arXiv subjects

Christine Gaßner

Publications and source records attributed to Christine Gaßner.

10 recordsLinked to original sources

arXiv2026

Abstract computation over first-order structures. Extras: From programs to decision trees I

Decisions and their consequences can be described and analyzed by means of decision trees. The decisions themselves depend on questions that, whenever possible, should be answered with yes or no. The original BSS machines over the real numbers are graphs with computation nodes and branching nodes for decisions. The evaluation of BSS machines and algebraic decision trees in computer-aided geometry which have been introduced for various types of numbers generally involves the evaluation of systems of literals of first-order logic. Flowchart-like representations and decision trees are also helpful for analyzing decisions made by BSS RAMs over first-order structures. All algorithms defined by machine-oriented programs of BSS RAMs can be illustrated using flowcharts, walks, and paths in program trees. The basic structures of these tools are graphs and their visual representations can help to characterize the behavior of individual BSS RAMs and other first-order machines. We offer a theoretical framework for linking various models. Here, we introduce program paths and transition systems for transforming partial configurations which are defined syntactically and can be extended and refined later. Finally, we define standard orders for program paths and present algorithms for enumerating finite program paths.

math.LO↗

arXiv2025

Abstract computation over first-order structures. Part IIb: Moschovakis' operator and other non-determinisms

BSS RAMs were introduced to provide a mathematical framework for characterizing algorithms over first-order structures. Non-deterministic BSS RAMs help to model different non-deterministic approaches. Here, we deal with different types of binary non-determinisms and study the consequences of the decidability of the identity relation and the decidability of finite sets consisting of one or two constants. We compare the binary non-determinism resulting from a non-deterministic branching process, the digital non-determinism resulting from the restriction of guesses to two constants, and some other non-determinisms resulting from the use of Moschovakis' operator applied to oracle sets restricted to tuples of constants. Moreover, we show that the performance capability and the efficiency of individual machines are influenced by the following properties. 1. The identity relation belongs to the underlying structure. 2. The identity is semi-decidable over the underlying structure. 3. Two single-element sets of constants are semi-decidable. 4. A set of two constants is semi-decidable. The order of these properties corresponds to the strength of their influence. In all cases mentioned, the semi-decidability of the sets implies their decidability.

math.LO↗

arXiv2025

Abstract computation over first-order structures. Part IIa: Moschovakis' operator and other non-determinisms

BSS RAMs over first-order structures help to characterize algorithms for processing objects by means of useful operations and relations. They are the result of a generalization of several types of abstract machines. We want to discuss whether this concept that allows a machine-oriented characterization of algorithms is sufficiently general for describing also other models of computation. Yiannis N. Moschovakis introduced a concept of abstract computability of functions on the basis of recursive definability over first-order structures. Moschovakis' search operator is the counterpart to the operator introduced by Stephen C. Kleene and suitable for structures without computable minima. To compare our concept with Moschovakis' generalization of the theory of recursive functions, we extend the abilities of BSS RAMs by an operator that makes it possible to provide information about computable functions and their inverses in a non-deterministic way. In Part IIb, we compare several non-determinisms, summarize effects resulting from the restriction of guesses to constants, and take into account properties such as the semi-decidability of oracle sets, the semi-decidability of the identity relation, and the recognizability of constants.

math.LO↗

arXiv2025

Abstract computation over first-order structures. Part I: Deterministic and non-deterministic BSS RAMs

Most ideas about what an algorithm is are very similar. Basic operations are used for transforming objects. The evaluation of internal and external states by relations has impact on the further process. A more precise definition can lead to a model of abstract computation over an arbitrary first-order structure. Formally, the algorithms can be determined by strings. Their meaning can be described purely mathematically by functions and relations derived from the operations and relations of a first-order structure. Our model includes models of computability and derivation systems from different areas of mathematics, logic, and computer science. To define the algorithms, we use so-called programs. Since we do this independently of their executability by computers, the so-called execution of our programs can be viewed as a form of abstract computation. This concept helps to highlight common features of algorithms that are independent of the underlying structures. Here, in Part I, we define BSS RAMs step by step. In Part II, we study Moschovakis' operator which is known from a general recursion theory over first-order structures. Later, we study hierarchies defined analogously to the arithmetical hierarchy by means of quantified formulas of an infinitary logic in this framework.

math.LO↗

arXiv2024

AC and the Independence of the Law of Trichotomy in Second-Order Henkin Logic

This paper focuses on the set HAC of 1-1 Ackermann axioms of choice in second-order predicate logic with Henkin interpretation (HPL). To answer a question posed by Michael Rathjen, we restrict the proof that the basic Fraenkel model of second order is a model of all n-m Ackermann axioms to the case where the Ackermann axioms are in HAC. In the second part, we show the independence of Hartogs' version of the law of trichotomy (TR) from HAC in HPL. A generalization of the latter proof implies the independence of TR from all Ackermann axioms in HPL. We conclude the paper with an open problem.

math.LO↗

arXiv2024

AC and the Independence of WO in Second-Order Henkin Logic, Part II

This article is concerned with the Axiom of Choice (AC) and the well-ordering theorem (WO) in second-order predicate logic with Henkin interpretation (HPL). We consider a principle of choice introduced by Wilhelm Ackermann (1935) and discussed also by David Hilbert and Ackermann (1938), by Günter Asser (1981), and by Benjamin Siskind, Paolo Mancosu, and Stewart Shapiro (2020). Our discussion is restricted to so-called Henkin-Asser structures of second order. Here, we give the technical details of our proof of the independence of WO from the so-called Ackermann axioms in HPL presented at the Colloquium Logicum in 2022. Most of the definitions used here can be found in Sections 1, 2, and 3 of Part I.

math.LO↗

arXiv2024

AC and the Independence of WO in Second-Order Henkin Logic, Part I

This article is concerned with the Axiom of Choice (AC) and the well-ordering theorem (WO) in second-order predicate logic with Henkin interpretation (HPL). We consider a principle of choice introduced by Wilhelm Ackermann (1935) and discussed also by David Hilbert and Ackermann (1938), by Günter Asser (1981), and by Benjamin Siskind, Paolo Mancosu, and Stewart Shapiro (2020). The discussion is restricted to so-called Henkin-Asser structures of second order. The language used is a many-sorted first-order language with identity. In particular, we give some of the technical details for a proof of the independence of WO from the so-called Ackermann axioms in HPL presented at the Colloquium Logicum in 2022.

math.LO↗

arXiv2024

Permutation Models of Second Order

Günter Asser (1981) introduced second-order permutation models. In this way, the Fraenkel-Mostowski-Specker method for defining models of ZFA was transferred to a new application area. To investigate the strength of second-order principles of choice in second-order predicate logic (PLII) with Henkin interpretation (HPL), we have extended the Fraenkel-Mostowski-Specker-Asser method. Here we discuss many details, including some useful references. Finally, we address a question raised by Stephen Mackereth.

math.LO↗

arXiv2024

Relationships between Principles of Choice in Second-Order Henkin Structures

We deal with the strength of classical second-order versions of the Axiom of Choice (AC) in second-order predicate logic (PLII) with Henkin interpretation (HPL). We use the known relationships between the so-called Zermelo-Asser axioms and the so-called Russell-Asser axioms and prove relationships between the so-called Ackermann axioms and the Zermelo-Asser axioms and between the so-called Asser axioms and the Zermelo-Asser axioms. In particular, we give the technical details of the proofs of our results presented at the DMV Annual Meeting 2022.

math.LO↗

arXiv2013

Strong Turing Degrees for Additive BSS RAM's

For the additive real BSS machines using only constants 0 and 1 and order tests we consider the corresponding Turing reducibility and characterize some semi-decidable decision problems over the reals. In order to refine, step-by-step, a linear hierarchy of Turing degrees with respect to this model, we define several halting problems for classes of additive machines with different abilities and construct further suitable decision problems. In the construction we use methods of the classical recursion theory as well as techniques for proving bounds resulting from algebraic properties. In this way we extend a known hierarchy of problems below the halting problem for the additive machines using only equality tests and we present a further subhierarchy of semi-decidable problems between the halting problems for the additive machines using only equality tests and using order tests, respectively.

cs.LO↗