The Naproche System Daniel K uhlwein University of Nijmegen - - PowerPoint PPT Presentation

Intro CNL PRS ATPs Future Work

The Naproche System

Daniel K¨ uhlwein

University of Nijmegen daniel.kuehlwein@gmail.com http://www.naproche.net

25th Mai 2011

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Outline

1

Introduction

2

The Naproche CNL

3

Proof Representation Structures

4

Checking Naproche Texts

5

Future Work

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The Naproche Project

The Naproche project (Natural language Proof Checking) is a joint project of the University of Bonn and the University of Duisburg-Essen. They study the semi-formal language of mathematics (SFLM) as used in journals and textbooks from the perspectives of linguistics and mathematics. A central goal of Naproche is to develop a controlled natural language (CNL) for mathematical texts and implement a system, the Naproche system, which can check texts written in this CNL for logical correctness using methods from computational linguistics and automatic theorem proving.

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The Naproche CNL and the Naproche system

The Naproche CNL is a controlled natural language for mathematical texts, i.e. a controlled subset of SFLM. The Naproche system translates Naproche CNL texts first into Proof Representation Structures (PRSs), an adapted version of Discourse Representation Structures. PRSs are further translated into lists of first-order formulas which are used for checking the logical correctness of a Naproche text using automated theorem provers (ATPs).

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Overview of the (old) Naproche system

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Outline

1

Introduction

2

The Naproche CNL

3

Proof Representation Structures

4

Checking Naproche Texts

5

Future Work

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An Example

Example (Euclid’s Elements) Theorem 2. Let b and c be distinct points. Assume a is a point such that a = b and a = c. Then there is a point v such that d(b, c) = d(a, v). Proof. By Theorem 1 there is a point d, such that d(d, a) = d(d, b) = d(a, b). Let M be the line such that b and d are on M. Let α be the circle such that b is the center of α, and c is on α. There is a point u such that u is on α, and u is on M and b is between d and u. Let N be the line L such that d and a are

• n L. Let β be the circle δ such that d is the center of δ, and u is
• n δ. There is a point v such that v is on β, and v is on N and a

is between v and d. Then d(b, c) = d(b, u). Hence, d(d, u) = d(d, v). By Lemma A d(b, u) = d(a, v). Then d(b, c) = d(a, v). qed.

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Features

Input in L

AT

EX. Axioms, theorems, definitions, lemmas, assumptions. Nouns, adjectives, verbs, natural language quantification and negation. Subclauses with such that. Natural language connectors like and, or, i.e., if and iff. Sentences in a proof can start with words like then, hence, therefore etc. Decent formula parser (e.g. d(d, a) = d(d, b) = d(a, b)). Plurals and definite descriptions. Basic induction.

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Collective vs. distributive readings of plurals

“a and b are foo” could mean either foo(a) ∧ foo(b) or foo(a, b). Usually, words have a preferred reading which is incorporated in the dictionary. The Naproche CNL can handle both collective and distributive readings of plurals:

Example

Let b and c be distinct points. The example is understood as point(b) ∧ point(c) ∧ distinct(b, c).

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Definite descriptions

A definite descriptions is a definite noun phrase referring to a single object by a restricting property whose extension contains exactly one object. abcdef “Let α be the circle such that b is the center of α” The presupposition of a singular definite description with the restricting property F (“the F”) is that there is a unique object with property F. This presupposition can be divided into two separate presuppositions:

An existential presupposition, claiming that there is at least one F A uniqueness presupposition, claiming that there is at most one F

Upon encountering a definite description, both presuppositions are verified.

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Induction

The Naproche CNL supports basic induction:

Example (Landau’s Grundlagen der Analysis)

Theorem 2: For all x x′ = x. Proof: By axiom 3, 1′ = 1. Suppose x′ = x. Then by theorem 1, (x′)′ = x′. Thus by induction, for all x x′ = x. Qed. The keywords “by induction ∀x ϕ(x)” trigger an induction verification: If nat(x), ϕ(1) and (nat(n) ∧ ϕ(n)) → ϕ(n + 1) holds, then ∀x nat(x) → ϕ(x).

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Outline

1

Introduction

2

The Naproche CNL

3

Proof Representation Structures

4

Checking Naproche Texts

5

Future Work

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Quantifiers in natural language

How should we translate natural language quantifiers?

Example

Every farmer is bald. → ∀x farmer(x) → bald(x) A farmer is bald. → ∃x farmer(x) ∧ bald(x) So “every” denotes universal quantification and “a” denotes existential quantification.

Example

Every farmer who owns a donkey beats it. → ∀x (farmer(x) ∧ ∃y (donkey(y)) ∧ owns(x, y)) → beats(x, y) The “right” translation is ∀x, y(farmer(x) ∧ donkey(y) ∧ owns(x, y)) → beats(x, y) The simple approach doesn’t work here.

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DRT, DRS and PRS

Discourse Representation Theory is a way to solve the problem of anaphora resolution. Non-anaphoric noun phrases introduce discourse referents which are used for the binding of anaphoric expressions. In DRT, discourse representation structures (DRS) are used to disambiguate anaphora. DRSs can easily be translated into first order logic. Proof Representation Structures (PRS) are an extension of DRSs for mathematical texts. PRSs are the linguistic format used in the Naproche System.

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Constituents of PRSs

A PRS has five constituents, which we display as “boxes”:

i d1, . . . , dm m1, . . . , mn c1 . . . cl r1, . . . , rk

i is the identifier of the PRS. d1, . . . , dm are discourse referents. m1, . . . , mn are mathematical referents. c1, . . . , cl are PRS conditions. r1, . . . , rk are textual references.

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A PRS Example

Show example on www.naproche.net.

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Outline

1

Introduction

2

The Naproche CNL

3

Proof Representation Structures

4

Checking Naproche Texts

5

Future Work

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Overview of the (old) Naproche system

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The checking algorithm: Basics

The checking algorithm keeps a list of first-order formulas considered to be true, called premises, which gets continuously updated during the checking process. The conditions of a PRS are checked sequentially. Each condition is checked under the currently active premises. According to the kind of condition, the Naproche system creates

• bligations in the TPTP format which have to be checked by an

ATP.

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An Example

Show example on www.naproche.net.

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Problems and possible solutions

Often, ATPs are unable to verify a proof obligation. This might mean that that particular proof step was wrong, or that the ATP was simply too weak to prove it. The Naproche system has several mechanisms to simplify the ATP proof tasks: Conjunction splitting and existential elimination. Skolemization. Premise Selection.

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Conjunction splitting and existential elimination

Two simple ways of simplifying proof tasks are conjunction splitting and existential elimination: If the conjecture is a conjunction ϕ ∧ ψ we split the proof obligation in two obligation c1 : ϕ and c2 : ψ. If the conjecture is of the form ∃x ϕ(x) ∧ x = a we can simplify it to ϕ(a). (Example: x is a point.)

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Skolemization

Another method to simplify proof tasks is by simplifying the premises: Assumptions can lead to long premises of the form ∀x ϕ → ∃y ψ1(y) ∧ ψ2(y)... ∧ ψn(y) By skolemizing the existential variables, we can split this formula into n formulas: ∀x ϕ → ψ1(skolem(x)), ∀x ϕ → ψ2(skolem(x))...

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Premise Selection

The number of premises determines the search space of the ATP. Each unnecessary premise increases the difficulty of finding a proof. We used the text structure to predict the usefulness of premises, namely:

Explicit References Textual Adjacency Logical Relevance

This ’heuristic’ approach to premise relevance ties with or

• utperforms the APRILS relevance order which is based on a latent

semantic analysis.

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Outline

1

Introduction

2

The Naproche CNL

3

Proof Representation Structures

4

Checking Naproche Texts

5

Future Work

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Future Work

Dynamic formula parsing. Better UI (e.g. saving and loading of unfinished proofs).

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Thanks for listening

Try out the web interface to Naproche: http://www.naproche.net

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