E. Allender, J. Jiao, M. Mahajan, and V. Vinay. Noncommutative arithmetic circuits: depth reduction and size lower bounds. Theoretical Computer Science 209:47--86, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....gates. Viewing the inputs to the circuit as taking on the values 0, 1 from the natural numbers, we obtain circuits which map naturally from the binary strings 1 # to the natural numbers. More recently, the arithmetic classes #AC , #BWBP , #NC , and #SAC have been defined and studied [1, 7, 3, 8, 4, 14]. Other than #BWBP , these classes are arithmetic versions of boolean classes typically defined by circuits, and arise from arithmetizing the corresponding boolean circuits. These classes obey the inclusion chain #AC # #BWBP #SAC , which essentially mirrors the known relationships # ....

E. Allender, J. Jiao, M. Mahajan, and V. Vinay. Non-commutative arithmetic circuits: depth reduction and size lower bounds. Theoretical Computer Science, 209:47--86, 1998.

....AND gates to # gates. Viewing the inputs to the circuit as taking on the values 0, 1 from the natural numbers, we obtain circuits which map naturally from 0, 1 # to the natural numbers. More recently, the arithmetic classes #AC 0 , #BP, #NC 1 , and #SAC 1 have been defined and studied [1, 7, 3, 8, 4, 14]. Other than #BP , these classes are arithmetic versions of boolean classes typically defined by circuits, and arise from arithmetizing the corresponding boolean circuits. These classes obey the inclusion chain #AC 0 # #BP # #NC 1 # #SAC 1 , which essentially mirrors the known ....

E. Allender, J. Jiao, M. Mahajan, and V. Vinay. Non-commutative arithmetic circuits: depth reduction and size lower bounds. Theoretical Computer Science, 209:47--86, 1998.

....Turing machines with logarithmic space bound and one way access to their input tape. More precisely, 1 UL Poly # 1 NL Poly, where the prefix 1 to the complexity classes indicates one way access to the input tape. This unpublished result is attributed to M. Dietzfelbinger by Allender et al. [5]. The article also contains an improved version of the result. Furthermore, the above fact can also be formulated in terms of restricted BPs. Analogous to the proof for one way Turing machines, one obtains that there is a sequence of functions which has nondeterministic OBDDs of polynomial size, ....

E. Allender, J. Jiao, M. Mahajan, and V. Vinay. Non-commutative arithmetic circuits: depth reduction and size lower bounds. Theoretical Computer Science, 209:47--86, 1998.

....on input w and C t (w) is a final configuration. The above definition is adapted in a straightforward way to define algebraic polytime Turing machines, algebraic logspace bounded Turing machines, and algebraic auxiliary space (and time) bounded pushdown automata. Note that Allender et al. [2] have introduced an auxiliary pushdown model similar to ours, called the generalized LOGCFL machine. Definition 12. 1. We define the generalized counting class S #P as the set of all functions f : Sigma S such that there is a a polytime algebraic Turing machine M over S which computes f . ....

....) is called uniform, if there is a logspace bounded deterministic Turing machine that given an input of length n produces a description of the nth circuit in the sequence. We prove the following characterization of S #LOGCFL in terms of arithmetic semi unbounded circuits. Note that Allender et al. [2] proved a similar result for height bounded auxiliary pushdown automata. Theorem 21. Let S be any finitely generated commutative semiring. The class of functions computed by logspace polytime algebraic auxiliary pushdown automata over S is equal to the class of functions computed by logspace ....

E. Allender, J. Jiao, M. Mahajan, and V. Vinay. Non-commutative arithmetic circuits: depth reduction and size lower bounds. Theoretical Computer Science, 209:47--86, 1998.

....fan in circuits. This was first proved in the nonuniform setting in [VSBR83] and related depth reduction results for uniform circuits were proved for the Boolean ring in [R81] and for N in [V91] A general proof that works in the uniform setting over any commutative semiring appears in [AJMV]. All of the functions in #SAC 1 can be computed by threshold circuits of logarithmic depth (known as TC 1 circuits) In fact, it is observed in [AJMV] that if gates for integer division (throwing the remainder away) are added to #SAC 1 circuits, then one obtains an exact characterization ....

.... for the Boolean ring in [R81] and for N in [V91] A general proof that works in the uniform setting over any commutative semiring appears in [AJMV] All of the functions in #SAC 1 can be computed by threshold circuits of logarithmic depth (known as TC 1 circuits) In fact, it is observed in [AJMV] that if gates for integer division (throwing the remainder away) are added to #SAC 1 circuits, then one obtains an exact characterization of TC 1 . On the other hand, nothing is known about the relative power of #SAC 1 and AC 1 (the class of problems accepted by logarithmic depth, ....

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E. Allender, J. Jiao, M. Mahajan, and V. Vinay. Non-commutative arithmetic circuits: depth reduction and size lower bounds. To appear in Theoretical Computer Science. Preliminary versions appeared as [AJ93a, MV94].

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E. Allender, J. Jiao, M. Mahajan, and V. Vinay. Noncommutative arithmetic circuits: depth reduction and size lower bounds. Theoretical Computer Science 209:47--86, 1998.

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E. Allender, J. Jiao, M. Mahajan, and V. Vinay. Noncommutative arithmetic circuits: Depth reduction and size lower bounds. Theoretical Computer Science, 209:47--86, 1998. 19

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E. Allender, J. Jiao, M. Mahajan, and V. Vinay, Non-commutative arithmetic circuits: depth reduction and size lower bounds, Theoretical Computer Science, 209:47--86, 1998.

....x 1 ; xn now take as values the natural numbers f0; 1g (instead of the Boolean values f0; 1g) and negated input literals x i now take on the value 1 Gamma x i . The counting classes that result in this way by arithmetizing the Boolean circuit classes SAC 1 and NC 1 were studied in [Vin91,AJMV98,CMTV96]. In this paper, we study #AC 0 . Definition 1. For any k 0, #AC 0 k is the class of functions f : f0; 1g N such that, for some polynomial p, for every n there is a depth k circuit Cn of size at most p(n) consisting of unbounded fan in ; Theta gates (the usual sum and product in N) ....

E. Allender, J. Jiao, M. Mahajan, and V. Vinay, Non-commutative arithmetic circuits: depth reduction and size lower bounds, Theoretical Computer Science, 209:47--86, 1998.

....#C) and (b) provides a natural notion of counting the number of proofs that C accepts. The arithmetic circuits corresponding to #L were studied further by Toda [Tod92b] The counting classes that result in this way by arithmetizing the Boolean circuit classes SAC 1 and NC 1 were studied in [Vin91, AJMV98, CMTV96]. In this paper, we study #AC 0 . Definition 1 For any k 0, #AC 0 k is the class of functions f : 0, 1 # # N such that, for some polynomial p, for every n there is a depth k circuit C n of size at most p(n) consisting of unbounded fan in , gates (the usual sum and product in N) ....

E. Allender, J. Jiao, M. Mahajan, and V. Vinay, Non-commutative arithmetic circuits: depth reduction and size lower bounds, Theoretical Computer Science, 209:47--86, 1998.

....this clean circuit characterization of LOGCFL stands in contrast to its somewhat messy machine definitions. Then came a flurry of papers, for example showing closure of LOGCFL under complementation [BCD 88] characterizing LOGCFL in terms of groupoids [BLM93] linking it to depth reduction [Vin96, AJ93], and studying the descriptive complexity of LOGCFL [LMSV99] Informatique et recherche op erationnelle, Universit e de Montr eal, C.P. 6128, Succ. Centre Ville, Montr eal (Qu ebec) H3C 3J7 Canada. Research performed while on leave at Universitat Tubingen. mckenzie iro.umontreal.ca y ....

E. Allender, J. Jiao, M. Mahajan and V. Vinay, Non-commutative arithmetic circuits: depth reduction and size lower bounds, Theoret. Comput. Sci. 209, pp. 47--86, 1998. 5 having only 1 steering bit and 2 data bits 6 by a condition which is in some way 'locally checkable'.

....and Bell Labs. DIMACS is an NSF Science and Technology Center, funded under contract STC 91 19999; and also receives support from the New Jersey Commission on Science and Technology. ABSTRACT Continuing a line of investigation that has studied the function classes #P [Val79b] #SAC 1 [Val79a, Vin91, AJMV], #L [AJ93b, Vin91, AO94] and #NC 1 [CMTV96] we study the class of functions #AC 0 . One way to define #AC 0 is as the class of functions computed by constant depth polynomial size arithmetic circuits of unbounded fan in addition and multiplication gates. In contrast to the preceding ....

E. Allender, J. Jiao, M. Mahajan, and V. Vinay. Non-commutative arithmetic circuits: depth reduction and size lower bounds. To appear in Theoretical Computer Science. Preliminary versions appeared as [AJ93a, MV94].

....1179 Piscataway, NJ 08855 1179,USA allender cs.rutgers.edu Samir Datta z Department of Computer Science Rutgers University P.O. Box 1179 Piscataway, NJ 08855 1179,USA sdatta paul.rutgers.edu Abstract Continuing a line of investigation that has studied the function classes #P [Val79b] #SAC 1 [Val79a, Vin91, AJMV], #L [AJ93b, Vin91, AO94] and #NC 1 [CMTV96] we study the class of functions #AC 0 . One way to define #AC 0 is as the class of functions computed by constantdepth polynomial size arithmetic circuits of unbounded fanin addition and multiplication gates. In contrast to the preceding ....

E. Allender, J. Jiao, M. Mahajan, and V. Vinay. Non-commutative arithmetic circuits: depth reduction and size lower bounds. To appear in Theoretical Computer Science. Preliminary versions appeared as [AJ93a, MV94].

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E. Allender, J. Jiao, M. Mahajan, and V. Vinay. Non-commutative arithmetic circuits: depth reduction and size lower bounds. Theoretical Computer Science, 209:47--86, 1998.

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