Candidate One-Way Functions and One-Way Permutations Based on Quasigroup String Transformations

In this paper we propose a definition and construction of a new family of one-way candidate functions ${\cal R}_N:Q^N \to Q^N$, where $Q=\{0,1,...,s-1\}$ is an alphabet with $s$ elements. Special instances of these functions can have the additional property to be permutations (i.e. one-way permutations). These one-way functions have the property that for achieving the security level of $2^n$ computations in order to invert them, only $n$ bits of input are needed. The construction is based on quasigroup string transformations. Since quasigroups in general do not have algebraic properties such as associativity, commutativity, neutral elements, inverting these functions seems to require exponentially many readings from the lookup table that defines them (a Latin Square) in order to check the satisfiability for the initial conditions, thus making them natural candidates for one-way functions.Comment: Submitetd to conferenc

Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?

Classical interpretations of Goedel's formal reasoning imply that the truth of some arithmetical propositions of any formal mathematical language, under any interpretation, is essentially unverifiable. However, a language of general, scientific, discourse cannot allow its mathematical propositions to be interpreted ambiguously. Such a language must, therefore, define mathematical truth verifiably. We consider a constructive interpretation of classical, Tarskian, truth, and of Goedel's reasoning, under which any formal system of Peano Arithmetic is verifiably complete. We show how some paradoxical concepts of Quantum mechanics can be expressed, and interpreted, naturally under a constructive definition of mathematical truth.Comment: 73 pages; this is an updated version of the NQ essay; an HTML version is available at http://alixcomsi.com/Do_Goedel_incompleteness_theorems.ht

Theories of analytic monads

We characterize the equational theories and Lawvere theories that correspond to the categories of analytic and polynomial monads on Set, and hence also the categories of the symmetric and rigid operads in Set. We show that the category of analytic monads is equivalent to the category of regular-linear theories. The category of polynomial monads is equivalent to the category of rigid theories, i.e. regular-linear theories satisfying an additional global condition. This solves a problem A. Carboni and P. T. Johnstone. The Lawvere theories corresponding to these monads are identified via some factorization systems.Comment: 29 pages. v2: minor correction

Potential infinity, abstraction principles and arithmetic (Leniewski Style)

This paper starts with an explanation of how the logicist research program can be approached within the framework of Leśniewski’s systems. One nice feature of the system is that Hume’s Principle is derivable in it from an explicit definition of natural numbers. I generalize this result to show that all predicative abstraction principles corresponding to second-level relations, which are provably equivalence relations, are provable. However, the system fails, despite being much neater than the construction of Principia Mathematica (PM). One of the key reasons is that, just as in the case of the system of PM, without the assumption that infinitely many objects exist, (renderings of) most of the standard axioms of Peano Arithmetic are not derivable in the system. I prove that introducing modal quantifiers meant to capture the intuitions behind potential infinity results in the (renderings of) axioms of Peano Arithmetic (PA) being valid in all relational models (i.e. Kripke-style models, to be defined later on) of the extended language. The second, historical part of the paper contains a user-friendly description of Leśniewski’s own arithmetic and a brief investigation into its properties

On Transformations of Interactive Proofs that Preserve the Prover's Complexity

Goldwasser and Sipser [GS89] proved that every interactive proof system can be transformed into a public-coin one (a.k.a., an Arthur-Merlin game). Their transformation has the drawback that the computational complexity of the prover's strategy is not preserved. We show that this is inherent, by proving that the same must be true of any transformation which only uses the original prover and verifier strategies as "black boxes". Our negative result holds even if the original proof system is restricted to be honest-verifier perfect zero knowledge and the transformation can also use the simulator as a black box. We also examine a similar deficiency in a transformation of Fürer et al. [FGM+89] from interactive proofs to ones with perfect completeness. We argue that the increase in prover complexity incurred by their transformation is necessary, given that their construction is a black-box transformation which works regardless of the verifier's computational complexity.Engineering and Applied Science