Binary Representation of Natural Numbers

This study was supported in part by JSPS KAKENHI Grant Numbers JP17K00182. The author would also like to express gratitude to Prof. Yasunari Shidama for his support and encouragement.Binary representation of integers [5], [3] and arithmetic operations on them have already been introduced in Mizar Mathematical Library [8, 7, 6, 4]. However, these articles formalize the notion of integers as mapped into a certain length tuple of boolean values.In this article we formalize, by means of Mizar system [2], [1], the binary representation of natural numbers which maps ℕ into bitstreams.Shinshu University, Nagano, JapanGrzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Donald E. Knuth. The Art of Computer Programming, Volume 1: Fundamental Algorithms, Third Edition. Addison-Wesley, 1997.Hisayoshi Kunimune and Yatsuka Nakamura. A representation of integers by binary arithmetics and addition of integers. Formalized Mathematics, 11(2):175–178, 2003.Gottfried Wilhelm Leibniz. Explication de l’Arithmétique Binaire, volume 7. C. Gerhardt, Die Mathematische Schriften edition, 223 pages, 1879.Robert Milewski. Binary arithmetics. Binary sequences. Formalized Mathematics, 7(1): 23–26, 1998.Yasuho Mizuhara and Takaya Nishiyama. Binary arithmetics, addition and subtraction of integers. Formalized Mathematics, 5(1):27–29, 1996.Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4 (1):83–86, 1993.26322322

A lower bound on branching programs reading some bits twice

AbstractBy (1, + k(n))-branching programs (b.p.'s) we mean those b.p.'s which during each of their computations are allowed to test at most k(n) input bits repeatedly. For a Boolean function computable within polynomial time a trade-off is presented between the size and the number of repeatedly tested input bits of any b.p. P computing the function. Namely, if at most k(n) repeated tests are allowed, where log2 n ⩽ k(n) ⩽ n(1000 log2 n), then the size of P is at least exp(Ω(n(k(n)log2 n))12). This is exponential whenever k(n) ⩽ nα for a fixed α < 1 and superpolynomial whenever k(n) = o(nlog32 n).The presented result is a step towards a superpolynomial lower bound for 2-b.p.'s which is an open problem since 1984 when the first superpolynomial lower bounds for 1-b.p.'s were proven (Wegener, 1988; Žák, 1984). The present result is an improvement on (Žák, 1995)

Enforcing equilibria in multi-agent systems

We introduce and investigate Normative Synthesis: a new class of problems for the equilibrium verification that counters the absence of equilibria by purposely constraining multi-agent systems. We show that norms are powerful enough to ensure a positive answer to every instance of the equilibrium verification problem. Subsequently, we focus on two optimization versions, that aim at providing a solution in compliance with implementation costs. We show that the complexities of our procedures range between 2exptime and 3exptime, thus that the problems are no harder than the corresponding equilibrium verification ones

Flight Guidance System Validation Using SPIN

To verify the requirements for the mode control logic of a Flight Guidance System (FGS) we applied SPIN, a widely used software package that supports the formal verification of distributed systems. These requirements, collectively called the FGS specification, were developed at Rockwell Avionics & Communications and expressed in terms of the Consortium Requirements Engineering (CoRE) method. The properties to be verified are the invariants formulated in the FGS specification, along with the standard properties of consistency and completeness. The project had two stages. First, the FGS specification and the properties to be verified were reformulated in PROMELA, the input language of SPIN. This involved a semantics issue, as some constructs of the FGS specification do not have well-defined semantics in CoRE. Then we attempted to verify the requirements' properties using the automatic model checking facilities of SPIN. Due to the large size of the state space of the FGS specification an exhaustive state space analysis with SPIN turned out to be impossible. So we used the supertrace model checking procedure of SPIN that provides for a partial analysis of the state space. During this process, we found some subtle errors in the FGS specification

Optimized Program Extraction for Induction and Coinduction

The paper proves soundness of an optimized realizability interpretationfor a logic supporting strictly positive induction and coinduction. Theoptimization concerns the special treatment of Harrop formulas whichyields simpler extracted programs. It is shown that wellfounded inductionis an instance of strictly positive induction and from this a newcomputationally meaningful formulation of the Archimedean property forreal numbers is derived. An example of program extraction in computableanalysis shows that Archimedean induction can be used to eliminatecountable choic