Designing a Proof GUIfor Non-Experts

Evaluation of an Experiment

Martin Homik, Andreas Meier

Presentation by Christoph Benzmüller

UITP 2005, Edinburgh

ActiveMath GroupGerman Research Center for Artificial Intelligence

DFKI GmbH, Saarbrücken

Motivation

Typical proof GUI design:• Proof system centered• Too specific; For experts only

Non-Expert proof GUI design:• User centered• Deliver what the user needs!

Motivation (2)

MIPPA Project goals:• Interactive learning tool for math. proof• Underlying proof engine: Proof planner MULTI

Target group:• Undergraduate students• A-level pupils

Expert GUI: Loui

Towards a User Centered GUI

First step:

• Paper&Pencil student experiment

• Primary task:– Observe basic user wants and needs

4 Groups• 2 students in eachBackground:• Computer Science, Math, LogicNo design restrictions:• creativity/underlying system• use/invent functionalities freely

Experiment Setting

Design (120 min) Presentation (15 + 10 min) Discussion

Example Theorem: Irrationality of √2Use:• Definitions• Term rewriting• Island introduction• Contradiction

Experiment Remarks

This is no HCI experiment:

• We let users design.

• Users were already familiar with PP/Loui.

• Users were restricted to certain tasks.

Why?

• First attempt: obtaining inspiration

Textbook Example: √2 is irrational

„Assume that √2 is rational. Then, there are integers n,m that satisfy √2=n/m and that have no common divisors. From √2=n/m follows that 2*m2=n2 (1), which results in the fact that n2 is even. Then, n is even as well and there is an integer k such that n=2*k. The substitution of n in (1) by 2*k results in 2*m2=4*k2 which can be simplified to m2=2*k2. Hence, m2 and m are even as well. This is a contradiction to the fact that n,m are supposed to have no common divisor.“

Group A: Text-based

Textual presentation of a proof.The same way as taught at school.

Textual presentation of a proof.The same way as taught at school.

There exist no two integers m and n:• m and n being coprime• √2 =m/n

√2 is irrational

check proof

complete proof automatically

feedback

no logical notation√2 is irrational

Statement access

Group A: Operator Application

There exist no two integers m and n:• m and n being coprime• √2 =m/n

√2 is irrational

• select operator (e.g. indirect proof)

• select operator (e.g. indirect proof)

There exist no two integers m and n:• m and n being coprime• √2 =m/n

• mark statement with mouse• click „Pick“ button

• mark statement with mouse• click „Pick“ button

Group B: Bridge Building

AssumptionsAssumptions

Forward ReasoningForward Reasoning

GoalsGoals

Backward ReasoningBackward Reasoning

Clear separation between:• Assumptions and Goals• Forward and Backward Reasoning

Clear separation between:• Assumptions and Goals• Forward and Backward Reasoning

Group B: Control Panel

HistoryHistory

System supportSystem support

Group B: Method Iconisation

(Definition-) Expansion

ContradictionInsert island

(Definition-) Collapse

Group B: Operator Application

Bridge Construction ExampleUpper bank

Lower bank

√2 is not rational

Action: definition application

There exist no two integers m und n:• m and n being coprime • √2 =m/n

(Hypotheses)

(Theorem)

There exist two integers m and n:• m and n being coprime• √2 =m/n

m2 =2*n2

ContradictionAction: indirect proof

Action: term rewriting

Placing Islands

There exist two integers m and n:• m and n being coprime• √2 =m/n

m2 =2*n2

m is even n is even

Group C: Masking Operator Names

√2 is irrational

√2 is rational

mn: √2=m/n m, n are coprime

Proof presented as trees of statementsEdges = Story tellers „next do … to get …

Proof presented as trees of statementsEdges = Story tellers „next do … to get …

Group C: Masking Operator Names

√2 is irrational

√2 is rational√2 is rational

• mn: √2=m/n• m, n are coprime

• √2 is irrational• contradtion

mn: √2=m/n m, n are coprime

Group D: Notebooks

√2 is irrational

We assume: √2 is rational

There exist two numbers n and m in Z,Being coprime, such that √2=n/m

2m=n2

n2 is even

n is even

Linear proof style:• arrows denote relations• arrows labeled by operators

Linear proof style:• arrows denote relations• arrows labeled by operators

√2 is irrational

We assume: √2 is rational

There exist two numbers n and m in Z,Being coprime, such that √2=n/m

2m=n2

n2 is even

n is even

Group D: Operator Application

Search Search

List allList all

Conclusion

• Used Argument: „As taught at school.“ (???)– A lot of „User Wants and User Needs“– Partly questionable

Discussion results:• Presentation

– Simplified, nested statements– Bridge construction paradigm– Proof structuring (notebook, expansion, collapsing)

• Standard interaction facilities– Copy&Paste, Drag&Drop, etc.

Conclusion (2): System Support

• Automation Support– Of simple steps– Verification of introduced islands– On demand completion of gaps– Copy&Paste for sub proofs – History

• Feedback– Check proof/operator arguments– Help (e.g. explanations of operators)

• Hints- General advice: „Derive a contradiction!“- Rank suggestions- Overcome failure (suggest suitable input arguments)

Future Work

… towards a User Centered GUI ?

• Prototype development

• HCI evaluation