Computational semantics for monadic quantifiers

The paper gives a survey of known results related to computational devices (finite and push–down automata) recognizing monadic generalized quantifiers in finite models. Some of these results are simple reinterpretations of descriptive—feasible correspondence theorems from finite–model theory. Additionally a new result characterizing monadic quantifiers recognized by push down automata is proven. The aim of the work is presentation of the state of knowledge and main research problems in computational approach to finite interpretations of monadic generalized quantifiers. We will concentrate on purely logical approach – that is we will consider mainly structures without standard linear orderings. However proper results for linearly ordered structures will be mentioned when relevant. Let us observe that monadic quantifiers – being the first natural class of quantifiers investigated from computational point of view – are not systematically treated in logical literature. In the first attempt [6] of surveying the subject of generalized quantifiers from computational point of view, monadic quantifiers are only shortly mentioned.