### CS704: Lecture 6 The Church-Rosser Theorem

, 2010

"... This lecture presents the Church-Rosser Theorem (i.e., −→α,β is confluent). 1 ..."

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This lecture presents the Church-Rosser Theorem (i.e., −→α,β is confluent). 1

### Logic and Computerisation in mathematics?

, 2009

"... – If you give me an algorithm to solve Π, I can check whether this algorithm really solves Π. – But, if you ask me to find an algorithm to solve Π, I may go on forever trying but without success. • But, this result was already found by Aristotle: Assume a proposition Φ. – If you give me a proof of Φ ..."

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– If you give me an algorithm to solve Π, I can check whether this algorithm really solves Π. – But, if you ask me to find an algorithm to solve Π, I may go on forever trying but without success. • But, this result was already found by Aristotle: Assume a proposition Φ. – If you give me a proof of Φ, I can check whether this proof really proves Φ. – But, if you ask me to find a proof of Φ, I may go on forever trying but without success. • In fact, programs are proofs and much of computer science in the early part of the 20th century was built by mathematicians and logicians. • There were also important inventions in computer science made by physicists (e.g., von Neumann) and others, but we ignore these in this talk. ISR 2009, Brasiliá, Brasil 1An example of a computable function/solvable problem • E.g., 1.5 chicken lay down 1.5 eggs in 1.5 days. • How many eggs does 1 chicken lay in 1 day? • 1.5 chicken lay 1.5 eggs in 1.5 days. • Hence, 1 chicken lay 1 egg in 1.5 days. • Hence, 1 chicken lay 2/3 egg in 1 day. ISR 2009, Brasiliá, Brasil 2Unsolvability of the Barber problem • which man barber in the village shaves all and only those men who do not shave themselves? • If John was the barber then – John shaves Bill ⇐ ⇒ Bill does not shave Bill – John shaves x ⇐ ⇒ x does not shave x – John shaves John ⇐ ⇒ John does not shave John • Contradiction. ISR 2009, Brasiliá, Brasil 3Unsolvability of the Russell set problem

### The evolution of types and logic in the 20th century: A journey through Frege, Russell and . . .

- ILLC ALUMNI EVENT, AMSTERDAM 2004
, 2004

"... ..."

### Approved by:

, 1987

"... This dissertation presents three implementation models for the Scheme Programming Language. The first is a heap-based model used in some form in most Scheme implementations to date; the second is a new stack-based model that is considerably more efficient than the heap-based model at executing most ..."

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This dissertation presents three implementation models for the Scheme Programming Language. The first is a heap-based model used in some form in most Scheme implementations to date; the second is a new stack-based model that is considerably more efficient than the heap-based model at executing most programs; and the third is a new string-based model intended for use in a multiple-processor implementation of Scheme. The heap-based model allocates several important data structures in a heap, including actual parameter lists, binding environments, and call frames. The stack-based model allocates these same structures on a stack whenever possible. This results in Jess heap allocation, fewer memory references, shorter instruction sequences, less garbage collection, and more efficient use of memory. The string-based model allocates versions of these structures right in the program text, which is represented as a string of symbols. In the string-based model, Scheme programs are translated into an FFP language designed specifically to support Scheme. Programs in this language are directly executed by the

### Heriot-Watt University Edinburgh, Scotland

, 2005

"... The evolution of types and logic in the 20th century ∗ ..."

### Um Ceclo de Computeraçao

"... Brasiliá 2010Welcome to the fastest developing and most influential subject: Computer Science • Computer Science is by nature highly applied and needs much precision, foundation and theory. • Computer Science is highly interdisciplinary bringing many subjects together in ways that were not possible ..."

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Brasiliá 2010Welcome to the fastest developing and most influential subject: Computer Science • Computer Science is by nature highly applied and needs much precision, foundation and theory. • Computer Science is highly interdisciplinary bringing many subjects together in ways that were not possible before. • Many recent scientific results (e.g., in chemistry) would not have been possible without computers. • The Kepler Conjecture: no packing of congruent balls in Euclidean space has density greater than the density of the face-centered cubic packing. • Sam Ferguson and Tom Hales proved the Kepler Conjecture in 1998, but it was not published until 2006. • The Flyspeck project aims to give a formal proof of the Kepler Conjecture.

### MathLang, a framework for computerising

, 2007

"... Can we formalise a mathematical text, avoiding as much as possible the ambiguities of natural language, while still guaranteeing the following four goals? 1. The formalised text looks very much like the original mathematical text (and hence the content of the original mathematical text is respected) ..."

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Can we formalise a mathematical text, avoiding as much as possible the ambiguities of natural language, while still guaranteeing the following four goals? 1. The formalised text looks very much like the original mathematical text (and hence the content of the original mathematical text is respected). 2. The formalised text can be fully manipulated and searched in ways that respect its mathematical structure and meaning. 3. Steps can be made to do computation (via computer algebra systems) and proof checking (via proof checkers) on the formalised text. 4. This formalisation of text is not much harder for the ordinary mathematician than L ATEX. Full formalization down to a foundation of mathematics is not required, although allowing and supporting this is one goal. (No theorem prover’s language satisfies these goals.) University of West of England, Bristol 1A brief history • There are two influencing questions:

### MathLang, a framework for computerising

, 2007

"... Saarbruecken, GermanyWhat is the aim for MathLang? Can we formalise a mathematical text, avoiding as much as possible the ambiguities of natural language, while still guaranteeing the following four goals? 1. The formalised text looks very much like the original mathematical text (and hence the cont ..."

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Saarbruecken, GermanyWhat is the aim for MathLang? Can we formalise a mathematical text, avoiding as much as possible the ambiguities of natural language, while still guaranteeing the following four goals? 1. The formalised text looks very much like the original mathematical text (and hence the content of the original mathematical text is respected). 2. The formalised text can be fully manipulated and searched in ways that respect its mathematical structure and meaning. 3. Steps can be made to do computation (via computer algebra systems) and proof checking (via proof checkers) on the formalised text. 4. This formalisation of text is not much harder for the ordinary mathematician than L ATEX. Full formalization down to a foundation of mathematics is not required, although allowing and supporting this is one goal. (No theorem prover’s language satisfies these goals.) Saarbruecken, Germany 1A brief history • There are two influencing questions:

### century: A journey through Frege, Russell and

, 2004

"... The evolution of types and logic in the 20th ..."