We prove a new combinatorial characterization of the determinant. The characterization yields a simple combinatorial algorithm for computing the determinant. Hitherto, all (known) algorithms for the determinant have been based on linear algebra. Our combinatorial algorithm requires no division, and works over arbitrary commutative rings. It also lends itself to e#cient parallel implementations. It has been known for some time now that the complexity class GapL characterizes the complexity of computing the determinant of matrices over the integers. We present a direct proof of this characterization.

We consider various types of unambiguity for logarithmic space bounded Turing machines and polynomial time bounded log space auxiliary push-down automata. In particular, we introduce the notions of (general), reach, and strong unambiguity. We demonstrate that closure under complement of unambiguous classes implies equivalence of unambiguity and "unambiguous fewness". This, as we will show, applies in the cases of reach and strong unambiguity for logspace. Among the many relations we exhibit, we show that the unambiguous linear contextfree languages, which are not known to be contained in LOGSPACE, nevertheless are contained in strongly unambiguous logspace, and, consequently, in LOGDCFL. In fact, this turns out to be the case for all logspace languages with reach unambiguous fewness. In addition, we show that general unambiguity and fewness of logspace classes can be simulated by reach unambiguity and fewness of auxiliary push-down automata. 1 Introduction Although the pow...

We analyze the complexity of simulating a PRAM (parallel random access machine) on a mesh structured distributed memory machine. By utilizing suitable algorithms for randomized hashing, routing in a mesh, and sorting in a mesh, we prove that simulation of a PRAM on p N \Theta p N (or 3 p N \Theta 3 p N \Theta 3 p N ) mesh is possible with O( p N ) (respectively O( 3 p N )) delay with high probability and a relatively small constant. Furthermore, with more sophisticated simulations further speed-ups are achieved; experiments show delays as low as p N + o( p N ) (respectively 3 p N + o( 3 p N )) per N PRAM processors. These simulations compare quite favorably with PRAM simulations on butterfly and hypercube. 1 Introduction PRAM 1 (Parallel Random Access Machine) is an abstract model of computation. It consists of N processors, each of which may have some local memory and registers, and a global shared memory of size m. A step of a PRAM is often seen to consist of...

. We show the equivalence of deterministic auxiliary pushdown automata to owner write PRAMs in a fairly large setting by proving that DAuxPDA-TISP \Gamma f O(1) ; log g \Delta and CROW-TIPR \Gamma log f; g O(1) \Delta coincide. Such, we provide the first circuit characterizations of depth O(log f) for deterministic sequential automata which are f time-bounded. 1 Introduction Parallel models provided fruitful extensions to structural complexity theory, a prominent example being the class NC . Ruzzo exhibited the tight connections between alternating Turing machines, auxiliary push-down automata, and boolean circuits [20]. By results of Stockmeyer and Vishkyn [21] and by the intertwining of the NC hierarchy with the polylogtime hierarchy of EREW-PRAMs these relations are also valid for the various types of PRAMs defined by the concurrent and the exclusive read and write feature. In 1986, Dymond and Ruzzo introduced the concept of owner write for PRAMs and showed close conn...

In the exponential case circuits of bounded depth characterize the polynomial hierachy. Using the notion of an unambiguous circuit we give a uniform framework to relate the various types of unambiguous polynomial hierarchies and to explain their differences.

CRCW-PRAMs can be characterized in terms of unbounded fan-in circuits. We introduce the notion of SELECT-gates. Combining this with the concept of an unambiguous circuit we are able to give a circuit equivalent of EREW-PRAMs, thus answering an open question of [SV84]. Moreover, circuits with SELECT-gates characterize CRCW-, CREW-, ERCW-, and EREW-PRAMs in a uniform manner. Introduction Parallel random access machines (PRAMs) and uniform circuit families are very important models in parallel complexity theory. Other important models are alternating Turing machines and auxiliary pushdown automata (see e.g. [Ruz80, Coo85]). NC , the class of efficiently parallel solvable problems, can be characterized in terms of all these models. For the design of efficient parallel algorithms, PRAMs are the most favorite model, while circuits are often used to achieve more theoretical results, such as lower bounds (see e.g. [Has86]). NC is in fact a hierarchy of classes that contains in particula...

We show that the class of context free languages CFL is not equal to ⊕1−P DA ( = ⊕CFL), the class of languages, which are recognized by a nondeterministic oneway push-down automaton equipped with parity acceptance. Furthermore we show that LOG(⊕CFL) = ⊕AuxP DApt contains all languages, which can be recognized by a uniform weak unambiguous AC 1-circuit introduced in [LR90a]. Therefore, it contains all languages, which can be recognized by a CREW-PRAM. We show, that L #Auxlog P DApt is contained in uniform T C 1, which sharpens a result in [Vin91], where inclusion in NC 2 was shown. At last we show, that AC k, SAC k and P can be characterized as the logarithmic closure of certain types of languages. These languages are either those given by linear alternating grammars or they are languages which are recognized by alternating pushdown automata with the restriction that their storage has to be empty if they make an alternating transition. In particular, we consider the cases when the depth of alternation is bounded by a constant or a polylogarithmic function. 1

We introduce a new variant of PC grammar systems, called PC grammar systems with terminal transmission, PCGSTT for short. We show that right-linear centralized PCGSTT have nice formal language theoretic properties: they are closed under gsm mappings (in particular, under intersection with regular sets and under and homomorphisms) and union; a slight variant is, in addition, closed under concatenation and star; their power lies between that of n-parallel grammars introduced by Wood and that of matrix languages of index n, and their relation to equal matrix grammars of degree n is discussed. We show that membership for these language classes is complete for NL. In a second part of the paper, we discuss questions concerning grammatical inference of these systems. More precisely, we show that PCGSTT whose component grammars are terminal distinguishable right-linear, a notion introduced by Radhakrishnan and Nagaraja in [29, 30], are identi able in the limit if certain data comm...

. This paper discusses a variant of the CREW PRAM model introduced by Dymond and Ruzzo called CROW (Concurrent Read Owner Write) PRAM. in which each global memory location may only be written by its assigned owner processor. We show that CROW PRAMs can sort in O(log n) parallel time using O(n log n) processors. 1 Introduction The PRAM (Parallel Random Access Machine) model for parallel computation was introduced as "a collection of synchronous processors executing in parallel and communicating via a global shared memory" by Fortune and Wyllie [FW78]. Their model satisfies the CREW (Concurrent-Read, Exclusive Write) convention for accesses to global memory. Dymond and Ruzzo proposed a natural and frequently occurring restriction of the CREW PRAM, called CROW (Concurrent Read Owner-Write) PRAM machines, for which each global memory location is assigned a unique owner processor and the owner is the only processor allowed to write into that cell throughout the course of a computation [D...

We investigate parallel algorithms to compute recursively defined functions. Our computational model are parallel random access machines (PRAM's). We preferably make use of the OROW-PRAM (owner read, owner write), a model supposed to be even weaker and more realistic than the EREW-PRAM (exclusive read, exclusive write) and that still provides the opportunities of a completely connected processor network. For OROW-PRAM's we show that our parallel algorithms are work-optimal.