and become 1 if all of these OR-gate become 1. Therefore, it is necessar that effective circuit include ANDgate that meet all these OR-gate which join all different variables like 2.8. Such AND-gate become 1 if and onl if input include different variables of one side of neighbor input pair. Each pair of different variables correspond to different vector, so the AND-gate correspond to different vector.

10 CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE 10 Figure 2.8. Different variables pair Therefore, NNF circuit have at least one unique AND-gate that correspond to different vector to differentiate neighbor input and boundar input. NNF circuit can emulate DTM in polnomial size, and NNF circuit include unique AND-gate that correspond to different vector. Therefore, we can measure problem compleit b counting different vector in problem s sandwich structure. 3. Negation HornSAT Consider different vector in actual problems. Let consider Negation HornSAT problem HornSAT. HornSAT can delete som negative literal which correspond definite clauses. This means that each HornSAT input are close each other in Hamming space. In fact, we can close neighbor distance within constant distance b devising HornSAT description.

11 CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE 11 Figure 2.9. Eample of a) Definition 3.1. We will use the term HornSAT as problem if and onl if Horn CNF is. In HornSAT, we use special description as following; i : Variables in HornSAT. i in i is variable code, and in i is constant code. Negative literal is constant code which length is same as. i : Disabled Variables in HornSAT. i = in HornSAT. in i is constant code which length is same as. : Ignored code in HornSAT. Input that include is same as another input that trim all. All another smbol () are also constant code which length is same as. Theorem 3.2.

12 CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE 12 In HornSAT, there is some sandwich structure which neighbor distance is at most constant size, and number of different vector is at most polnomial size. Proof. Let t = i ( i ) HornSAT. can reduce another t = i ( i ) HornSAT because we can delete all literal i b using definite clauses i. Neighbor distance between t, t is constant because difference between i and i is constant part of,. Because all i =, we can reduce all i b overwriting at most constant size in each steps, and each neighbor distance are at most constant. That is, we can reduce t to t = i ( ) HornSAT with overwriting constant distance. The other hand, we can appl above steps all reduction of negative literals. When some clauses have no variables like ( ), we can overwrite an code in formula because the formule is. Therefore, all of HornSAT have neighbor input that distance is at most constant. Consider number of HornSAT different vectors. Let different distance is constant k. Because different distance is k, number of different vector is combination of different variables n k n k 2 k = n! k! (n k)! 2k O ( n k) and conbination of variables pair in constant code 2 k. Therefore we obtain theorem. 4. Negation CNFSAT Consider Negation CNFSAT problem CN F SAT. CN F SAT dose not have definite clauses, so we cannot delete negative literal like HornSAT. However, CNF SAT is smmetr that permutate truth value assignment, so CNF SAT is smmetr that permutate all literal of one variables. We can analze CN F SAT lower limit of different vector b using this literal smmetr. Definition 4.1. We will use the tenrm;

13 CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE 13 CNF SAT as problem if and onl if CNF is. In CNF SAT, we use special description like as HornSAT, and all of positive / negative literal described with flag code. That is, we can change literal positive / negative b overwriting constant code. To smplif, CNF SAT does not include same clauses. We treat CNF as set of clauses, and also treat clause as set of literals. f as permutate to in f. f f,, as permutate all literal, to, in f. as proper partical permutate, to, in f. (f do not decide specific permutation. So f permutation. ) is, Neighbor variable as variables that f, f f, f [f as f CNF SAT, f,,,,,,, means several partical is neighbor input. That / CNF SAT that add positive free literal in some f clause(s). g] as [g that add negative free literal in some g clause(s). [f g] as [f g]. However, each variables in [f, g],[f g] do not bind another variables, we appl alpha equivalence at f, g. f ( = ), f ( = ) as formula that appl =, in f. MUC and Minimal Unsatisfiable CNF as subset of CNF SAT that an CNF which delete an clauses do not become CNF SAT. That is, MUC = { f f CNF SAT, h f h / CNF SAT }

14 CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE 14 Figure 4.1. [f g] clauses Figure 4.1 shows eample of [f g] clauses. Theorem 4.2. f CNF SAT, f f CNF SAT Proof. It is trivial because CNF SAT is smmetric with permutation of truth value assignment. Theorem 4.3. [f ( = ) = f, [f ( = ) f g] ( = ) g, g] ( = ) = g f, g CNF SAT ( [f g] CNF SAT ) Proof. It is trivial that binding relation of adding literal, in [f, g].

15 CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE 15 Theorem 4.4. f, g MUC, [f g] CNF SAT Proof. Mentioned above 4.3, f [f g] g [f g] ( = ) CNF SAT Therefore we obtain theorem. ( = ) CNF SAT Theorem 4.5. f, g MUC, [f g] / CNF SAT Proof. Mentioned above , (considering smmetr of ) [f g] is 3 cases that; [f g] [f g] [f g] In [f g] case, if = then [f g] ( = ) = [f ( = ) g] ( = )

16 CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE 16 Some clauses in [f [f So ( = ) f f = f \ [f then [f ( = ) ( = ) = f \ f f and also g] g] ( = ) ( = ) = g \ g g include literal, then Because f, g MUC and f, g do not have same variables, there are some truth value assignment t that; [f g] = (f \ f g \ g ) (t) = ( = ) (t) Thatis [f g] Another case [f g] / CNF SAT [f g] also [f g] ( = )

17 CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE 17 [f g] ( = ) and [f g] / CNF SAT [f g] / CNF SAT Therefore we obtain theorem. Theorem 4.6. If f, g MUC, then variable in [f g] is neighbor variable. Proof. Mentioned above 4.4, if f, g CNF SAT then [f g] CNF SAT. Mentioned above 4.5, if f, g MUC then [f g] / CNF SAT.,, Therefore, if f, g MUC then [f g], [f g] and is neighbor variable. is neighbor input, Theorem 4.7. In CNF SAT, there is some sandwich structure which number of different vector is over polnomial size. Proof. Mentioned above 4.6, We can make [f g] that neighbor input is [f g] b using f, g MUC. We can select an clauses which add,, and [f g] include man clauses that number is over logarithm. So number of [f g] case is over polnomial size. The other hand, different vector of [f g], [f g] other if clauses that include is different. is different each

18 CIRCUIT COMPLEXITY AND PROBLEM STRUCTURE IN HAMMING SPACE 18 Therefore, [f g] make different vector that number is over polnomial, and we obtain theorem. References [Sipser] Michael Sipser, (translation) OHTA Kazuo, TANAKA Keisuke, ABE Masauki, UEDA Hiroki, FUJIOKA Atsushi, WATANABE Osamu, Introduction to the Theor of COMPUTATION Second Edition, 2008 [Ajtai] [Furst] Ajtai, M. Komlós, J. and Szemerédi, E.: An 0(n log n) sorting network, STOC(1983). Furst, M. Sae, J.B. and Sipser, M.: Parit, circuits, and the polnomial-time hierarch Mathematical Sstems Theor(1984) [Razborov] Razborov, A.: Lower bounds on the monotone compleit of some Boolean functions. Mathematics of the USSR(1985)