E. Allender. The permanent requires large uniform threshold circuits. To appear in Chicago Journal on Theoretical Computer Science. A preliminary version of this paper appeared as [All96].  Home/Search   Document Details and Download   Summary   Related Articles   Check

....is the counting hierarchy CH, defined in [Wag86b, Wag86a] as CH = PP [ PP [ PP [ Delta Delta Delta . Finer results are given in [Vol98] Building on leaf language characterizations, the circuit class TC (where we require logtime uniformity) was separated from the counting hierarchy in [CMTV98, All99]. Theorem 3.14. TC 6= PP. Proof sketch. First, suppose that TC = CH. Then we obtain the inclusion chain = CH = BLeaf (TC (CH) BLeaf (P) EXPTIME, thus P EXPTIME, which is a contradiction; hence TC 6= CH. Suppose now that TC = PP, then PP PP PP TC ; ....

E. Allender. The permanent requires large uniform threshold circuits. Chicago Journal of Theoretical Computer Science, 1999. To appear. A preliminary version appeared in Proceedings 2nd Computing and Combinatorics Conference, vol. 1090 of Lecture Notes in Computer Science, pages 127--135. Springer Verlag, 1990.

....majority of its successors are. A TTM also has a read only input tape with random access via an index tape to allow for sub linear runtimes. In the following, all TTMs are required to perform only constantly many threshold operations on each computation path. The following was noted by Allender [2]: Proposition 1. The class of languages accepted in time O(t(n) on a TTM coincides with TC 0 (2 O(t(n) for every complexity function t(n) 473 n) Thus TC 0 ( k (n) is equal to k 1 (log n) time on a TTM , and in particular, TC 0 is equal to O(log n) time on a TTM , and ....

....referee for the paper [10] suggested that R 2 0 = C 0 2 might be RSUV isomorphic to a certain subtheory of D 0 1 , which was the starting point of this paper. Peter Clote brought [18] to our attention, and Eric Allender referred us to the concept of Threshold Turing Machines and his [2]. ....

E. Allender. The permanent requires large uniform threshold circuits. Manuscript. Preliminary Version appeared in COCOON'96, 1996.

....somewhat involved definition. DLogTime uniform TC 0 is a fairly natural complexity class: it is characterized by first order logic with majority quantifiers on ordered finite models [3] in Descriptive Complexity Theory, or by acceptance in polynomialtime on so called Threshold Turing Machines [2], or by the machine independent characterization below, which is most convenient for our purposes. Whenever we speak of TC 0 in the following without further qualification, we mean DLogTime uniform TC 0 . For a complexity class C, the class C=poly is defined as follows: A predicate A(x) is in ....

E. Allender. The permanent requires large uniform threshold circuits. To appear in Chicago Journal of Theoretical Computer Science. Preliminary Version appeared in COCOON'96, 1998.

....by Turing [Tur36] to show that there existed computably enumerable problems that were not computable. The seminal paper in computational complexity [HS65] used diagonalization to give time and space hierarchies. Diagonalization works In Section 2. 1 we describe Allender s diagonalization proof [All99] that the permanent is not computable by uniform constant depth threshold circuits. Without diagonalization, can we even show that the halting problem does not have such circuits Razborov and Rudich [RR97] has developed the concept of natural proofs that capture the known techniques for ....

....results to give a taste and history of the technique. 2.1 Permanent is not in uniform TC 0 A wonderful example of the diagonalization technique is the result showing that the permanent cannot be computed by uniform constant depth threshold circuits. This result was proven by Allender [All99] building on work of Caussinus, McKenzie, Th erien and Vollmer [CMTV98] and Allender and Gore [AG94] We sketch the proof in this section. Consider threshold machines: These are like alternating machines except that instead of asking existential and universal questions they ask Do a majority of ....

E. Allender. The permanent requires large uniform threshold circuits. Chicago Journal of Theoretical Computer Science, 1999(7), August 1999.

....majority of its successors are. A TTM also has a read only input tape with random access via an index tape to allow for sub linear runtimes. In the following, all TTMs are required to perform only constantly many threshold operations on each computation path. The following was noted by Allender [2]: Proposition 1. The class of languages accepted in time O(t(n) on a TTM coincides with TC 0 (2 O(t(n) for every complexity function t(n) Omega Gamma473 n) Thus TC 0 ( k (n) is equal to k Gamma1 (log n) time on a TTM , and in particular, TC 0 is equal to O(log n) time on a ....

....referee for the paper [10] suggested that R 2 0 = C 0 2 might be RSUV isomorphic to a certain subtheory of D 0 1 , which was the starting point of this paper. Peter Clote brought [18] to our attention, and Eric Allender referred us to the concept of Threshold Turing Machines and his [2]. ....

E. Allender. The permanent requires large uniform threshold circuits. Manuscript. Preliminary Version appeared in COCOON'96, 1996.

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E. Allender. The permanent requires large uniform threshold circuits. To appear in Chicago Journal on Theoretical Computer Science. A preliminary version of this paper appeared as [All96].

....#P or C=P, etc. In order to prove our main result we will need the following theorem, which can be proven by standard diagonalization techniques using the fact that a multitape probabilistic Turing machine running in time t can be simulated on a two tape probabilistic Turing machine in time O(t) [8, 2]. Theorem 2 (Hierarchy Theorem) Let T and t be time constructible functions. If t # o(T ) then PrTime(t) # PrTime(T ) As usual, TC 0 denotes the class of languages decidable by constant depth, polynomial size threshold circuits, i.e. Boolean circuits that consist of unbounded fanin ....

E. Allender. The permanent requires large uniform threshold circuits. Chicago Journal of Theoretical Computer Science, article 7, 1999.

.... NC 1 correspond to logarithmic time on an alternating Turing machine [29] and uniform AC 0 correspond to logarithmic time on an alternating Turing machine making O(1) alternations [10] Similarly, uniform TC 0 corresponds to logarithmic time and O(1) alternations on a threshold machine [26, 1]. Further support for this uniformity condition comes from a series of striking connections to finite model theory. A language is in uniform AC 0 if and only if it can be viewed as the class of finite models of a first order formula. That is, a single formula (with existential and universal ....

E. Allender. The permanent requires large uniform threshold circuits. Chicago Journal of Theoretical Computer Science, article 7, 1999.

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E. Allender. The permanent requires large uniform threshold circuits. Technical Report TR 97-51, DIMACS, September 1997. Submitted for publication.

.... as multiplication, division, and sorting, as well as being a computational model for neural nets [RT92, CSV84, PS88] It remains an open question as to whether every function in #P has TC 0 circuits (although it is at least known that not all #P functions have Dlogtime uniform TC 0 circuits [All]) The main contribution of this paper is to present a new connection between AC 0 and TC 0 . We characterize TC 0 as being the class of languages that arises in several ways from counting the number of accepting subtrees of AC 0 circuits. Equivalently, we characterize TC 0 in terms of ....

.... It is currently an open question whether a given bit of the permanent can be computed as the high order bit of Dlogtime uniform #AC 0 functions, although if the inclusion PAC 0 TC 0 holds also in the Dlogtime uniform setting, then a negative answer would follow from the lower bound of [All]. The main obstacle to proving a Dlogtime uniform analog to Theorem 18 seems to be the problem of finding a generator for the multiplicative group Z p . Note in this regard that our proof uses the fact that a logspace machine can check if the graph on f1; pg with edges i ig(modp) is a ....

E. Allender. The permanent requires large uniform threshold circuits. Submitted. A preliminary version of this paper appeared as [All96].

No context found.

E. Allender. The permanent requires large uniform threshold circuits. Submitted. A preliminary version of this paper appeared as [All96].

.... as multiplication, division, and sorting, as well as being a computational model for neural nets [RT92, CSV84, PS88] It remains an open question as to whether every function in #P has TC 0 circuits (although it is at least known that not all #P functions have Dlogtimeuniform TC 0 circuits [All]) The main contribution of this paper is to present a new connection between AC 0 and TC 0 . We characterize TC 0 as being the class of languages that arises in several ways from counting the number of accepting subtrees of AC 0 circuits. Equivalently, we characterize TC 0 in terms of ....

.... It is currently an open question whether a given bit of the permanent can be computed as the high order bit of Dlogtime uniform #AC 0 functions, although if the inclusion PAC 0 TC 0 holds also in the Dlogtime uniform setting, then a negative answer would follow from the lower bound of [All]. The main obstacle to proving a Dlogtime uniform analog to Theorem 17 seems to the problem of finding a generator for the multiplicative group Z p . Note in this regard that our proof uses the fact that a logspace machine can check if the graph on f1; pg with edges i ig(modp) is a ....

E. Allender. The permanent requires large uniform threshold circuits. Submitted. A preliminary version of this paper appeared as [All96].

No context found.

E. Allender. The permanent requires large uniform threshold circuits. Chicago Journal of Theoretical Computer Science, 1999. To appear. A preliminary version appeared in Proceedings 2nd Computing and Combinatorics Conference, vol. 1090 of Lecture Notes in Computer Science, pages 127--135. Springer Verlag, 1990.

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