M. Otto. Ptime canonization for two variables with counting. In Proc. 10th IEEE Symp. on Logic in Computer Science, pages 342-352, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....are of the order O(n) there are constructions known that raise this to at least 2 . Once again, we use the fact that our signatures are purely relational. It should also be pointed out that the case of k = 2 is solved, as linear lower and upper bounds are essentially established in [17]. Thus, we adopt the following proviso for the rest of the paper: Wherever k refers to the number of variables, we assume that k 3. 4 Trees The exponential lower bounds on the functions d k and e k can be established by considering the class of complete binary trees. We introduce this class ....

M. Otto. Ptime canonization for two variables with counting. In Proc. 10th IEEE Symp. on Logic in Computer Science, pages 342-352, 1995.

....of L(A) from page 349) FINITE VARIABLE LOGICS IN DESCRIPTIVE COMPLEXITY THEORY 369 We define an inverse f of I k # by letting f(I) be the first structure A with respect to # of size at most g( I ) with I k (f(I) I if such a structure exists, and arbitrary otherwise. Otto [62] proved that for k = 2 the answer to Questions 1 and 2 is yes . Actually, he proved more: Theorem 5.1 (Otto [62] For each vocabulary #, there is a PTIME compu table inverse for I 2 # . If we are only interested in a recursive rather than a PTIME computable inverse of I 2 # , we can ....

....f of I k # by letting f(I) be the first structure A with respect to # of size at most g( I ) with I k (f(I) I if such a structure exists, and arbitrary otherwise. Otto [62] proved that for k = 2 the answer to Questions 1 and 2 is yes . Actually, he proved more: Theorem 5. 1 (Otto [62]) For each vocabulary #, there is a PTIME compu table inverse for I 2 # . If we are only interested in a recursive rather than a PTIME computable inverse of I 2 # , we can proceed as follows, as has been observed by Flum and Ziegler [23] and independently by Otto [64] Given the invariant ....

[Article contains additional citation context not shown here]

M. Otto, Ptime canonization for two variables with counting, Proceedings of the 10th IEEE symposium on logic in computer science, 1995, pp. 342--352.

....for all structures A, the function f maps I k (A) to a structure that has the same L k theory as A If such an f exists, we say that I k is recursively invertible on structures. Equivalently, we can ask if the image of the class of structures under I k is a recursive set. Otto [14] proved that I 2 is invertible for all vocabularies , even in polynomial time. It is easy to see that I 1 , though somewhat pathological, is also invertible in polynomial time. Theorem 1. Let k 3 and a vocabulary that contains at least one (k 1) ary relation symbol. Then I k is not ....

....that contains at least one (k 1) ary relation symbol. Then there is no recursive function f such that for all L k theories T of vocabulary we have f k size(T) min jAj A j= Tg: The best previously known lower bound for such a function was simply exponential [5] Otto [14] showed that there is a linear bound on the size of the smallest model of an L 2 theory in terms of its 2 size. Dawar [3] pointed out that another corollary of Theorem 1 is the separation of the intersection of partial fixed point logic with P from the intersection of L 1 with P. Our ....

[Article contains additional citation context not shown here]

M. Otto. Ptime canonization for two variables with counting. In Proceedings of the 10th IEEE Symposium on Logic in Computer Science, pages 342--352, 1995.

....of the finite sets L(A) f0; 1g (recall the definition of L(A) from page 4) We define an inverse f of I k by letting 20 MARTIN GROHE f(I) be the first structure A with respect to of size at most g(jI j) with I k (f(I) I if such a structure exists, and arbitrary otherwise. Otto [62] proved that for k = 2 the answer to Questions 1 and 2 is yes . Actually, he proved more: Theorem 5.1 (Otto [62] For each vocabulary , there is a PTIME computable inverse for I 2 . If we are only interested in a recursive rather than a PTIME computable inverse of I 2 , we can ....

.... by letting 20 MARTIN GROHE f(I) be the first structure A with respect to of size at most g(jI j) with I k (f(I) I if such a structure exists, and arbitrary otherwise. Otto [62] proved that for k = 2 the answer to Questions 1 and 2 is yes . Actually, he proved more: Theorem 5. 1 (Otto [62]) For each vocabulary , there is a PTIME computable inverse for I 2 . If we are only interested in a recursive rather than a PTIME computable inverse of I 2 , we can proceed as follows, as has been observed by Flum and Ziegler [23] and independently by Otto [64] Given the invariant I ....

[Article contains additional citation context not shown here]

M. Otto. Ptime canonization for two variables with counting. In Proceedings of the 10th IEEE Symposium on Logic in Computer Science, pages 342--352, 1995.

....interesting here because they approximate isomorphism in the following sense: With increasing k, L k equivalence becomes finer. The intersection of all L k equivalence relations is precisely equivalence in first order logic, which is the same as isomorphism on finite structures. Martin Otto [14] proved that L 2 equivalence admits P canonization and thus that there is a logic capturing L 2 ( equivalence) invariant P. Our main result shows that this approach seems to fail for more than two variables. A definition of non uniform P, denoted by P=poly, will be given in the ....

....Lindell, and Weinstein [4] suggested to invert I k in P to obtain a canonization function for L k equivalence and thus a logic that captures L k invariant P. An inversion of I k is a mapping f such that for all structures A the structure f(I k (A) is L k equivalent to A. Otto [14] obtained his aforementioned result for L 2 along these lines. Unfortunately, for k 3 the mapping I k is not even recursively invertible [8] But Dawar et al. 4] had already observed that it is not possible, but also not necessary, to invert I k in time polynomial in the size of the given ....

M. Otto. Ptime canonization for two variables with counting. In Proceedings of the 10th IEEE Symposium on Logic in Computer Science, pages 342--352, 1995.

....of the finite sets L(A) f0; 1g (recall the definition of L(A) from page 4) We define an inverse f of I k by letting 20 MARTIN GROHE f(I) be the first structure A with respect to of size at most g(jI j) with I k (f(I) I if such a structure exists, and arbitrary otherwise. Otto [62] proved that for k = 2 the answer to Questions 1 and 2 is yes . Actually, he proved more: Theorem 5.1 (Otto [62] For each vocabulary , there is a PTIME computable inverse for I 2 . If we are only interested in a recursive rather than a PTIME computable inverse of I 2 , we can ....

....k by letting 20 MARTIN GROHE f(I) be the first structure A with respect to of size at most g(jI j) with I k (f(I) I if such a structure exists, and arbitrary otherwise. Otto [62] proved that for k = 2 the answer to Questions 1 and 2 is yes . Actually, he proved more: Theorem 5. 1 (Otto [62]) For each vocabulary , there is a PTIME computable inverse for I 2 . If we are only interested in a recursive rather than a PTIME computable inverse of I 2 , we can proceed as follows, as has been observed by Flum and Ziegler [23] and independently by Otto [64] Given the invariant I ....

[Article contains additional citation context not shown here]

M. Otto. Ptime canonization for two variables with counting. In Proceedings of the 10th IEEE Symposium on Logic in Computer Science, pages 342--352, 1995.

....f maps I k (A) to a structure that has the same L k theory as A More formally, we ask for a recursive function f such that I k ffi f ffi I k = I k . If such an f exists, we say that I k is recursively invertible. Equivalently, we can ask if the image of I k is a recursive set. Otto [8] proved that I 2 is invertible, even in polynomial time. It is easy to see that I 1 , though somewhat pathological, is also invertible in polynomial time. Our main result Theorem 1.1 implies that for k 3 the invariant I k is not recursively invertible. There are two main reasons for the ....

....R be a (k Gamma 1) ary relation symbol. Then there is no recursive function f such that for all L k theories T of vocabulary fRg we have f Gamma k size(T ) Delta min Phi jAj fi fi A j= Tg: The best previously known lower bound for such a function was simply exponential [4] Otto [8] showed that there is a linear bound on the size of the smallest model of an L 2 theory in terms of its 2 size. Otto [9] also noticed that the questions of whether there is a recursive bound on the size of the smallest model of an L k theory in terms of its k size and whether the ....

[Article contains additional citation context not shown here]

M. Otto. Ptime canonization for two variables with counting. In Proceedings of the 10th IEEE Symposium on Logic in Computer Science, pages 342--352, 1995.

....interesting here because they approximate isomorphism in the following sense: With increasing k, L k equivalence becomes finer. The intersection of all L k equivalence relations is precisely equivalence in first order logic, which is the same as isomorphism on finite structures. Martin Otto [14] proved that L 2 equivalence admits P canonization and thus that there is a logic capturing L 2 ( equivalence) invariant P. Our main result shows that this approach seems to fail for more than two variables. A definition of non uniform P, denoted by P=poly, will be given in the ....

....Lindell, and Weinstein [4] suggested to invert I k in P to obtain a canonization function for L k equivalence and thus a logic that captures L k invariant P. An inversion of I k is a mapping f such that for all structures A the structure f(I k (A) is L k equivalent to A. Otto [14] obtained his aforementioned result for L 2 along these lines. Unfortunately, for k 3 the mapping I k is not even recursively invertible [8] But Dawar et al. 4] had already observed that it is not possible, but also not necessary, to invert I k in time polynomial in the size of the given ....

M. Otto. Ptime canonization for two variables with counting. In Proceedings of the 10th IEEE Symposium on Logic in Computer Science, pages 342--352, 1995.

....for all structures A, the function f maps I k (A) to a structure that has the same L k theory as A If such an f exists, we say that I k is recursively invertible on structures. Equivalently, we can ask if the image of the class of structures under I k is a recursive set. Otto [14] proved that I 2 is invertible for all vocabularies , even in polynomial time. It is easy to see that I 1 , though somewhat pathological, is also invertible in polynomial time. Theorem 1. Let k 3 and a vocabulary that contains at least one (k Gamma 1) ary relation symbol. Then I k is ....

....least one (k Gamma 1) ary relation symbol. Then there is no recursive function f such that for all L k theories T of vocabulary we have f Gamma k size(T) Delta min Phi jAj fi fi A j= Tg: The best previously known lower bound for such a function was simply exponential [5] Otto [14] showed that there is a linear bound on the size of the smallest model of an L 2 theory in terms of its 2 size. Dawar [4] pointed out that another corollary of Theorem 1 is the separation of the intersection of partial fixed point logic with P from the intersection of L 1 with P. Our ....

[Article contains additional citation context not shown here]

M. Otto. Ptime canonization for two variables with counting. In Proceedings of the 10th IEEE Symposium on Logic in Computer Science, pages 342--352, 1995.

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