We introduce the recently started Leo-III project — a Higher-Order Logic Theorem Prover and successor to LEO-II.

*Correspondence: [email protected] 1College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, China Full list of author information is available at the end of the article Abstract In this work, a prey-predator model with both state-dependent impulsive harvesting and constant rate harvesting is investigated, where the replenishment rate of prey and ...

It is surprising that despite the simplicity of its syntax, the λ-calculus hosts a large body of notation, abbreviations, naming conventions, etc. Our aim, as far as the notation throughout this work is concerned, is to remain consistent, clear and unambiguous as much as possible. By and large we adhere to the notation set down in Church’s Calculi of Lambda Conversion [4], and Barendregt’s The ...

Data Types In this chapter we give an introduction to the algebraic specification of abstract data types by means of a set of equations. See [10, 80] for lucid overviews of this field. 2.1 Algebraic specification We start with an example. Example 2.1 As a standard example we specify the natural numbers with addition and multiplication. The signature consists of the function 0 of arity zero, the...

Let T be a first order theory formulated in the language L and P a new relation symbol not in L. Let φ(P ) be an L ∪ {P}-sentence. Let us say that φ(P ) defines P implicitly in T if T proves φ(P ) ∧ φ(P ) → ∀x(P (x) ↔ P (x)). Beth’s definability theorem states that if φ(P ) defines P implicitlly in T then P (x) is equivalent to an L-formula. However, if we consider implicit definability in a gi...

It is well known that the Ackermann function can be defined via diagonalization from an iteration hierarchy (of Grzegorczyk type) which is built on a start function like the successor function. In this paper we study for a given start function g iteration hierarchies with a sublinear modulus h of iteration. In terms of g and h we classify the phase transition for the resulting diagonal function...

Assume λ is a singular limit of η supercompact cardinals, where η ≤ λ is a limit ordinal. We present two forcing methods for making λ+ the successor of the limit of the first η measurable cardinals while the tree property holding at λ+. The first method is then used to get, from the same assumptions, tree property at אη2+1 with the failure of SCH at אη2 . This extends results of Neeman and Sina...

We start with a simple proof of Leivant's normal form theorem for 1 1 formulas over nite successor structures. Then we use that normal form to prove the following: (i) over all nite structures, every 1 2 formula is equivalent to a 1 2 formula whose rst-order part is a boolean combination of existential formulas, and (ii) over nite successor structures, the Kolaitis-Thakur hierarchy of minimizat...

The goal of this paper is to extend the general solution to the ramification problem for the situation calculus to the case where causal rules specifying indirect effects of actions may form cycles or cycles through negation. We formulate causal dependencies as rules of inductive definitions. Then we give a semantics to a logic for representing inductive definitions (ID-logic) by translating de...

In this paper we study the definability and decidability of binary predicates for time granularity in monadic languages interpreted over finitely and infinitely layered structures. We focus our attention on the equi-level (resp. equi-column) predicate constraining two time points to belong to the same layer (resp. column) and on the horizontal (resp. vertical) successor predicate relating a tim...