Establishing static scope name binding and direct superclassing in the external language of the object oriented Java with inner classes is a difficult and subtle task

In [IP02] an axiomatic approach towards the semantics of FJI, Featherweight Java with Inner classes, essentially a subset of Java-programming language, is presented. In this way the authors contribute to an ambitious project: to give an axiomatic definition of the semantics of programming language Java. A similar project with a partly axiomatic flavour, with so called Abstract State Machines ASM, was initiated by E. Boerger and his colleagues [Boe01] in 2001, but did not yet include inner classes. At a first glance the approach of reducing Java's semantics to that of FJI seems promising. We are going to show that several questions have been left unanswered. It turns out that the theory how to elaborate or bind types and thus to determine direct superclasses as proposed in [IP02] has different models. Therefore the suggestion that the formal system of [IP02] defines the (exactly one) semantics of Java is not justified. We present our contribution to the project showing that it must be attacked from another starting point. Quite frequently one encounters a set of inference rules and a claim that a semantics is defined by the rules. Such a claim should be proved. One should present arguments: $1^0$ that the system has a model and hence it is a consistent system, and $2^0$ that all models are isomorphic. Sometimes such a proposed system contains a rule with a premise which reads: \underline{there is no proof of something}. One should notice that this is a metatheoretic property. It seems strange to accept a metatheorem as a premise, especially if such a system does not offer any other inference rules which would enable a proof of the premise. We are going to study the system in [IP02]. We shall show that it has many non-isomorphic model. We present a repair of Igarashi's and Pierce's calculus such that their ideas are preserved as close as possible.

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