Stability of the 7-3 Compressor Circuit for Wallace Tree. Part I

SummaryTo evaluate our formal verification method on a real-size calculation circuit, in this article, we continue to formalize the concept of the 7-3 Compressor (STC) Circuit [6] for Wallace Tree [11], to define the structures of calculation units for a very fast multiplication algorithm for VLSI implementation [10]. We define the circuit structure of the tree constructions of the Generalized Full Adder Circuits (GFAs). We then successfully prove its circuit stability of the calculation outputs after four and six steps. The motivation for this research is to establish a technique based on formalized mathematics and its applications for calculation circuits with high reliability, and to implement the applications of the reliable logic synthesizer and hardware compiler [5].

Formalization of mathematics through proof assistants

The formalization of mathematics in practice relies heavily on proof assistants and automatic theorem provers, therefore we studied what are the state of the art proof assitants and their limitations to understand what are the main challenges in making formalized mathematics common practice among mathematicians. We found out that curretly the two major dificulties in formalizing mathematics with proof assistants are due to steep learning curves in how to use these tools and due to a wide gap between the notation employed in these proof assistants and the currently used mathematical notation. We also developed a C++ library to develop proof assistants with great notational flexibility.

An experimental library of formalized Mathematics based on the univalent foundations

This is a short overview of an experimental library of Mathematics formalized in the Coq proof assistant using the univalent interpretation of the underlying type theory of Coq. I started to work on this library in February 2010 in order to gain experience with formalization of Mathematics in a constructive type theory based on the intuition gained from the univalent models (see Kapulkin et al. 2012).

Probability on Finite Set and Real-Valued Random Variables

Probability on Finite Set and Real-Valued Random Variables In the various branches of science, probability and randomness provide us with useful theoretical frameworks. The Formalized Mathematics has already published some articles concerning the probability: [23], [24], [25], and [30]. In order to apply those articles, we shall give some theorems concerning the probability and the real-valued random variables to prepare for further studies.

MoMM — FAST INTERREDUCTION AND RETRIEVAL IN LARGE LIBRARIES OF FORMALIZED MATHEMATICS

MoMM (in the narrower sense) is a tool allowing fast interreduction of a high number of clauses, dumping and fast-loading of the interreduced clause sets, and their use for real-time retrieval of matching clauses in an interactive mode. MoMM's main task is now providing these services for the world's largest body of formalized mathematics — the Mizar Mathematical Library (MML), which uses a richer formalism than just pure predicate logic. This task leads to a number of features (strength, speed, memory efficiency, dealing with the richer Mizar logic, etc.) required from MoMM, and we describe the choices taken in its implementation corresponding to these requirements. An important part of MoMM (in the wider sense) are the tools exporting the richer logic of MML into the clause-like format suitable for fast interreduction, and the tools allowing the use of MoMM as an interactive advisor for the authors of Mizar articles. These tools and choices taken in their implementation are also described here. Next we present some results of the interreduction of MML, which provide an interesting information about subsumption and repetition in the MML and can be used for its refactoring. This interreduction reveals that more than 2 percent of the main MML theorems are subsumed by others, and that for more than 50 percent of the internal lemmas proved by Mizar authors MoMM can provide useful advice for their justification. Finally some problems and possible future work are discussed.