6/3/2015Andrew Frank1 Communication: Information Content Andrew U. Frank Geoinformation TU Vienna Overheads at:

6/3/2015Andrew Frank2 Overview Case: games Case: Real World – Driving Conclusions

6/3/2015Andrew Frank3 Question: How to measure information? Shannon & Weaver: 1 bit = 1 binary decision (e.g. head or tail) Works for data. How to apply to information?

6/3/2015Andrew Frank4 Definitions Data : (machine readable) signs Information: answers to questions Descriptions of real world situations Used to decide on actions Information is derived from data.

6/3/2015Andrew Frank5 Case Board Games e.g. checkers (Dame) Differentiate: The game of checkers ( game ) The game I played last Wednesday evening ( play ) We analyzed the game by XX in 1995 ( match )

6/3/2015Andrew Frank6 Game as algebra: The game with rules is an algebra: class BoardGames g where initialize:: g move :: Player -> Position -> Position -> g -> g isFinished:: g -> Bool winner:: g -> Maybe Player This signature seems to fir for most board games. The difference is in the instantiation, where e.g. ‘move’ checks for legality of a move according to the rules.

6/3/2015Andrew Frank7 Play as a sequence of operations Playing a game is carrying out a sequence of operations: My partner and I moved yesterday evening alternatively pieces on my board. Algebraically: g0 = initialize g2 = move Black A2 A3. Move White A7 A6 $ g0 … g17 =move …. g16 w2 = winner g17

6/3/2015Andrew Frank8 Match as an abstract description of a play separated from instantiation Weiß: Kasparov Gary (2595) Schwarz: Pribyl Josef (2395) 1.d4 Sf6 2.c4 g6 3.Sc3 d5 4.cxd5 Sxd5 5.e4 Sxc3 6.bxc3 Lg7...

6/3/2015Andrew Frank9 Descriptions of a Match: Representation as observable physical phenomena (including mental representation) Information is linked to a representation, Information cannot exist independent of a representation

6/3/2015Andrew Frank10 Alternative Descriptions of a Match -Natural Language -Alternative Formal Descriptions Different encoding -A play (instantiation) -… All are equivalent (in terms of the algebra) They describe the same abstract match.

6/3/2015Andrew Frank11 Information as an Equivalence Class All descriptions which instantiate to the same play are equivalent. The information is the equivalence class homomorphism between the representations

6/3/2015Andrew Frank12 Measure of the Information Measure a representation (data) How many binary decisions are necessary for the representation? = Logarithm base 2 of the number of different messages Information content of all non-redundant representations must be equal.

6/3/2015Andrew Frank13 Chess: How many different messages in a ‘move’ operation: Player -> Position -> Position -> … 2 * 64 * 64 = = 2**7 7 bits per move This encodes legal and illegal situations; it is the information content of arbitrary positioning and moving of pieces on a board.

6/3/2015Andrew Frank14 Information Content Depends on the algebra underlying. The 7 bit calculation is the free algebra: All possible moves are legal Real play: many moves are illegal, -> algebra with axioms -> less possible cases -> less information Many moves are stupid; good chess players consider only reasonable moves -> less information again

6/3/2015Andrew Frank15 Redundancy Representations can contain more bits than the minimum necessary for a specific game. The additional information can help to guard against transmission errors (but is not necessarily effective) Better: use non-redundant representation and add redundancy systematically

6/3/2015Andrew Frank16 Conclusions: Information is the content of the class of equivalent representations – equivalent with respect to the algebra of the game. The equivalence class contains only the non- redundant information (with respect to a given algebra).

6/3/2015Andrew Frank17 Real World Situation: Driving in City A friend gives me driving instructions: Follow Rechte Wienzeile, turn into Schleifmuehlgasse … I check the web Rechte Wienzeile 0.7 km Schleifmuehlgasse 0.1 km … Or get a map:..

6/3/2015Andrew Frank18 Driving as an Algebra Driving in a city is like a game and formalized as an algebra: class Driving d where startAt :: Location -> Driver -> d -> d move :: Driver -> Location -> d -> d isAt :: Driver -> d -> Location This is a different model than the one used in the driving instructions.

6/3/2015Andrew Frank19 Equivalence of Instructions The instructions are equivalent if they lead me correctly along the same path. This is equivalence with respect to the algebra defined. Information content can be measured the same way than for a a game!

6/3/2015Andrew Frank20 Different drivers – different algebra: Possible instructions: move :: Location -> Location -> d -> d Drive to intersection Rechte Wienzeile/Schleifmuehlgasse, Drive to intersection Schleifmuehlgasse/Margaretenstrasse …

6/3/2015Andrew Frank21 Alternatives Web instructions: move :: Street -> Distance -> d -> d Rechte Wienzeile 0.7 km Schleifmuehlgasse 0.1 km … Oral instructions move :: Location -> Location -> d -> d turn :: Direction -> d -> d Follow Rechte Wienzeile to the intersection with Schleifmuehlgasse, turn right Follow Schleifmuehlgasse to the intersection with Margaretenstrasse, turn right

6/3/2015Andrew Frank22 Ontological commitment: Information is linked to a representation, Information cannot exist independent of a representation

6/3/2015Andrew Frank23 Information content Relative to an algebra: Minimal representation of description of a sequence of actions.

6/3/2015Andrew Frank24 Redundancy A representation can contain more data than necessary for the algebra, but not more information.

6/3/2015Andrew Frank25 Different algebra – different information content (from the same data) Information content is different relative to the algebra: Different people play different games; their algebras are different (e.g. more knowledge available)

6/3/2015Andrew Frank26 Information for sender and receiver is not always the same What is redundant for one may be necessary for the other. Information + Redundancy = constant

6/3/2015Andrew Frank27 Conclusion No simple answer to ‘how much information’ Algebra gives framework in which questions can be posed and answered for specific cases. Insight gained corresponds with experience.