We treat theoretically the problem of the specificity of interaction between nucleic acid and an oligonucleotide, its analog or its mimic (such as peptide nucleic acid, or PNA). We consider simplest models with only essential details using numerical solutions of kinetic equations and the kinetic Monte Carlo method. In our first model, describing the formation of complementary duplex, we demonstrate anti-correlation between specificity and affinity for nucleic acid/oligonucleotide interaction. We analyze in detail one notable exception. Homopyrimidine PNAs exhibit very high affinity to DNA forming extraordinarily stable DNA/(PNA)2 triplexes with the complementary DNA strand. At the same time, such PNAs show remarkable sequence specificity of binding to duplex DNA. We formulate a theoretical model for the two-step process of PNA interaction with DNA. The calculations demonstrate that two-stage binding may secure both high affinity and very high specificity of PNA interaction with DNA. Our computer simulations define the range of parameter values in which high specificity is achieved. These findings are of great importance for numerous applications of PNA and for design of future drugs which specifically interact with DNA.