Results

**1 - 5**of**5**### Formal Proof: Reconciling Correctness and Understanding

"... A good proof is a proof that makes us wiser. Manin [41, p. 209]. Abstract. Hilbert’s concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of pr ..."

Abstract
- Add to MetaCart

A good proof is a proof that makes us wiser. Manin [41, p. 209]. Abstract. Hilbert’s concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of proof assistants was followed by an impressive record of deep mathematical theorems formally proved. Formal proof is practically achievable. With formal proof, correctness reaches a standard that no pen-and-paper proof can match, but an essential component of mathematics — the insight and understanding — seems to be in short supply. So, what makes a proof understandable? To answer this question we first suggest a list of symptoms of understanding. We then propose a vision of an environment in which users can write and check formal proofs as well as query them with reference to the symptoms of understanding. In this way, the environment reconciles the main features of proof: correctness and understanding. 1

### Automated Discovery and Proof in Three Combinatorial Problems

"... In this Ph.D. disseration, I will go over advances I have made in three combinatorial problems. The running theme throughout these three problems is the novel use of computers to aid not only in the discovery of the theorems proved, but also in the proofs themselves. The first problem involves the e ..."

Abstract
- Add to MetaCart

In this Ph.D. disseration, I will go over advances I have made in three combinatorial problems. The running theme throughout these three problems is the novel use of computers to aid not only in the discovery of the theorems proved, but also in the proofs themselves. The first problem involves the enumeration of spanning trees in grid graphs- graphs of the form G×Pn (or Cn) for arbitrary G. An enumeration scheme is developed based on the partitions of [n], yielding an algorithmic method to completely solve the sequence for any G. These techniques yield a surprising consequence: sequences obtained in this manner are divisibility sequences. The second problem concerns the quantity f∆(n), defined as the size of the largest subset of [n] avoiding differences in ∆. Originally motivated by the Triangle Conjecture of Schützenberger and Perrin, we again define an enumeration scheme that will find, and prove automatically, the sequence f∆(n) ∞ n=1 for any prescribed ∆. Although the Triangle Conjecture has long been refuted, we present an asymptotic version of it and prove it. The final problem is the firefighter problem, a dynamic graph theory problem modeling the spread of diseases, information,

### Efficient Rough Set Theory Merging

"... Abstract. Theory exploration is a term describing the development of a formal (i.e. with the help of an automated proof-assistant) approach to selected topic, usually within mathematics or computer science. This activityhoweverusuallydoesn’t reflecttheviewofscience consideredasa whole, notas separat ..."

Abstract
- Add to MetaCart

Abstract. Theory exploration is a term describing the development of a formal (i.e. with the help of an automated proof-assistant) approach to selected topic, usually within mathematics or computer science. This activityhoweverusuallydoesn’t reflecttheviewofscience consideredasa whole, notas separated islands ofknowledge. Merging theoriesessentially has its primary aim of bridging these gaps between specific disciplines. As we provided formal apparatus for basic notions within rough set theory (as e.g. approximation operators and membership functions), we try to reuse the knowledge which is already contained in available repositories of computer-checked mathematical knowledge, or which can be obtained in a relatively easy way. We can point out at least three topics here: topological aspects of rough sets – as approximation operators have properties of the topological interior and closure; lattice-theoretic approach giving the algebraic viewpoint (e.g. Stone algebras); possible connections with formal concept analysis. In such a way we can give the formal characterization of rough sets in terms of topologies or orders. Although fully formal, still the approach can be revised to keep the uniformity all the time.

### Computer-Driven Searching for Axiomatization of Rough Sets

"... Abstract. The formalization of rough sets in a way understable by machines seems to be far beyond the test phase. For further research, we try to encode some problems within RST and as the testbed of already developed foundations – and in the same time as a payoff of the established framework, we sh ..."

Abstract
- Add to MetaCart

Abstract. The formalization of rough sets in a way understable by machines seems to be far beyond the test phase. For further research, we try to encode some problems within RST and as the testbed of already developed foundations – and in the same time as a payoff of the established framework, we shed some new light on the well-known question of generalization of rough sets and the axiomatization of approximation operators in terms of (various types of) relations. 1

### Chapter 1 Checking proofs

"... Argumentative practice in mathematics evidently takes a number of shapes. An important part of understanding mathematical argumentation, putting aside its special subject matters (numbers, shapes, spaces, sets, functions, etc.), is that mathematical argument often tends toward formality, and it ofte ..."

Abstract
- Add to MetaCart

Argumentative practice in mathematics evidently takes a number of shapes. An important part of understanding mathematical argumentation, putting aside its special subject matters (numbers, shapes, spaces, sets, functions, etc.), is that mathematical argument often tends toward formality, and it often has superlative epistemic goals: