Foundation Model of Interaction Theory of Interaction Yuxi Fu BASICS, Shanghai Jiao Tong University Talk at BASICS 2009, Shanghai, 12-16 October, 2009
Foundation Model of Interaction Models for Modern Computing Modern computing is all about interaction : Turing’s Computation Closed systems of computation (interaction within) Milner’s Interaction Open systems of interaction (interaction without)
Foundation Model of Interaction Theory of Interaction aims to provide a model theory for both computation and interaction , in a model independent manner.
Foundation Model of Interaction The goal of this talk is to formulate the Thesis on Interaction that extends the Thesis on Computation (Church-Turing Thesis) that deals with computational completeness, which answers to the issue of interactional completeness.
Foundation Model of Interaction Synopsis 1. Foundation 2. Model of Interaction 3. Theory of Equality 4. Theory of Expressiveness 5. Theory of Completeness 6. Future Work
Foundation Model of Interaction I. Foundation
Foundation Model of Interaction Content 1. Motivation 2. Principle 3. Number and Function 4. Computation Model as Interaction Model
Foundation Model of Interaction 1. Motivation
Foundation Model of Interaction Motivation 1. To define completeness, we need a universal expressiveness relation; 2. To define the universal expressiveness relation, everything has to be model independent; 3. To be able to talk model independently, we must assume that all models share something in common; 4. To assume that all models share something in common, we lay down some principles.
Foundation Model of Interaction 2. Principle
Foundation Model of Interaction Object I. Principle of Object . There are two kinds of objects, the names and the interactants.
Foundation Model of Interaction Object The names are the formalizations of the interfaces . All interactions are via interfaces. All models make use of the same set of interfaces.
Foundation Model of Interaction Object Assumption on the names: N : the set of names , ranged over by a , b , c , d , e , f , g , h ; N v : the set of name variables , ranged over by u , v , w , x , y , z ; N ∪ N v : ranged over by l , m , n , o , p , q .
Foundation Model of Interaction Action II. Principle of Action . There are two aspects of the atomic actions, the internal aspect and the external aspect.
Foundation Model of Interaction Action τ → P ′ indicates that P evolves to P ′ by performing an internal P − action. → P indicates that P evolves to P ′ by performing the external λ − P action λ .
Foundation Model of Interaction Action → P ′ such that P = P ′ . τ A computation is an internal action P − → P ′ such that P � = P ′ . τ A change of state is an internal action P −
Foundation Model of Interaction Observation The notion of observation is model independent.
Foundation Model of Interaction Observation The only way to make an observation on an interactant is to interact with it. There is absolutely no other way. It follows that the observers and the observees must be in reciprocal positions in a closed model. Consequently the observers have the same observing power as the observees, no more, no less.
Foundation Model of Interaction Observation Interaction is the reason for Composition . Composition enables Interaction .
Foundation Model of Interaction Observation Non-Interaction is the reason for Localization . Localization disables Interaction .
Foundation Model of Interaction Observation III. Principle of Observation . There are two universal operators, the composition operator and the localization operator.
Foundation Model of Interaction Observation The environment that makes the observation is of the form ( � c )( | O ) The composition operator allows the observer to observe. The localization operator forbid the observer to observe.
Foundation Model of Interaction Consistency From the point of view of computation, what could be the sharpest difference between two interactants?
Foundation Model of Interaction Consistency The difference can not be sharper than that one always terminates, and the other does not always terminate.
Foundation Model of Interaction Consistency From the point of view of interaction, what could be the sharpest difference between two interactants?
Foundation Model of Interaction Consistency The difference can not be sharper than that one can interact, and the other can not.
Foundation Model of Interaction Consistency Classification in terms of Computation: An interactant P is terminating if it does not have any infinite τ τ internal action sequence P − → − → . . . , it is divergent otherwise. Classification in terms of Interaction: ⇒ λ An interactant P is observable if ∃ λ. P = − → , it is unobservable otherwise.
Foundation Model of Interaction Consistency IV. Principle of Consistency . A terminating interactant is never equal to a divergent interactant. An observable interactant is never equal to an unobservable interactant.
Foundation Model of Interaction Completeness Since every model is a model of computation, every model must be computationally complete (Turing complete). Since every model is a model of interaction, every model must be interactionally complete.
Foundation Model of Interaction Completeness V. Principle of Completeness . There exists a least expressive model of interaction.
Foundation Model of Interaction Five Foundational Principles I. Principle of Object . There are two kinds of objects. II. Principle of Action . There are two kinds of actions. III. Principle of Observation . There are two universal operators. IV. Principle of Consistency . There are two unequal interactants. V. Principle of Completeness . There is a least model.
Foundation Model of Interaction 3. Number and Function
Foundation Model of Interaction Content of Communication By Principle of Object, there are only three possibilities: The contents of communications are processes The contents of communications are names Anything but processes/names
Foundation Model of Interaction Communication Based Process Calculi Consequently, three kinds of communication based process calculi Process passing calculi Name passing calculi Value passing calculi must refer to a theory of numbers and a theory of recursive functions
Foundation Model of Interaction First Order Logic P { r 1 ( t 1 1 , . . . , t 1 k 1 ) / X 1 , . . . , r n ( t n 1 , . . . , t n B k n ) / X n } E1 t = t E2 s = t ⇒ t = s E3 r = s ∧ s = t ⇒ r = t t 1 = t ′ 1 ∧ . . . ∧ t k = t ′ k ⇒ f( t 1 , . . . , t k ) = f( t ′ 1 , . . . , t ′ S1 k ) t 1 = t ′ 1 ∧ . . . ∧ t k = t ′ k ⇒ r( t 1 , . . . , t k ) ⇒ r( t ′ 1 , . . . , t ′ S2 k ) FO1 ∀ x .φ ⇒ φ { t / x } FO2 φ ⇒ ∀ x .φ FO3 ( ∀ x . ( ϕ ⇒ ψ )) ⇒ ( ∀ x .ϕ ⇒ ∀ x .ψ )
Foundation Model of Interaction Peano Number Theory: PA PA1 ∀ x . (succ( x ) � = 0) ∀ xy . (succ( x ) = succ( y ) ⇒ x = y ) PA2 PA3 ∀ x . ( x = 0 ∨ ∃ y . succ( y ) = x ) ∀ x . ( x < succ( x )) PA4 PA5 ∀ xy . ( x < y ⇒ succ( x ) ≤ y ) PA6 ∀ xy . ( ¬ ( x < y ) ⇔ y ≤ x ) PA7 ∀ xy . (( x < y ) ∧ ( y < z ) ⇒ x < z ) The axioms that are missing: axioms for arithmetic operators, natural induction.
Foundation Model of Interaction Peano Number Theory: PA Proposition . The theoremhood of the boolean expressions of PA is decidable. This is important for the value passing calculi.
Foundation Model of Interaction Peano Number Theory: PA The term model ( { 0 , 1 , 2 , . . . } , 0 , succ , <, =)
Foundation Model of Interaction Recursive Function Model: RF Grammar: f( � x ) := s( x ) n i ( � x ) p j i ( � x ) x , x ′ , x ′′ ) , f ′ ( � rec(f( � x )) µ z . f( � x , z ) f(f 1 ( � x ) , . . . , f l ( � x ))
Foundation Model of Interaction Recursive Function Model: RF Reductional Semantics : → s( n ) n +1 n k ( n 1 , . . . , n i ) → n p j → i ( n 1 , . . . , n i ) n j x , x ′ , x ′′ ) , h( � rec(f( � x ))( n 1 , . . . , n i , 0) → h( n 1 , . . . , n i ) x , x ′ , x ′′ ) , h( � → rec(f( � x ))( n 1 , . . . , n i , n +1) R x , x ′ , x ′′ ) , h( � where R ≡ f( n 1 , . . . , n i , rec(f( � x ))( n 1 , . . . , n i , n ) , n ).
Foundation Model of Interaction Recursive Function Model: RF Reductional Semantics : µ f( n 1 ,..., n i , i ) [g] whose evaluation strategy is given by the following reduction rules: µ f( n 1 ,..., n i , i ) [g ′ ] if g → g ′ µ f( n 1 ,..., n i , i ) [g] → µ f( n 1 ,..., n i , i ) [0] → i µ f( n 1 ,..., n i , i ) [ l +1] → µ f( n 1 ,..., n i , i +1) [f( n 1 , . . . , n i , i +1)] Now the evaluation of µ x . f( n 1 , . . . , n i , x ) is identified to the evaluation of µ f( n 1 ,..., n i , 0) [f( n 1 , . . . , n i , 0)].
Foundation Model of Interaction Recursive Function Model: RF Reductional Semantics : If g j → g ′ j , where 1 ≤ j ≤ i , then f( n 1 , . . . , n j − 1 , g j , . . . , g i ) → f( n 1 , . . . , n j − 1 , g ′ j , . . . , g i )
Recommend
More recommend
