J. Gergov. Time-space tradeoffs for integer multiplication on various types of input-oblivious sequential machines. Information Processing Letters, 51 (1994), pp. 265-269.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....program of linear length for the problem of directed s t connectivity requires exponential size; in [KMW92] similar lower bounds are proved for such programs with nondeterminism added. Using a lemma from [AM88] and the communication complexity arguments outlined in Section 2.5. 1, Gergov [Ge94] proves that computing MULT requires size for arbitrary oblivious programs of linear length, even with nondeterministic AND, OR, or PARITY nodes. There has also been great success in proving lower bounds on the size of read once programs. Many of the functions that require exponential size are ....

....to these variables is a smaller version of the original problem, and hence the known linear lower bound on the communication complexity applies. To demonstrate, we give an easy lower bound which has not appeared in the literature. The proof is very similar to the lower bounds of [BSSW95] and [Ge94]. Recall that [BSSW93] showed ACH 62 C OBDD; we will show that ACH 62 C IBDD. Theorem 2 ACH 62 C IBDD. Proof: Consider a k IBDD G computing ACH. We will show that G has size at least n=k2 . Recall from Section 2.4.2 that ACH(x; y; z) V 1jn (x j y j z ) if z 6= 0, and W 1jn (x j y j ) ....

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J. Gergov. Time-space tradeoffs for integer multiplication on various types of input-oblivious sequential machines. Information Processing Letters, 51 (1994), pp. 265-269.

....OBBD computing integer multiplication. During last decade there were several attempts to find generalizations of OBDDs model for hardware verification, strong enough to compute efficiently integer multiplication. But again the results showed that multiplication remained hard for these models ([11, 15]) In [4] a randomized model of branching programs was introduced. The importance of this model was highlighted by the fact that there is a function which is hard for deterministic OBDDs but is easy for randomized OBDDs [4] During the last couple of years new examples of such function were ....

....syntactic read k times branching programs [5] but is simple for randomized OBDDs [18, 20] See [21] for another example. It was proved that randomized and nondeterministic models of OBDD are incomparable [2] So there was still hope (note that multiplication is hard for nondeterministic OBDD [11]) that randomized OBDDs can compute integer multiplication in polynomial size. Our results show that randomized OBDDs can test integer multiplication in polynomial size but integer multiplication itself requires exponential size. Up to now it was not clear what is harder to multiply or to test ....

[Article contains additional citation context not shown here]

J. Gergov, Time-space tradeoffs for integer multiplication on various types of input oblivious sequential machines, Information Processing Letters, 51, (1994), 265-269.

....is a kIBDD where the variable orderings have to be identical for all layers. Apart from practical issues, these restricted types of branching programs are also interesting as objects of theory. The first exponential lower bounds for OBDDs are due to Bryant [8] Jukna [16] Krause [21] and Gergov [11] have proven exponential lower bounds even for kOBDDs. Bollig, Sauerhoff, Sieling, and Wegener [5] have shown that the classes of sequences of functions representable in polynomial size by kIBDDs and by kOBDDs form a proper hierarchy with respect to k. Here we are concerned with the ....

J. Gergov. Time-space tradeoffs for integer multiplication on various types of input oblivious sequential machines. Information Processing Letters, 51:265 -- 269, 1994.

.... synthesis and satisfiability test (see Chapter 1) on POBDDs. Hence, also the equivalence check can be done deterministically in polynomial time. An exponential lower bound on the size of MOD 2 OBDDs (and hence, also POBDDs) for the middle bit of multiplication has been proven by Gergov [41]. For this, Gergov has used the rank method of communication complexity theory and Ramsey theoretic arguments of Alon and Maass [10] as tools. 28 2.2 Randomized Branching Programs Analogously to the last section, we introduce randomized variants of general as well as restricted branching ....

J. Gergov. Time-space tradeoffs for integer multiplication on various types of input oblivious sequential machines. Information Processing Letters, 51:265 -- 269, 1994.

....by a family of polynomial size OBDDs is equivalent to the class of languages accepted by log space bounded nondeterministic on line Turing machines [10] In this paper, we develop new methods for proving lower bounds on the size of OBDDs, OBDDs and Phi OBDDs. Although Gergov s method [5] can be also applied for proving lower bounds on the size of those OBDDs, our methods are more simple and intuitive. We show the relation between the cardinality of 1 fooling sets and the size of OBDDs and the one between the cardinality of 0 fooling sets and the size of OBDDs. Moreover, we ....

J. Gergov, "Time-Space Tradeoffs for Integer Multiplication on Various Types of Input Oblivious Sequential Machines," Information Processing Letters, vol.51, no.5, pp.265--269, 1994.

....such example is the integer multiplication. Up to now all the attempts to verify 32 bit multiplicator have failed. Bryant (1991) explained this failure by proving that the n th bit in the binary representation of x Delta y; is hard for oblivious read once only programs. Bollig et al. 1993) and Gergov (1994) observed that Bryant s result combined with known lower bounds arguments for oblivious branching programs implies that the multiplication is hard also for oblivious read k times programs even with k = o(log n) The requirement for a program to be oblivious is very strong restriction: in each s t ....

Gergov, J. Time-space tradeoffs for integer multiplication on various types of input oblivious sequential machines. Manuscript, March,1994.

....is a kIBDD where the variable orderings have to be identical for all layers. Apart from practical issues, these restricted types of branching programs are also interesting as objects of theory. The first exponential lower bounds for OBDDs are due to Bryant [8] Jukna [16] Krause [21] and Gergov [11] have proven exponential lower bounds even for kOBDDs. Bollig, Sauerhoff, Sieling, and Wegener [5] have shown that the classes of sequences of functions representable in polynomial size by kIBDDs and by kOBDDs form a proper hierarchy with respect to k. Here we are concerned with the ....

J. Gergov. Time-space tradeoffs for integer multiplication on various types of input oblivious sequential machines. Information Processing Letters, 51:265 -- 269, 1994.

....OBBD computing integer multiplication. During last decade there were several attempts to find generalizations of OBDDs model for hardware verification, strong enough to compute efficiently integer multiplication. But again the results showed that multiplication remained hard for these models ([11, 15]) In [4] a randomized model of branching programs was introduced. The importance of this model was highlighted by the fact that there is a function which is hard for deterministic OBDDs but is easy for randomized OBDDs [4] During the last couple of years new examples of such function were ....

....syntactic read k times branching programs [5] but is simple for randomized OBDDs [18, 20] See [21] for another example. It was proved that randomized and nondeterministic models of OBDD are incomparable [2] So there was still hope (note that multiplication is hard for nondeterministic OBDD [11]) that randomized OBDDs can compute integer multiplication in polynomial size. Our results show that randomized OBDDs can test integer multiplication in polynomial size but integer multiplication itself requires exponential size. Up to now it was not clear what is harder to multiply or to test the ....

[Article contains additional citation context not shown here]

J. Gergov, Time-space tradeoffs for integer multiplication on various types of input oblivious sequential machines, Information Processing Letters, 51, (1994), 265-269.

....Email: marek cs.uni bonn.de , URL: http: theory.cs.uni bonn.de marek . multiplication (cf. e.g. P95] It is well known that computing the integer multiplication requires exponential size on deterministic read k times ordered branching programs even if k = o(log n) cf. B91] BSSW93] and [G94]. Ablayev and Karpinski [AK98b] succeeded in designing a small (polynomial size) randomized OBDDs for testing the function of integer multiplication, and proving at the same time an exponential lower bound on the size of any randomized OBDD computing exactly the integer multiplication. ....

J. Gergov, Time-Space Tradeoffs for Integer Multiplication on Various Types of Input Oblivious Sequential Machines, Information Processing Letters 51 (1991), pp. 265--269.

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J. Gergov, Time-Space Tradeoffs for Integer Multiplication on Various Types of Input Oblivious Sequential Machines, Information Processing Letters 51 (1991), pp. 265--269.

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