Is Constructive Logic relevant for Computer Science? Thorsten - - PowerPoint PPT Presentation

Is Constructive Logic relevant for Computer Science?

Thorsten Altenkirch University of Nottingham

BCTCS 05 – p.1/16

Birth of Modern Mathematics

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Birth of Modern Mathematics

Isaac Newton (1642 - 1727)

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Birth of Modern Mathematics

Isaac Newton (1642 - 1727) 1687: Philosophiae Naturalis Principia Mathematica

BCTCS 05 – p.2/16

19/20th century: Foundations?

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19/20th century: Foundations?

Frege (1848-1925) Russell (1872-1970)

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• 1925: ZF set theory

Zermelo (1871-1953) Fraenkel (1891-1965)

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• 1925: ZF set theory

Zermelo (1871-1953) Fraenkel (1891-1965)

End of story ?

BCTCS 05 – p.4/16

Mathematics is universal

The foundations which are good for mathematical reasoning within natural sciences are equally useful in Computer Science.

BCTCS 05 – p.5/16

Constructivism?

BCTCS 05 – p.6/16

Constructivism?

Computer Science focusses on constructive solutions to problems.

BCTCS 05 – p.6/16

Constructivism?

Computer Science focusses on constructive solutions to problems.

Classical Mathematics is based on the platonic idea of truth.

BCTCS 05 – p.6/16

Constructivism?

Computer Science focusses on constructive solutions to problems.

Classical Mathematics is based on the platonic idea of truth.

Constructive Mathematics is based on the notion of evidence or proof.

BCTCS 05 – p.6/16

BHK: Programs are evidence

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BHK: Programs are evidence

Brouwer (1881-1966) Heyting (1898-1980) Kolmogorov (1903-1987)

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, classically

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, classically

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, classically

The same truth table shows that

BCTCS 05 – p.8/16

BHK semantics

BCTCS 05 – p.9/16

BHK semantics

Evidence for

is given by pairs:

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BHK semantics

Evidence for

is given by pairs:

Evidence for

is tagged evidence for

• r

.

BCTCS 05 – p.9/16

BHK semantics

Evidence for

is given by pairs:

Evidence for

is tagged evidence for

• r

.

Evidence for

is a program constructing evidence for

from evidence for

.

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, constructively

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, constructively

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, constructively

The program is invertible, because the right hand sides are patterns.

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, constructively

The program is invertible, because the right hand sides are patterns.

This shows that the types are isomorphic.

BCTCS 05 – p.10/16

Predicate logic

BCTCS 05 – p.11/16

Predicate logic

Evidence for

is a function

which assigns to each

evidence for

.

BCTCS 05 – p.11/16

Predicate logic

Evidence for

is a function

which assigns to each

evidence for

.

Evidence for

is a pair

where

and

.

BCTCS 05 – p.11/16

Predicate logic

Evidence for

is a function

which assigns to each

evidence for

.

Evidence for

is a pair

where

and

.

We need dependent types!

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Propositions = Types

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Propositions = Types

Per Martin-L¨

• f

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Propositions = Types

Per Martin-L¨

• f

Martin-L¨

• f Type Theory

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Propositions = Types

Per Martin-L¨

• f

Martin-L¨

• f Type Theory

Implementations: NuPRL, LEGO, ALF , COQ, AGDA, Epigram . . .

BCTCS 05 – p.12/16

BCTCS 05 – p.13/16

We cannot prove

, where

, for an undecided proposition

.

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We cannot prove

, where

, for an undecided proposition

.

• ■

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We cannot prove

, where

, for an undecided proposition

.

• ■

is provable, i.e. Prime is decidable.

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We cannot prove

, where

, for an undecided proposition

.

• ■

is provable, i.e. Prime is decidable.

Indeed, the proof is the program which decides Prime.

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We cannot prove

, where

, for an undecided proposition

.

• ■

is provable, i.e. Prime is decidable.

Indeed, the proof is the program which decides Prime.

BCTCS 05 – p.13/16

We cannot prove

, where

, for an undecided proposition

.

• ■

is provable, i.e. Prime is decidable.

Indeed, the proof is the program which decides Prime.

is not provable, because Halt is undecidable.

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The classical Babelfish

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The classical Babelfish

Classical reasoner says: Babelfish translates to:

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The classical Babelfish

Classical reasoner says: Babelfish translates to:

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The classical Babelfish

Classical reasoner says: Babelfish translates to:

BCTCS 05 – p.14/16

The classical Babelfish

Classical reasoner says: Babelfish translates to:

BCTCS 05 – p.14/16

The classical Babelfish

Classical reasoner says: Babelfish translates to:

BCTCS 05 – p.14/16

The classical Babelfish

Classical reasoner says: Babelfish translates to:

Negative translation

BCTCS 05 – p.14/16

The classical Babelfish

Classical reasoner says: Babelfish translates to:

Negative translation

is traslated to

BCTCS 05 – p.14/16

The classical Babelfish

Classical reasoner says: Babelfish translates to:

Negative translation

is traslated to

which is constructively provable.

BCTCS 05 – p.14/16

The classical Babelfish

Classical reasoner says: Babelfish translates to:

Negative translation

is traslated to

which is constructively provable.

A classical reasoner is somebody who is unable to say anything positive.

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The Axiom of Choice ?

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The Axiom of Choice ?

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The Axiom of Choice ?

is provable constructively.

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The Axiom of Choice ?

is provable constructively.

However, its negative translation:

is not.

BCTCS 05 – p.15/16

The Axiom of Choice ?

is provable constructively.

However, its negative translation:

is not.

There is empirical evidence that CAC is consistent.

BCTCS 05 – p.15/16

Summary

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