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a b s t r a c tIn this paper we continue previous studies on the computational efficiency of spiking neural P systems, under the assumption that some pre-computed resources of exponential size are given in advance. Specifically, we give a deterministic solution for each of two well known PSPACE-complete problems: QSAT and Q3SAT. In the case of QSAT, the answer to any instance of the problem is computed in a time which is linear with respect to both the number n of Boolean variables and the number m of clauses that compose the instance. As for Q3SAT, the answer is computed in a time which is at most cubic in the number n of Boolean variables.

We continue the investigations concerning the possibility of using spiking neural P systems as a framework for solving computationally hard problems, addressing two problems which were already recently considered in this respect: Subset Sum and SAT: For both of them we provide uniform constructions of standard spiking neural P systems (i.e., not using extended rules or parallel use of rules) which solve these problems in a constant number of steps, working in a non-deterministic way. This improves known results of this type where the construction was non-uniform, and/or was using various ingredients added to the initial definition of spiking neural P systems (the SN P systems as defined initially are called here ''standard''). However, in the Subset Sum case, a price to pay for this improvement is that the solution is obtained either in a time which depends on the value of the numbers involved in the problem, or by using a system whose size depends on the same values, or again by using complicated regular expressions. A uniform solution to 3-SAT is also provided, that works in constant time.

We investigate sets of Mutually Orthogonal Latin Squares (MOLS) generated by Cellular Automata (CA) over finite fields. After introducing how a CA defined by a bipermutive local rule of diameter d over an alphabet of q elements generates a Latin square of order q d−1 , we study the conditions under which two CA generate a pair of orthogonal Latin squares. In particular, we prove that the Latin squares induced by two Linear Bipermutive CA (LBCA) over the finite field F q are orthogonal if and only if the polynomials associated to their local rules are relatively prime. Next, we enumerate all such pairs of orthogonal Latin squares by counting the pairs of coprime monic polynomials with nonzero constant term and degree n over F q . Finally, we present a construction of MOLS generated by LBCA with irreducible polynomials and prove the maximality of the resulting sets, as well as a lower bound which is asymptotically close to their actual number.

We consider the problem of evolving a particular kind of shift-invariant transformation-namely, Reversible Cellular Automata (RCA) defined by conserved landscape rules-using GA and GP. To this end, we employ three different optimization strategies: a single-objective approach carried out with GA and GP where only the reversibility constraint of marker CA is considered, a multi-objective approach based on GP where both reversibility and the Hamming weight are taken into account, and a lexicographic approach where GP first optimizes only the reversibility property until a conserved landscape rule is obtained, and then maximizes the Hamming weight while retaining reversibility. The results are discussed in the context of three different research questions stemming from exhaustive search experiments on conserved landscape CA, which concern 1) the difficulty of the associated optimization problem for GA and GP, 2) the utility of conserved landscape CA in the domain of cryptography and reversible computing, and 3) the relationship between the reversibility property and the Hamming weight.

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Starting from an extended nondeterministic spiking neural P system that solves the Subset Sum problem in a constant number of steps, recently proposed in a previous paper, we investigate how different properties of spiking neural P systems affect the capability to solve numerical NP-complete problems. In particular, we show that by using maximal parallelism we can convert any given integer number from the usual binary notation to the unary form, and thus we can initialize the above P system with the required (exponential) number of spikes in polynomial time. On the other hand, we show that this conversion cannot be performed in polynomial time if the use of maximal parallelism is forbidden. Finally, we show that by selectively using nondeterminism and maximal parallelism (that is, for each neuron in the system we can specify whether it works in deterministic or nondeterministic way, as well as in sequential or maximally parallel way) there exists a uniform spiking neural P system that solves all the instances of Subset Sum of a given size.

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