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Revised proofs of Kenneth Arrow's impossibility theorem have been presented in prose form, incorporating novel ideas such as decisive sets and pivotal voters. This study develops another approach to proving the theorem. Using a proof…
The authors' ATR programming formalism is a version of call-by-value PCF under a complexity-theoretically motivated type system. ATR programs run in type-2 polynomial-time and all standard type-2 basic feasible functionals are ATR-definable…
Iterative equation is an equality with an unknown function and its iterates. There were not found a result on iterative equations with multiplication of iterates of the unknown function on $\mathbb{R}$. In this paper we use an exponential…
An approach for encoding abstract dialectical frameworks and their semantics into classical higher-order logic is presented. Important properties and semantic relationships are formally encoded and proven using the proof assistant…
Some iterative techniques are defined to solve reversible inverse problems and a common formulation is explained. Numerical improvements are suggested and tests validate the methods.
We propose a notion of iterating functions $f:X^{k}\rightarrow X$ in a way that represents recurrence relations of the form $a_{n+k}=f(a_{n},a_{n+1},...,a_{n+k-1})$. We define a function as $n$-involutory when its $n$th iterate is the…
There are two possible computational interpretations of second-order arithmetic: Girard's system F or Spector's bar recursion and its variants. While the logic is the same, the programs obtained from these two interpretations have a…
Many automatic theorem-provers rely on rewriting. Using theorems as rewrite rules helps to simplify the subgoals that arise during a proof. LCF is an interactive theorem-prover intended for reasoning about computation. Its implementation of…
Existing computer algebra packages do not fully support quantum mechanics calculations in Dirac's notation. I present the foundation for building such support: a mathematical system for the symbolic manipulation of expressions used in the…
We develop the Akhiezer iteration, a generalization of the classical Chebyshev iteration, for the inner product-free, iterative solution of indefinite linear systems using orthogonal polynomials for measures supported on multiple, disjoint…
We present variants of Goodstein's theorem that are equivalent to arithmetical comprehension and to arithmetical transfinite recursion, respectively, over a weak base theory. These variants differ from the usual Goodstein theorem in that…
In this paper, several modifications are introduced to the functional approximation method iterLap to reduce the approximation error, including stopping rule adjustment, proposal of new residual function, starting point selection for…
To reversify an arbitrary sequential algorithm $A$, we gently instrument $A$ with bookkeeping machinery. The result is a step-for-step reversible algorithm that mimics $A$ step-for-step and stops exactly when $A$ does. Without loss of…
We report on the mechanization of (preference-based) conditional normative reasoning. Our focus is on Aqvist's system E for conditional obligation, and its extensions. Our mechanization is achieved via a shallow semantical embedding in…
In constraining iterative processes, the algorithmic operator of the iterative process is pre-multiplied by a constraining operator at each iterative step. This enables the constrained algorithm, besides solving the original problem, also…
The authors' ATR programming formalism is a version of call-by-value PCF under a complexity-theoretically motivated type system. ATR programs run in type-2 polynomial-time and all standard type-2 basic feasible functionals are ATR-definable…
Recent analysis of classical algorithms resulted in their axiomatization as transition systems satisfying some simple postulates, and in the formulation of the Abstract State Machine Theorem, which assures us that any classical algorithm…
In this work the numerical solution of acoustic tomography problem based on the iterative and functional-analytical algorithms is considered. The mathematical properties of these algorithms were previously described in works of R.G.Novikov…
We propose a formalization of generic algorithms that includes analog algorithms. This is achieved by reformulating and extending the framework of abstract state machines to include continuous-time models of computation. We prove that every…
Term rewriting has a significant presence in various areas, not least in automated theorem proving where it is used as a proof technique. Many theorem provers employ specialised proof tactics for rewriting. This results in an interleaving…