Doktordisputats, Roskilde universitet.

Argumentation and proof have played a fundamental role in mathematics education in recent years. The author of this dissertation would like to investigate the development of the proving process within a dynamic geometry system in order to support tertiary students understanding the proving process. The strengths of this dynamic system stimulate students to formulate conjectures and produce arguments during the proving process. Through empirical research, we classified different levels of proving and proposed a methodological model for proving. This methodological model makes a contribution to improve students’ levels of proving and develop their dynamic visual thinking. We used Toulmin model of argumentation as a theoretical model to analyze the relationship between argumentation and proof. This research also offers some possible explanation so as to why students have cognitive difficulties in constructing proofs and provides mathematics educators with a deeper understanding on the proving process within a dynamic geometry system. / Argumentation und Beweis haben eine fundamentale Rolle in der Mathematikdidaktik in den letzten Jahren gespielt. Der Autor der vorliegenden Arbeit möchte die Entwicklung des Prozesses beweisen, in einer dynamischen Geometrie-System zu untersuchen, um das Verständnis der Studierenden im Tertiärbereich beweisen Prozess zu unterstützen. Die Stärken dieses dynamische System stimulieren Studierenden Vermutungen zu formulieren und Argumente zu produzieren während des Beweisprozesses. Durch empirische Forschung, klassifiziert wir verschiedene Niveaustufen zu beweisen und schlugen ein methodisches Modell für Beweisprozesse. Dieser methodologische Modell leistet einen Beitrag zur studentischen Niveaustufen des Beweises zu verbessern und entwickeln ihre dynamische-visuelle Denken. Wir verwendeten das Argumentationsmodell von Toulmin als theoretisches Modell, die Beziehung zwischen Argumentation und Beweis zu analysieren. Diese Forschung bietet auch einige mögliche Erklärung dafür, warum so Studierenden haben kognitive Schwierigkeiten bei der Beweis-Konstruktion und liefert Pädagogen mit einem tieferen Verständnis auf der Beweisprozess in einem dynamischen Geometriesystem.

Univ., Habil.-Schr. u.d.T.: Wansing, Heinrich: Proof-theoretic aspects of intensional and non-classical logics--Leipzig, 1997.

University, Diss., 2005--Münster (Westfalen).

In this thesis we see deductive systems for classical propositional and predicate logic which use deep inference, i.e. inference rules apply arbitrarily deep inside formulas, and a certain symmetry, which provides an involution on derivations. Like sequent systems, they have a cut rule which is admissible. Unlike sequent systems, they enjoy various new interesting properties. Not only the identity axiom, but also cut, weakening and even contraction are reducible to atomic form. This leads to inference rules that are local, meaning that the effort of applying them is bounded, and finitary, meaning that, given a conclusion, there is only a finite number of premises to choose from. The systems also enjoy new normal forms for derivations and, in the propositional case, a cut elimination procedure that is drastically simpler than the ones for sequent systems.

This thesis studies the design of deep-inference deductive systems. In the systems with deep inference, in contrast to traditional proof-theoretic systems, inference rules can be applied at any depth inside logical expressions. Deep applicability of inference rules provides a rich combinatorial analysis of proofs. Deep inference also makes it possible to design deductive systems that are tailored for computer science applications and otherwise provably not expressible. By applying the inference rules deeply, logical expressions can be manipulated starting from their sub-expressions. This way, we can simulate analytic proofs in traditional deductive formalisms. Furthermore, we can also construct much shorter analytic proofs than in these other formalisms. However, deep applicability of inference rules causes much greater nondeterminism in proof construction. This thesis attacks the problem of dealing with nondeterminism in proof search while preserving the shorter proofs that are available thanks to deep inference. By redesigning the deep inference deductive systems, some redundant applications of the inference rules are prevented. By introducing a new technique which reduces nondeterminism, it becomes possible to obtain a more immediate access to shorter proofs, without breaking certain proof theoretical properties such as cutelimination. Different implementations presented in this thesis allow to perform experiments on the techniques that we developed and observe the performance improvements. Within a computation-as-proof-search perspective, we use deepinference deductive systems to develop a common proof-theoretic language to the two fields of planning and concurrency.

This thesis studies the design of deep-inference deductive systems. In the systems with deep inference, in contrast to traditional proof-theoretic systems, inference rules can be applied at any depth inside logical expressions. Deep applicability of inference rules provides a rich combinatorial analysis of proofs. Deep inference also makes it possible to design deductive systems that are tailored for computer science applications and otherwise provably not expressible. By applying the inference rules deeply, logical expressions can be manipulated starting from their sub-expressions. This way, we can simulate analytic proofs in traditional deductive formalisms. Furthermore, we can also construct much shorter analytic proofs than in these other formalisms. However, deep applicability of inference rules causes much greater nondeterminism in proof construction. This thesis attacks the problem of dealing with nondeterminism in proof search while preserving the shorter proofs that are available thanks to deep inference. By redesigning the deep inference deductive systems, some redundant applications of the inference rules are prevented. By introducing a new technique which reduces nondeterminism, it becomes possible to obtain a more immediate access to shorter proofs, without breaking certain proof theoretical properties such as cutelimination. Different implementations presented in this thesis allow to perform experiments on the techniques that we developed and observe the performance improvements. Within a computation-as-proof-search perspective, we use deepinference deductive systems to develop a common proof-theoretic language to the two fields of planning and concurrency.

In this thesis we see deductive systems for classical propositional and predicate logic which use deep inference, i.e. inference rules apply arbitrarily deep inside formulas, and a certain symmetry, which provides an involution on derivations. Like sequent systems, they have a cut rule which is admissible. Unlike sequent systems, they enjoy various new interesting properties. Not only the identity axiom, but also cut, weakening and even contraction are reducible to atomic form. This leads to inference rules that are local, meaning that the effort of applying them is bounded, and finitary, meaning that, given a conclusion, there is only a finite number of premises to choose from. The systems also enjoy new normal forms for derivations and, in the propositional case, a cut elimination procedure that is drastically simpler than the ones for sequent systems.

In this thesis I study several deductive systems for linear logic, its fragments, and some noncommutative extensions. All systems will be designed within the calculus of structures, which is a proof theoretical formalism for specifying logical systems, in the tradition of Hilbert's formalism, natural deduction, and the sequent calculus. Systems in the calculus of structures are based on two simple principles: deep inference and top-down symmetry. Together they have remarkable consequences for the properties of the logical systems. For example, for linear logic it is possible to design a deductive system, in which all rules are local. In particular, the contraction rule is reduced to an atomic version, and there is no global promotion rule. I will also show an extension of multiplicative exponential linear logic by a noncommutative, self-dual connective which is not representable in the sequent calculus. All systems enjoy the cut elimination property. Moreover, this can be proved independently from the sequent calculus via techniques that are based on the new top-down symmetry. Furthermore, for all systems, I will present several decomposition theorems which constitute a new type of normal form for derivations.

In this thesis I study several deductive systems for linear logic, its fragments, and some noncommutative extensions. All systems will be designed within the calculus of structures, which is a proof theoretical formalism for specifying logical systems, in the tradition of Hilbert's formalism, natural deduction, and the sequent calculus. Systems in the calculus of structures are based on two simple principles: deep inference and top-down symmetry. Together they have remarkable consequences for the properties of the logical systems. For example, for linear logic it is possible to design a deductive system, in which all rules are local. In particular, the contraction rule is reduced to an atomic version, and there is no global promotion rule. I will also show an extension of multiplicative exponential linear logic by a noncommutative, self-dual connective which is not representable in the sequent calculus. All systems enjoy the cut elimination property. Moreover, this can be proved independently from the sequent calculus via techniques that are based on the new top-down symmetry. Furthermore, for all systems, I will present several decomposition theorems which constitute a new type of normal form for derivations.