Andrzej Salwicki

On algorithmic theory of stacks

Algorithmic Theory of Stacks ATS formalizes properties of relational systems of stacks. It turns out that apart of previously known axioms, the new axiom of algorithmic nature [while - empty(s)do s := pop(s)]true is in place. The representation theorem stating that every relational system of stacks is isomorphic to a system of finite sequences of elements is proved. The connections between ATS and a type STACKS declaration /written in LOGLAN programming language/ are shown.

On an algorithm determining direct superclasses in Java and similar languages with inner classes---Its correctness, completeness and uniqueness of solutions

Some object oriented programming languages allow inner classes. All of them admit inheritance. This combination of inner classes and inheritance is very fruitful however less known. On the other hand it creates a serious problem: how to determine the direct superclass of a given class C, i.e. the class which class C directly inherits from. For there may be several classes of the same name in one program. A specification of the problem and a non-deterministic algorithm are provided. We prove that the algorithm is correct w.r.t. the specification and complete, i.e. if the algorithm signals an error then no solution exists. We show that the specification itself has at most one solution, in other words, it is a complete specification. This proves also that the corresponding parts of Java Language Specification are consistent and define uniquely a fragment of Java semantics.

Procedures, Formal Computations and Models

Starting from the logical point of view we conceive procedures as formulas of a formalized algorithmic language defining functions and/or relations. The notion of formal computation is introduced in a way resembling formal proofs. Computations may serve to extend the original interpretation of the language onto symbols defined by procedures. The main result is: if a system of procedures is consistent then the computed extension of a given interpretation is the smallest model of the system. From this the principle of recursion induction can be proved. A technique transforming any system of procedures to a consistent system of conditional recursive definitions is shown.

Applied algorithmic logic

This is a survey of the last years work of the group of algorithmic logic. Our studies have concetrated on two (not disjoint) tasks: design of programming language LOGLAN 77. studies of computational complexity.

Critical Remarks on MAX Model of Concurrency

A few critical remarks on a mathematical model of concurrency are discussed. As the result we were able to improve the model, to state new problems to be studied later on and to remark that the ideas of our model can be applied to the tasking of ADA, CSP and CCS. It means that apart of standard semantics of ADA there is another one and it is not obvious which semantics is more natural.

On Axiomatic Definition of Max-Model of Concurrency

The thesis we present in this paper states that for every concurrent program π there exists a set of modal formulas, also called the axioms Ax(π), such that a) the structure of admissible parallel executions of the program π is a Kripke model of the set Ax(π) and, b) any Kripke model of the axioms Ax(π) is an extension of the structure of all admissible distributed (i.e. parallel) executions of the program π.

On the Algorithmic Theory of Dictionaries

A class of algebraic structures called dictionaries is defined. Among the properties (axioms) of dictionaries we find the algorithmic property : for every s the program while empty(s) do s := delete(amember(s),s) terminates. Starting with this observation we define and develop the algorithmic theory of dictionaries ATD. We are proving the representation theorem: every dictionary structure is isomorphic to the family of all finite subsets of some set. The complexity of the set of theorems of various extensions of ATD can vary from a hiperarithmetical set to the complement of a recursively enumerable set.

Some Methodological Remarks Inspired by the Paper “On inner classes” by A. Igarashi and B. Pierce

In [14] an axiomatic approach towards the semantics of FJI, Featherweight Java with Inner classes, essentially a subset of the 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. At a first glance the approach of reducing Javas 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 [14] has different models. Therefore the suggestion that the formal system of [14] 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° that the system has a model and hence it is a consistent system, and 2° that all models are isomorphic. Sometimes such a proposed system contains a rule with a premise which reads: 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 [14]. We shall show that it has many non-isomorphic models. We present a repair of Igarashis and Pierces calculus such that their ideas are preserved as close as possible.

The Algebraic Specifications do not have the Tennenbaum Property

It is commonly believed that a programmable model satisfying the axioms of a given algebraic specification guarantees good properties and is a correct implementation of the specification. This convinction might be related to the Tennenbaums property[Ten] of the arithmetic: every computable model of the Peano arithmetic of natural numbers is isomorphic to the standard model. Here, on the example of stacks, we show a model satisfying all axioms of the algebraic specification of stacks which can not be accepted as a good model in spite of the fact that it is defined by a program. For it enables to ”pop” a stack infinitely many times.

On a hierarchy of file types and a tower of their theories

We present here a general method of software development which uses both: algorithmic logic and object programming. The approach will be presented in the form of a case study.