We show that Tarski’s inductive definitions admit evidence-based interpretations of the first-order Peano Arithmetic PA that allow us to define the satisfaction and truth of the quantified formulas of PA constructively over the domain N of the natural numbers in two essentially different ways: (1) in terms of algorithmic verifiabilty; and (2) in terms of algorithmic computability. We argue that...

In the eld of inductive theorem proving syntactical diierences between the induction hypothesis and induction conclusion are used in order to guide the proof BvHS91, Hut90, Hut]. This method of guiding induction proofs is called rippling / coloring terms and there is considerable evidence of its success on practical examples. For equality reasoning we use these annotated terms to represent synt...

This paper provides a bridge in the gap between the model-theoretic and the algebraic side of modal correspondence theory. We give a new, algebraic proof of the classical Sahlqvist correspondence theorem, as well as a new, algebraic proof of the analogous result for the atomic inductive formulas, which form a proper extension of the Sahlqvist class.

This paper describes a technique of inductive proof based on model checking. It differs from previous techniques that combine induction and model checking in that the proof is fully mechanically checked and temporal variables (process identifiers, for example) may be natural numbers. To prove ∀n.φ(n) inductively, the predicate φ(n − 1) ⇒ φ(n) must be proved for all values of the parameter n. It...

Logic of discovery was developed in 1970’s as an answer to questions ”Can computers formulate and justify scientific hypotheses?” and ”Can they comprehend empirical data and process it rationally, using the apparatus of modern mathematical logic and statistics to try to produce a rational image of the observed empirical world?”. Logic of discovery is based on observational and theoretical langu...

Discrete Green’s functions are the inverses or pseudo-inverses of combinatorial Laplacians. We present compact formulas for discrete Green’s functions, in terms of the eigensystems of corresponding Laplacians, for products of regular graphs with or without boundary. Explicit formulas are derived for the cycle, torus, and 3-dimensional torus, as is an inductive formula for the t-dimensional toru...

In this paper we discuss combinatorial questions about lattice polytopes motivated by recent results on minimum distance estimation for toric codes. We also include a new inductive bound for the minimum distance of generalized toric codes. As an application, we give new formulas for the minimum distance of generalized toric codes for special lattice point configurations. 2010 MSC: 14M25, 14G50,...

In this paper we study automatic veri cation of proofs in process algebra Formulas of process algebra are represented by types in typed calculus Inhabitants terms of these types represent proofs The speci c typed calculus we use is the Calculus of Inductive Constructions as implemented in the interactive proof construction program COQ