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Formal and Computer Models of C-K Design theory

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### Formal and Computer Models of C-K Design theory

Akin Osman Kazakci

Centre de Gestion Scientifique

www.cgs.ensmp.fr

Plan

- Models of C-K theory : current work, state of art
- Mathematics of design
A Kripke-style first-order logical model of C-K theory based on Intuitionist Logic

- Design in Mathematics
A C-K theoretic Analysis of “Creative Subject” of Brouwer and Intuitionist Mathematics

- JouJou
A simulation package for reproducing, studying C-K type reasoning processes

- Social Design
And its modelling with C-K

A concept-knowledge theory of design

Concepts

Knowledge

- Design as the interaction of two spaces
- Knowledge space; the knowledge available to the designer
- Concept space; propositions
- that are neither true, nor false in K
- whose terms are interpretable in L

- Interaction through operators KK, KC, CK, CC

(Hatchuel and Weil, 1999, 2002, 2003, 2007, 2009)

Current work on the formal development of CK – Forcing and Set theory

- A recent development: Design as Forcing

Hatchuel, A., B. Weil, and Le Masson, P. Design as Forcing:

Deepening the foundations of CK theory.

@ ICED 2007

Forcing is a method invented by Paul Cohen(1963) for the creation of new sets

And proving independence resuts

- Set theory as the K space; designing new numbers

Design of C-K design theory: So far…

- - Two spaces
- Operators

Simon

- Branch&Bound

- Bounded Rationality

- Axiom of choice rejected (99,02,03)

Field observations

&models

Models of K

- CKE (05)

Situated Cognition

Clancey, Suchman,...

- Forcing (07)

- Set Theory

- Cohen on Forcing

Fluid Concepts (Hofstadter 84,95)

- What else?
- nDim?
- ID?

Implemented by Wang algebras (2005,7)

Wang algebras

On which K?

Logical models

Computer models

C-K type reasoning and Formal Models In formal siences , « reasoning » seen as quasi-equivalent to « Logic »

- C-K is first and foremost a theory of reasoning
- Design ⇒ From « Reasoning about knowledge » to « Reasoning about unknown »

➨Are there formal/logical approaches enabling « Reasoning about unknown »?

C-K make use of logical notions

- A concept is a proposition of the type
« ∃x. P1(x)∧P2(x) ∧…∧Pn(x) » such that….

- K space contains propositions with a logical status (true or false) are knowledge

Preliminaries and notations on FOL

- The language of first-order logic allows us to express/reason about:
- Properties of objects (P, P1, P2, Q, Q1,…)
- Relations between objects (R, R1,…)
- Ex. Red(apple), OnTopOf (apple, table)

- Logical constants:
- &, et
- ⋁, ou,
- ⟶, implication
- ¬, négation
- Eventuellement, des quantificateurs,
- ∀, pour tout,
- ∃, il existe

Inference:

p, p⟶q ⊢ q

Notion of « Model »

Tarski (1933, 1935,1956)

- Given a (first-order) language L, a model is a couple < D, I > where
- D is a set of objets (d1, d2, …)
- I is a truth function.
Ex. I(Red(apple))= 1 (true)

Alfred Tarski (1901-1983)

- A model M= < D, I > force (or satisfy) a proposition P(d) if I(P(d))=1 for d∈D.
- We note M ⊨ P(a).

A model for Set Theory: Objects are sets; the only relation is « ∈ ».

Can we use classical FOL?

How to define the K space?

- Propositions with a logcial status: Propositions P such that a model M ⊨ P or M ⊨ ¬P
- The K space is a model M.

How to define concepts?

Neither true, nor false with respect to K space:

P such that neither M ⊨ P, nor M ⊨ ¬P The reject of LEM.

- Whichisonly possible if knowledgeisincomplete
- Design is not possible in purelyPlatonistuniverse
- whereeverytingthatisknowableisknown.

Intuitionist Logic P(a) means « I have a method for constructing an onject a that has a property P ».

The logic of Intuitionist Mathematics of Brouwer: Mathematics as an activityof construction of mental objects throughout time

- Temporal aspect recognizing the possibility of unknown and learning.

L.E.J Brouwer (1881-1966)

- A constructive interpretation of « truth » Reject of LEM.

¬P(a) : « I have a method for proving that each tentative for constructing a sucht that P(a) will lead to absurdity »

What model for IL? - Kripke semantics

Partial order on states of knowledge (K spaces)

- K is a set of possible worlds
- < is an accessibility relation
- DK domain function for each world K ∈ K
- IK interpretation function for each world K ∈ K

Saul Kripke (1940-…)

- Let K, K’ ∈ K besuchthat K < K’
- K’ contains more information than K
- DK ⊆ DK’ : more objects are known K’
- New objects and new propertiesappear over the time.

The original example of Kripke (1963)

With a single propositions P…

The Rieger–Nishimura lattice. Source: Wikipedia

Its nodes are the propositional formulas in one variable up to intuitionistic logical equivalence, ordered by intuitionistic logical implication.

But…

Extending Kripke’s model : DUAL Expansion

Where is the C space

What guides the learning?

C driven K expansion

Void based reasoning

Yes. (Kazakci, 2009; Hendriks and Kazakci, ?)

Can we add a C space?

A designstage in a design process is a s = <K, C>, where K∪{C} is a set of formulas (C is a concept with respect to K).

A design space is a partially ordered set of stages, where

<K0, C0> ⩽ <K1, C1>

⇔

K0* ⊆ K1* and K1, C1⊢ C0

<K1, C1>

<K1, C0>

<K0, C1>

<K0, C0>

Further definitions…

s = <K, C> is called

consistent⇔ K |/- ¬C (and inconsistent o.w.)

closed⇔ K |- C

feasible ⇔ s is consistent and closed

open⇔ s is consistent and not closed

s0 and s1 are equivalent, s0≡ s1, if s0⩽ s1 and s1⩽ s0

s0 < s1 if s0⩽ s1 and ¬(s0≡ s1)

1.<K, C> is open iff K |/- C and K |/- ¬C

2. ⩽ is reflexive: s0⩽ s0

3. ⩽ is transitive: s0⩽ s1 and s1⩽ s2 then s0⩽ s2

4. if s consistent then s either feasible or open

5.if s0⩽ s1 and s0 and s1 share the same body of knowledge then s1 open if s0open

6.If <K0, C0> ⩽ <K1, C1> and <K1, C1> feasible then <K1, C0> feasible

7.<K0, C0> equivalent <K1, C1> iff K0* = K1* and K0 |- C0<-> C1

Some facts and properties- - 2 -3 state that ⩽ is in fact a partial ordering and so the design space is a partially ordered set.
- 5 implies that only refining the concept without extending the knowledge base will never provide a feasible solution in the design process.
- - 6 proves that a design path ending in a feasible design stage also provides a feasible solution for the original concept C0.

Let s = <K, C> and s’ = <L, D> be design stages. The product of s and s’, s⨂s’ is defined as: s⨂s’ = <K∪L, C∧D>

- Definition: Design Move
- Let s = <K, Cx> and s’ = <T, Dx> be design stages. The product of s and s’, s⨂s’ is a design move from s if:
- s < s⨂s’
- If Dx above is a literal (an atomic formula or the negation of an atomic formula) s⨂s’ is called an atomic partition.

Types of design moves

- pure K-extending if Dx = T, where T is a constant

- pure C-extending if T*⊆K* (if σ(T) ∩σ(K) = ∅ then expansion)

- dual, if neither pure K- or C-extending

- Let s ⨂ s’:
- If s’ is pure K-extending then:
- 1.s’ is open (consistent) ⇒ s is open (consistent)
- 2.s feasible and s’ consistent ⇒ s’ is feasible

- If s’ is pure C-extending then:
- 3.s and s’ share the same body of knowledge
- 4.s’ closed ⇒ s closed
- 5.s open and s’ consistent ⇒ s’ open
- 6.s’ feasible ⇒ s feasible

How does the spaces interact?

- Definition (Proper Partitioning)Definition 10. Let C be a concept for K, then Cx ∧ Qxy is a proper partition of C if:
- Q is a predicate in LK (with free variables in the union of the sets x, y)
- Q is related to C
- Cx ⟶ Qxy ∉ K
- Cx ∧ Qxy ⟶¬ Cx ∉ K

- If for some property P in C, ∃x (Px ∧ Qx) ∈ K, restrictive.

Definition (K-query) This is an operation that takes as argument a concept c, a predicate pi occurring in c and that returns a predicate q such that K ⊨ ∃x.(pi∧q) or, NIL, if no such q exists. From any concept of the C space, the designer can query the K space in order to retrieve related knowledge in the K space.

Definition (K-validation) The K-validation of a concept c is an operation that takes a knowledge space K and a concept c as argument and returns true, if K ⊨ ∃x.c or K ⊨ ¬∃x.c and false, if not.

Remarks

In traditional logical models, although learning is considered, creating new ideas have not been considered. C space is missing. However, we begin to see that it is possible to add a concept space and model dual expansion.

In this model,

Object construction takes place in K space; Definition construction takes place in C space

C is an intensional space; K is intensional and extensional.

C space is not a tree (it is a partial order) unless the order of partitions implies a kind of priority

The reject of LEM is not in C space; but in K space. C space has no semantics (only rules of definition formation which are identical in K space; pertaining to the logic itself)

There is no axiomatic associated with C space. No need. meaning is in K space.

Perspectives

Intuitionist Logic has given rise to Martin-Lof Type theory (Intuitionist Type Theory, 1984)

Interactive proof assistants and (functional programming languages)

Several implementations: NuPRL, LEGO, ALF, COQ, AGDA, Epigram

Martin-Lof, 1948…

…all implemented without a C space

« would you rather prove this other formula? »

Brouwer and Intuitionist Mathematics

Famous for his contribution to topology(!)

Fixed-point theorem (1909).

Father of Intuitionist Mathematics

Mathematics is an activity of construction

…Of mental mathematical objects

…by a creative subject!

L.E.J. Brouwer (1881-1966)

But is this not contradictory with Universal and Objective conception of mathematics?

The seemingly contradiction will dissappear once we review Brouwer’s work from the viewpoint of design theory:

Brouwer describe mathematics as a conceptive reasoning process.

Pre-intuitionists on existence of objects and definability

Reactions to Cantorian Maths: Transfinite and the abstract choice principles are strongly contested …by Borel, Poincaré, Hadamard, Baire, Lebesgue…

Russell set out to save the mathematics using logic: Logical foundations for the totality of mathematics.

Strongly supported by Hilbert

Again contested by pre-intuitionists: “The syllogism cannot reveal anything fundamentally new. If all mathematical propositions can be derived ones from others, how would mathematics not be reduced to an immense tautology?” Poincaré.

Fundamental role of Intuition: “The logic is sure, but creates nothing; the intuition is creative but fallible.” Poincaré.

What is definable is what is conceivable,

A mathematical object exists if we can conceive the means necessary to construct it

First act of Intuitionism (1907,8)

Mathematics is independent of any outside reality.

It is an activity of mental construction.

“[…] completely separates mathematics from mathematical language, in particular from the phenomena of language which are described by theoretical logic, and recognizes that intuitionist mathematics is an essentially languageless activity of the mind…

…having its origin in the perception of a move of time, i.e. of the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory. If the two-ity thus born is divested of all quality, there remains the empty form of the common substratum of all two-ities. It is this common substratum, this empty form, which is the basic intuition of mathematics. “ (Brouwer, 1907-8)

Since mathematics is a mental construction based on the step-by-step activity of an idealized mathematician, any notion of infinity is potential – not actual.

Existence of Mathematical Objects

[…] for mathematical objects, “to exist” means “to be constructed” (Heyting)

… If this is not the case, then “to exist” should have a metaphysical meaning…

Although Brouwer or his followers have no objection to any particular metaphysical theory, they believe that the study of mathematics cannot be related to these;

[…] it is something much simpler, more immediate than metaphysics – the study of mental mathematical constructions

Law of the Excluded Middle (LEM) is not tenable for the kind of mathematics Brouwer proposed, since, to be able to say that P(a) or its negation, ~P(a), is true for every proposition and every object a, we must have a general method for constructing any object having any property – that which we do not have (Heyting, 1983, reprint)

Second act of intuitionism

Contrary to the first act, which is critical and destructive, the second act of intuitionism is creative and constructive (Largeault, 1993).

Infinitely proceeding sequences – sequences of mathematical objects that proceed to infinity but never achieve it.

E.g. by means of classes of equivalence of Cauchy sequences for defining sets of real numbers.

Lawlike sequences: Yet, such sequences determined from the beginning (by a law or an algorithm)

Lawlike sequences are highly limitative: such sequences are always countable and the real numbers that can be defined by means of such sequences can only offer a countable infinity and a reduced continuum.

Second act of intuitionism

Free choice sequences (or, lawlesssequences):

infinitely proceeding sequences of mathematical objects whose construction are not fixed by a predetermined law or algorithm but whose terms can be chosen arbitrarily at any stage of their construction (by a creative subject).

Allowing an act of free choice at any moment and the possibility to break away from any algorithm or law allows the consideration of partially determined objects and their undecided properties.

“Brouwer’s universe does not get beyond ω1 (there is no transfinite and actual infinity). But, what it lacks in ‘height’ is compensated in ‘width’ by the extra fine structure that is inherent to the intuitionist approach and its logic.” (Van Dalen, 2005)

Generative power of the Second Act

“the second act recognizes the possibility of generating new mathematical entities, …

firstly, in the form of infinitely proceeding sequences a1, a2, …, whose terms are chosen more or less freelyfrom mathematical entities previously acquired; […]

the choices of the av’s themselves, at any stage may be made to depend on possible future mathematical experiences of the creating subject;

secondly, in the form of mathematical species, i.e. properties supposable for mathematical entities previously acquired, and satisfying the condition that, if they hold for a certain mathematical entity, they also hold for all mathematical entities that have been defined to be equal to it.” Brouwer (1976,77; collected works)

- Two kind of sets!
- Spreads : Extensional
- Species: Intensional

Spreads are on the side of ‘becoming’ whereas species are on the side of ‘being’ (Largeault, 1993)

Spreads may be used to construct new objects which have some properties yet to be decided.

An example: The full binary spread

Ex, the species P(S,n) : sequences such that “the sum of n first terms is pair”.

A spread can be narrowed down by an operation called hemmung (restriction, in german) to more specific spreads (e.g. the full binary spread can be narrowed down to sequences starting with < 0, 0, 1>).

While free choices allow constructing unprecedented objects, hemmung may be used to get more and more specific types of objects.

Theory of Creative Subject (Kreisel, 1963)

Notation: SnP

At stage n, I have a proof of P.

(∃nSnP ) P

(SmP and n > m) SnP

P ~ ~∃nSnP

P (∃nSnP)

- If we have a proof of P at some stage n, then P is true.
- If at some stage m we prove (and learn) P, then at later stages n > m, we still know P.
- If P is true, it is absurd to say that S will never have a proof of P.
- If P is true, then S will certainly have a proof of it in time.

Third and fourth formulae reflect the idea that if an object with a certain property can be constructed, then, eventually the creative subject will get there.

If every proposition P will certainly be proved at some point, then, where is the creativity of the creative subject?

Althoughthe learning over the time aspect is included in TCS model, the “creativity” of the subject appears nowhere.

TCS – A missing C space

Partially defined objecty gives the theory the possibility to expand the repertory of known objects in new ways.

However, this does not explain how the idea of such objects appear, neither how these ideas develop over time.

Although Brouwer consider the possibility of free choice in constructing new objects, he does not recognize that such free choices go hand in hand with the choice of new concept of objects.

It is not considered in Intuitionist Mathematics how the choice of the properties to be proved may affect the next stages of the subject’s creative activity.

For Brouwer, this is only normal: Brouwer’s intention was to develop and expose an alternative way of seeing the object of mathematical investigation;

Brouwer never set out to explain creativity itself.

By contrast to C-K…

But concepts « do » exist in Brouwer’s work

How to prove that LEM is not an acceptable principle? (1908)

An object o with the property of « not respecting LEM »

?

Infinitelyproceedingsequences

o Is an object containing undecided propositions in its definition

Why do ipsshouldbelawlike? Wecanallow free choices

Thesechoicesmaydepend on future outcomes

o Is a real number such that…

Objectswhosepropertiesdepend on future outcomeswill have some of theirpropertiesundecideduntil the future outcomesrevealthemselves

But concepts « do » exist in Brouwer’s work

An object o with the property of « not respecting LEM »

P: “Let dn be the n-th digit in the decimal expansion of π and m = kn if in the growing decimal expansion of π at dm it is the case that for the n-th time the part dmdm+1 . . . dm+9 of this

decimal expansion forms a sequence 0123456789.

o Is an object containing undecided propositions in its definition

o Is a real number such that…

Let R be a positively convergent sequence to 0, if P is true, negatively convergent sequence if P false.

Since Rconvergent anyway, it defines a real number.

R neitherpositively convergent, nornegatively convergent until P isknown LEM fallsapart.

Remarks and Discussion Mathematics

Brouwer describes Mathematics as conceptive reasoning process pretty much like C-K describes design process (Hatchuel and Weil, 1999-2009)

Intuitionist mathematics provides a constructive alternative to Forcing for C-K’s foundations.

Choice sequences and Generic sets: intuitionist free choice sequences can be called potentially generic (Dummett, Largeault, Nikeus…)

Choice, decision and design: The free choice Brouwer introduces emphasizes an important aspect in the creative process: the creative subject can choose how to construct an object.

An overwhelming majority of the work in Decision Sciences has studied decisions in the context of « extentionnaly and completely defined objects » Intuitionist mathematics about partially determined objects may allow to consider preferences about “intensionnally and partially defined” objects in a dynamic way.

JouJou Mathematics

Demo.

CKE: an extension of MathematicsCK theory

Espace E

action

Interprétation

Espace K

Espace C

- Conception at the heart of design,
- Design as the externalisation of concepts

Kazakci,Tsoukias, 2005

What happens if more than one designer?

Adding E-space

- Situated designers
- Perception, conception, action

CKE: an extension of MathematicsCK theory

- Duality between creativity/ comprehension (Kazakci, 2007):
- Modelling misunderstanding (quiprosquos) in design (Szpirglas, 2005,)
- Can we reinterpret/rediscuss
- Design collaboration and synergy,
- Shared understanding,
- Skecthing
- Group creativity

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