Search When the Lie Depends on the Target

Let us start with the basic model of Search Theory. An n-element set S is given, one of its elements, say x, is distinguished, the goal is to find x. Questions of type ”x ∈ A?” can be asked, where A is a subset of S. The unknown x should be determined on the base of the answers to these questions. In general one cannot use every subset A. A family A ⊂ 2 is given, the question sets A can be chosen only from A. We show some ”practical examples”. 1.1. Twenty question. Alfred chooses a person x, Paul has to find out who he/she is and he can ask questions like ”is x a man?”, ”is x alive?”, and so on... Alfred answers honestly, and Paul has to determine the person based on the answers to his questions. Obviously, S is the set of persons, A is the set of men, or the set of living persons, etc. 1.2. Chemical analysis. It is known that the given solution contains exactly one metal. This should be determined by chemical tests. Then S is the set of all metals, x ∈ S is the one contained in the solution. A chemical test is e.g. when a certain other specific chemical is added to the solution. If x ∈ A where A is a subset of S then the solution turns red, otherwise it does not. ∗The first author was supported by the Hungarian National Foundation for Scientific Research, grant number NK78439. †The work of the second author was supported by the European Union and co-financed by the European Social Fund (ELTE TÁMOP-4.2.2/B-10/1-2010-0030).