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Second-harmonic generation and parametric down-conversion processes have been studied as the basis of all-optical logic gates. All possibilities that are obtainable with both the low and high conversion efficiencies of such processes have been analyzed here. XOR and AND gates are also experimentally proven by using 1-ps pulses at 800 nm within a beta-BaB2O4 crystal, reaching conversion efficiencies of as high as 80%. Based on these phenomena, complex algebraic operations are proposed for performing several different logic functionalities particularly concerning network switching and arithmetic calculation.