AGI 08 March 1-3, University of Memphis Hybrid Reasoning and the Future of Iconic Representations Catherine RECANATI LIPN UMR 7030 Université Paris 13

1.Icons or diagrammatic objects can be used as first class citizens (=normal syntactical objects) in safe inferential systems 2.Diagrammatic representations have a limited power of abstraction but are computationally very efficient 3.Diagrammatic and logico-linguistic representations having dual and complementary properties, their combining in HRS is very promising Three points

1.Icons or diagrammatic objects can be used as first class citizens (=normal syntactical objects) in safe inferential systems 2.Diagrammatic representations have a limited power of abstraction but are computationally very efficient 3.Diagrammatic and logico-linguistic representations having dual and complementary properties, their combining in HRS is very promising Three points

Closure under constraints “ Homer is on the left of Lisa ”

Closure under constraints “ Homer is on the left of Lisa ” “ Lisa is on the left of Bart ” The fact that Homer is on the left of Bart is directly accessible (explicit) on the diagrammatic representation

Barwise and Etchemendy (1990) Logic as a theory of valid inferences independent of the modes of representation  Shin (1991) : two graphical systems inspired by those of Venn and Peirce (for solving syllogisms)

Euler “ All A is B ” A B 1768

Euler “ All B is C ” B C 1768

Euler … therefore “ All A is C ” A B C 1768

Venn “ All A is B ” AB 1894

Peirce “ All A is B ” and “ There is a B which is not an A ” AB o x 1933

Peirce “ All A is B ” or “ There is a B which is not an A ” AB o x 1933

Shin (Venn-Peirce) A 1 A2A2 A 3 D1 D2 A 6 A 4 A 5 D3

Properties of diagrammatic systems for Barwise and Etchemendy the main properties of diagrammatic systems are derived from the existence of a syntactical homomorphism between icons (and icons types) used and the properties of the objects

In paradigmatic cases, these systems exhibit the property of Closure under constraints requiring that all logical consequences of requiring that all logical consequences of the represented situation be explicit in the representation. Properties of diagrammatic systems

 Easy treatment of conjunctions  But difficulties with disjunctions and abstract relations (negation, implication …)  Contradictions cannot be represented and each representation corresponds to a genuine situation Properties of diagrammatic systems

1.Icons or diagrammatic objects can be used as first class citizens (=normal syntactical objects) in safe inferential systems 2.Diagrammatic representations have a limited power of abstraction but are computationally very efficient 3.Diagrammatic and logico-linguistic representations having dual and complementary properties, their combining in HRS is very promising Three points

Closure under constraints “ Homer is on the left of Lisa ” “ Lisa is on the left of Bart ” The fact that Homer is on the left of Bart is directly accessible (explicit) on the diagrammatic representation

A linguistic representation “ Homer is-on-the-left-of Lisa ” “ Lisa is-on-the-left-of Bart ” needs a supplementary step and the use of a rule of transitivity to get that “Homer is-on-the-left-of Bart ”. Transitivity rule: if A is-on-the-left-of B, and if B is-on-the-left-of C, then A is-on-the-left-of C.

Linguistic reasoning requires (1) the representation of initial facts (2) an explicit representation of the abstract properties of the objects (3) a computational mechanism linking the two sources of information

Diagrammatic reasoning (2) No explicit representation of the abstract properties of the objects  these properties are automatically taken into account by syntactic constraints on the representation of the objects (3) No computational mechanism  the representations have only to be inspected to check whether the new fact is or not represented there This makes these systems computationally very efficient

What is Closure under constraints ? Stenning and Oberlander (1995) (C) = Specificity requires information of a certain kind to be specified in all interpretable representation Classes of Representational Systems MARS < LARS < UARS diagrammatic systems are LARS

Minimum Abstraction Representational System In a MARS a representation corresponds to a unique model of the world. P1 P2 P3 Obj Obj [ B B Y Y R ]

Limited Abstraction Representational System You can abstract on a minimal representation to quantify over the dimensions, by adding new symbols P1 P2 P3 Obj Obj [ B _ Y _ R ]

Specificity and limited abstraction for Stenning and Oberlander Specificity requires information of a certain kind to be specified in all interpretable representation = closure under constraints of B&E

What is Closure under constraints ? Perry and Macken (1996) only Berkeley’s notion of determined character is required –the representation of an object as having a particular property requires a specified value for this property. Ex: colored objects (C) = Localization (or unique token property) + Iconicity + a constraint and systematic homomorphism

Iconicity and Richly Grounded Meaning Iconic symbols have richly grounded meanings :  RIM – Readily Inferable Meaning  ERM – Easily Remembered Meaning  IMM – Internally Modifiable Meaning

Partially implicit Situation Explicit consequences Algorithm in a particular language Evaluation Mapping Syntax to Semantics

1.Icons or diagrammatic objects can be used as first class citizens (=normal syntactical objects) in safe inferential systems 2.Diagrammatic representations have a limited power of abstraction but are computationally very efficient 3.Diagrammatic and logico-linguistic representations having dual and complementary properties, their combining in HRS is very promising Three points

Z X, y => z O(x) G(y) annotated aspects Hybrid Representation Systems

No need of inter-lingua What makes these systems correctly tied together is just that They denote the same objects in the world.

New computational perspectives through the “diagonalization” of proofs computations reputed to be bounded by a minimal cost may be turned out to be less costly in a hybrid system

P1 cost 1 P2 2 P3 7 P4 P1 cost 8 Q2 2 Q3 1 P4 2

HRS : new perspectives in AI and Cognitive Science  Reasoning  Semantics  Natural Language