Improvements and Extensions of the EG Interface

1. Improvements and Extensions of the EG Interface Fall 2002

2. Improvements and ExtensionsAreas of Interest • The interface should satisfy demands of the following 3 areas: • Educational: The interface should be something that serves an educational purpose, i.e. it teaches logical concepts to the user. • User-friendliness: The interface should be easy to understand and use. • Functionality: The interface should have capabilities and features to serve a wide variety of purposes. • There will both be overlap as well as interference between these areas. E.g. increasing functionality may decrease user-friendliness (too much to choose from) as well as its educational value (cheating).

3. Improvements and ExtensionsEducational Musts • The user needs to be able to load, save, and edit proofs • Explicit reference to what graph is inferred from what graph by what justification • User can go to any step in the proof • User can ‘see’ the proof • Students can hand in proofs as HW • User should be able to make mistakes • User can insert or delete any part of a graph, indicate alleged justification, and program provides appropriate feedback (see Educational Modes)

4. Improvements and ExtensionsEducational Modes • Different modes (see WinKE software) • Unsupervised (user can do anything) • Supervised (user can only perform valid steps) • Assistant (individual steps are performed for user) • Expert (automated theorem proving) • Other?

5. Improvements and ExtensionsUser-Friendly Goodies • Helpful Buttons/Menus • Buttons for ‘P’, ‘Q’, etc. • Buttons for ‘’, ‘’, ‘’, ‘’, ‘’, and ‘’ • Example: For ‘If P then Q’ graph, click ‘’ button, then ‘P’, and then ‘Q’ (needs automatic drawing routine). • Undo • Right Button brings up Context Menu for direct manipulation of graphs

6. Improvements and ExtensionsRelation between EG and PL • Translate between PL and EG • Problem: This is a many-to-many-more mapping • Solution: Automatically generate possible generations; User picks preferred one • Or: have user do all the translation, and have system merely check translation (more educational) • Can this be used for automatic proof generation? • Buttons can reflect relation between PL and EG (and English): • Button says ‘If … then’ or ‘’, but when clicked the EG graph appears.

7. Improvements and ExtensionsProofs as Movies • Ability to view and edit proofs • User can play, rewind, stop, fast-forward • The proof can be edited • Have window that shows miniature snapshots of proof • Maybe have separate Proof Player that ‘polishes’ proofs

8. Improvements and ExtensionsAutomatic Drawing Routine • Tree to Graph automatic drawing routine • Educational: • Clean up user mess • Exploit 2 dimensional visual space • Facilitates visual comparison between graphs and proofs • Automatic Drawing vs User Drawing mode? • User-Friendliness: • Facilitates graph generation (see User-Friendly Goodies) • User does not have to create room before inserting things • Facilitates Inference Patterns (see Inference Patterns) • Facilitates Lift and Replace (see Lift and Replace) • Greatly facilitates proof editing (e.g. gluing parts of proofs together) as visual operations can now be handled by operations over tree data structures (see Proofs as Movies)

9. Improvements and ExtensionsLift and Replace • ‘Lift and Replace’ Technique: • To work on a subset of premises among many, ‘lift’ them from the main proof area, transform, and have the result replace the original section. • This feature would be greatly facilitated by an automatic drawing routine. • This feature facilitates doing explicit subproofs.

10. Improvements and ExtensionsInference Patterns • Apply common patterns to graph • Reduce tedium and repetition • Patterns can be made like any other proof but saved as patterns • Use greek letters to represent arbitrary subgraphs in patterns • Pattern can be applied to graph to perform part of proof. • Using Lift and Replace, patterns can be applied to parts of graph • Application of all this requires automatic drawing routine • Facilitates Inference Graph (see Inference Graph)

11. Improvements and ExtensionsWorking Backwards • All rules of EG can easily be reversed: • Double Cut and (De)Iteration are rules of equivalence, so they go both ways already • The inverse of Erasure is Insertion on an even level • The inverse of Insertion is Erasure from an odd level • Therefore, we can work backwards from the conclusion to the premises by applying transformation rules • Traditional systems do not have this feature!

12. Improvements and ExtensionsConsequence Rule • Routine that checks whether a statement is a logical consequence of a set of statements (see Fitch). • Useful for user, but care needs to be taken that this does not get abused. • Application for Automated Theorem Proving

13. Improvements and ExtensionsInference Graphs • An Inference Graph displays what (according to the user) can be inferred from what. • Existential Graphs are nodes in the Inference Graph, and the (directed) edges are the inferences. • User can zoom in on node of Inference Graph to get to Existential Graph • The inference graph allows the user to try multiple lines of reasoning. • A proof is a path from the premises to the conclusion in the inference graph. • Since there can be multiple paths, there can be multiple proofs that can be directly compared. • The Inference Pattern feature is helpful here as sequences of steps that occur in one part of the Inference Graph can be copied to other parts in the Inference Graph. • Inference Graph allows invalid steps to be retained and stored, thus providing important data for instructor (human or artificial)