No abstract

In the article, we consider verification of programs with mutual recursion in the data driven functional parallel language Pifagor. In this language the program could be represented as a data flow graph, that has no control connections, and has only data relations. Under these conditions it is possible to simplify the process of formal verification, since there is no need to analyse resource conflicts, which are present in the systems with ordinary architectures. The proof of programs correctness is based on the elimination of mutual recursions by program transformation. The universal method of mutual recursion of an arbitrary number of functions elimination consists in constructing the universal recursive function that simulates all the functions in the mutual recursion. A natural number is assigned to each function in mutual recursion. The universal recursive function takes as its argument the number of a function to be simulated and the arguments of this function. In some cases of the indirect recursion it is possible to use a simpler method of program transformation, namely, the merging of the functions code into a single function. To remove mutual recursion of an arbitrary number of functions, it is suggested to construct a graph of all connected functions and transform this graph by removing functions that are not connected with the target function, then by merging functions with indirect recursion and finally by constructing the universal recursive function. It is proved that in the Pifagor language such transformations of functions as code merging and universal recursive function construction do not change the correctness of the initial program. An example of partial correctness proof is given for the program that parses a simple arithmetic expression. We construct the graph of all connected functions and demonstrate two methods of proofs: by means of code merging and by means of the universal recursive function.

Nowadays, due to software sophistication, programs correctness is more often proved by means of formal verification. The method of deduction based on Hoare logic could be used for any programming language and it has the capability of partial automation of the proof process. However, the method of deduction is not widely used for verification of parallel programs because of high complexity of the process. The usage of the functional data-flow paradigm of parallel programming allows to decrease the complexity of the proof process. In this article a proof process of correctness of functional data-flow parallel programs in the Pifagor language is considered. The proof process of a program correctness is considered as a tree where each node is a program data-flow graph, whose edges are marked with formulas in a specification language. The tree root is the initial program data-flow graph with a precondition and a postcondition, which describe restrictions on input variables and correctness conditions of the result of the program execution, respectively. Basic transformations of the data-flow graph are edge marking, equivalent transformation, splitting, folding of the program. By means of these transformations the data-flow graph is transformed and finally is reduced to a set of formulas in the specification language. If all these formulas are identically true, the program is correct. Several modules is distinguished in the system: "Program correctness prover", "Axioms and theorems library management system" and "Errors analysis and output of information about errors". According to this architecture, the toolkit for supporting formal verification was developed. The main functionality of the system implementation is considered.

A statically typed version of the data driven functional parallel computing model is proposed. It enables a representation of dynamically changing parallelism by means of asynchronous serial data ows. We consider the features of the syntax and semantics of the statically typed data driven functional parallel programming language Smile that supports asynchronous sequential ows. Our main idea is to apply the Hoar concept of communicating sequential processes to the computation control on the data readiness. It is assumed that on the data readiness a control signal is emi ed to inform the processes about the occurrence of certain events. e special feature of our approach is that the model is extended with the special asynchronous containers that can generate events on their partial lling. ese containers are a stream and a swarm, each of which has its own speci cs. A stream is used to process data which have identical type.e data comes sequentially and asynchronously at arbitrary time moments.e number of the incoming data elements is initially unknown, so the processing completes on the signal of the end of the stream. A swarm is used to contain independent data of the same type and may be used for the massive parallel operations performing. Unlike a stream, the swarm's size is xed and known in advance. General principles of the operations with the asynchronous sequential ows with an arbitrary order of data arrival are described. e use of the streams and the swarms in various situations is considered. We propose the language constructions which allow us to operate the swarms and streams and describe the speci cs of their application. We provide the sample functions to illustrate the use of the di erent approaches to description of the parallelism: recursive processing of the asynchronous ows, processing of the ows in an arbitrary or prede ned order of operations, direct access and access by the reference to the elements of the streams and swarms, pipelining of calculations. We give a preliminary parallelism assessment which depends on the ratio of the rates of data arrival and their processing. e proposed methods can be used in the development of the future languages and tool-kits of architecture-independent parallel programming.