Formal Models of Timed Musical Processes Doctoral Defense

Transcription

1 Formal Models of Timed Musical Processes Doctoral Defense Gerardo M. Sarria M. Advisor: Camilo Rueda Co-Advisor: Juan Francisco Diaz Universidad del Valle AVISPA Research Group September 22, 2008

2 Motivation

3 Motivation Music composition, performance and improvisation are complex tasks of defining and controlling real-time concurrent activities. It is necessary a formal model capable of specifying real-time, concurrent, (a)sychronous and constrained systems to be used as a mathematical platform for musical composition and improvisation.

4 Motivation Various formalisms have found use in practical musical situations. Those formalisms were not intended originally to be used in music. Process Calculi are popular contemporary formalisms for modeling and analyzing concurrent systems. Effective models of concurrency High degree of abstraction Intuitive programming language feel

5 Contributions A new CCP calculus, called rtcc, for modeling real-time reactive systems. Strict extension of ntcc Explicit notions of time and resources Precise way of delay processes Strong preemption and default behaviour New approach to denotacional semantics (Chu Spaces) True Concurrency Intended for real-time multimedia interaction

6 Agenda 1 Preliminars CCP Constraint System NTCC 2 The rtcc Calculus Syntax Operational Semantics Denotational Semantics Real-time Logics 3 Concluding Remarks and Future Work

7 Concurrent Constraint Programming Concurrent constraint programming (CCP) is a model for specifying concurrent systems in terms of constraints. A constraint is a first-order formulae representing partial information about shared variables. Von Neumann s store is replaced by a store of partial information (i.e. x 20) Read and write operators are replaced by ask and tell

8 Concurrent Constraint Programming when y < 30 do tell x < 5 tell x < 8 x :: [0, max] y :: [0, max] z :: [0, max] when y < 7 do tell x < 3 when x < 7 do tell z < 10 tell y < 20

9 Concurrent Constraint Programming when y < 30 do tell x < 5 skip x :: [0, 7] y :: [0, max] z :: [0, max] when y < 7 do tell x < 3 when x < 7 do tell z < 10 tell y < 20

10 Concurrent Constraint Programming when y < 30 do tell x < 5 skip x :: [0, 7] y :: [0, 19] z :: [0, max] when y < 7 do tell x < 3 when x < 7 do tell z < 10 skip

11 Concurrent Constraint Programming skip skip x :: [0, 4] y :: [0, 19] z :: [0, max] when y < 7 do tell x < 3 when x < 7 do tell z < 10 skip

12 Concurrent Constraint Programming skip x :: [0, 4] y :: [0, 19] z :: [0, 9] Remains blocked when y < 7 do tell x < 3 skip skip

13 Concurrent Constraint Programming Partial Information: Variables, Domains and Constraints Concurrency: Multiple agents Synchronization: via store

14 Constraint System A constraint system specifies what kind of constraints handle the store. Formally, it is a tuple Σ,, where Σ is a signature (set of constraints, functions and predicate symbols) and is a consistent first-order theory over Σ. Constraints are first-order formulae over Σ The conjunction of all constraints accumulated monotonically is called the store The entailment relation c d holds iff c d is valid on. c d iff c d and d c

15 The ntcc calculus is CCP calculi proposed for modeling and programming temporal reactive systems. NTCC

16 NTCC The ntcc calculus is CCP calculi proposed for modeling and programming temporal reactive systems. Stimulus Store Process Residual Process Time Unit 1 Time Unit 2

17 NTCC The ntcc calculus is CCP calculi proposed for modeling and programming temporal reactive systems. Stimulus Store Resting Point Residual Process Time Unit 1 Time Unit 2

18 NTCC The ntcc calculus is CCP calculi proposed for modeling and programming temporal reactive systems. Stimulus Store Response Resulting Store Resting Point Residual Process Time Unit 1 Time Unit 2

19 NTCC The ntcc calculus is CCP calculi proposed for modeling and programming temporal reactive systems. Stimulus Store Resting Point Residual Process Time Unit 1 Time Unit 2

20 NTCC The ntcc calculus is CCP calculi proposed for modeling and programming temporal reactive systems. Stimulus Store Resting Point Residual Process Time Unit 1 Time Unit 2 Stores are not automatically transferred from a time unit to the next one.

21 NTCC Syntax P, Q = tell(c) Tell i I when c i do P i Nondeterminism P Q Parallel Composition local x in P Local Behavior next P Unit Delay unless c next P Time-Out P Asynchrony! P Infinite Behavior

22 NTCC Example: M Conductor def =! when Go = 1 do i Notes tell Note = i unless End = 1 next (tell Go = 1) def = tell Go = 1 (tell End = 1)

23 NTCC Disadvantages: No explicit metric notion of time is available. Time units are not homogeneous. There are no facilities to account for resource usage. Execution of processes is not preemptive.

24 The rtcc Calculus Extends ntcc in these ways: Real-Time Delta Delay Strong Preemption Default Behaviour

25 The rtcc Calculus Extends ntcc in these ways: Real-Time Delta Delay Strong Preemption Default Behaviour Resources: A natural number indicating how many resources was provided by the environment. Each process P takes some of these. When P is finished, it releases them.

26 The rtcc Calculus Extends ntcc in these ways: Real-Time Delta Delay Strong Preemption Default Behaviour Bounded Time: Each time unit is divided in a discrete sequence of minimal units called ticks. The number of ticks will be the available time that processes have to execute.

27 The rtcc Calculus Extends ntcc in these ways: Real-Time Delta Delay Strong Preemption Default Behaviour Now a process can be delayed for at least certain number of ticks.

28 The rtcc Calculus Extends ntcc in these ways: Real-Time Delta Delay Strong Preemption Default Behaviour A process can be interrupted if a signal is given.

29 The rtcc Calculus Extends ntcc in these ways: Real-Time Delta Delay Strong Preemption Default Behaviour If the execution of a process is preempted then another process might be launched.

30 The rtcc Calculus Stimulus Store,Resources,Duration Process Residual Process Time Unit 1 Time Unit 2

31 The rtcc Calculus Stimulus Store Resting Point Residual Process Time Unit 1 Time Unit 2

32 The rtcc Calculus Stimulus Store Response Resulting Store, Max Resources Used, Time Consumed Resting Point Residual Process Time Unit 1 Time Unit 2

33 The rtcc Calculus Stimulus Store Resting Point Residual Process Time Unit 1 Time Unit 2

34 The rtcc Calculus Stimulus Store Resting Point Residual Process Time Unit 1 Time Unit 2 Stores are not automatically transferred from a time unit to the next one

35 Syntax P, Q :== tell(c) Tell i I when c i do P i Nondeterminism P Q Parallel Composition local x in P Local Behavior next P Unit Delay delay P until δ Delta Delay unless c next P Weak Time-Out catch c in P finally Q Strong Time-Out P Asynchrony! P Replication

36 Operational Semantics Operational semantics is based on configurations P, d, t. We have internal ( r ) and observable transitions ( α,α )

37 Operational Semantics tell(c), d, t t Φ T (c, d) 0 1 skip, d c, t Φ T (c, d) Tell Operation Φ T (c, d): Time needed to post constraint c in store d. The tell construct uses only 1 resource.

38 Operational Semantics t Φ A (c j, d) 0 d c j, j I i I when c i do P i, d, t 1 P j, d, t Φ A (c j, d) Ask Operation Φ A (c, d): Time needed to query if constraint c can be entailed by store d. The ask construct uses only 1 resource.

39 Operational Semantics P, d, t sp P, d p, t p s p r s p P Q, d, t P Q, d p, t p Q, d, t sq Q, d q, t q s q r s q P Q, d, t P Q, d q, t q P, d, t sp P, d p, t p Q, d, t sq Q, d q, t q s p + s q r P Q, d, t s p+s q P Q, d p d q, min(t p, t q) Parallel Composition Two above: Overlapping version The one below: Truely parallel version

40 Operational Semantics t Φ A (c, d) 0 catch c in P finally Q, d, t d c 1 Q, d, t Φ A (c, d) P, d, t Φ A (c, d) s P, d, t d c s catch c in P finally Q, d, t catch c in P finally Q, d, t Strong Time-Out d c: Process P is stopped and Q is launched. d c: Process P continues (if it can) its execution to P. Process P is now guarded by c.

41 Operational Semantics delay P until δ, d, t δ T t t > 0 0 delay P until δ, d, t 1 δ < T t delay P until δ, d, t 0 P, d, t Delta Delay Operation T : Total duration of the time unit (given by the environment). In each transition the available time is reduce one tick until the delay is greater than or equal to the current time.

42 Operational Semantics P, c, t S Q, d, t P ( c,r,t, d,max(s),t t ) R if R F (Q) Residual Process F (Q) = R if Q = next R or Q = unless c next R F (Q 1) F (Q 2) if Q = Q 1 Q 2 catch c in F (R) finally S if Q = catch c in R finally S local x in F (R) if Q = local x, c in R skip Otherwise

43 Operational Semantics Observations made to Processes: P = P 1 (ι 1,o 1 ) === P 2 (ι 2,o 2 ) === P 3 (ι 3,o 3 ) ===... denoted as P (α,α ) === ω with α = ι 1 ι 2..., α = o 1 o 2... input-output behaviour of P io(p) = {(α, α ) P (α,α ) ω }

44 Denotational Semantics A denotation is a labelled K-valued Chu space. Chu Spaces A matrix C = A, X, λ over a set K: A: The events X : The states λ: The label K: The alphabet

45 Denotational Semantics Events perform actions. States record event ocurrence. Labels are functions λ A Act, where each element of Act, the set of possible actions, is composed by: the actual information (constraints or time) tell posting information a tag: ask querying for information time time this event actually occurs

46 Denotational Semantics The elements of K are the possible values of an event in a given state. K = 0 before it has not yet started during it is happening 1 after it is finishing instead it has been canceled

47 Denotational Semantics tell(c) = c 0 1 λ(c) c, tell when c do P = tell(c) ; P λ(c) c, ask P Q = P Q catch c in P finally Q = ( tell(c) (A A) Q ) ( tell(c) = 0 P B) λ(c) c, ask

48 Denotational Semantics delay P until δ = DELAY ; P next P = CLOCK ; P DELAY = delay 0 1 λ(delay) = δ, time CLOCK = clock 0 1 λ(clock) = T, time

49 Denotational Semantics Example 1: Let P def = tell(a 1 ) and Q def = when a 2 do tell(a 3 ). Then P + Q = a a a λ(a 1 ) = a 1, tell λ(a 2 ) = a 2, ask λ(a 3 ) = a 3, tell

50 Denotational Semantics Example 2: Let P def = catch r in tell(b) and Q def = tell(b). Then: P Q = r b b λ(r) = r, ask λ(b 1 ) = b, tell λ(b 2 ) = b, tell

51 Denotational Semantics Denotational Inputs: Let Ins N D N N be the function that given a time unit index, it will return a tuple consisting of a constraint representing the initial store, a number of resources and an amount of time, for that time unit. We denote Ins(i).j the jth component (j {1, 2, 3}) of Ins(i).

52 Denotational Semantics Definition Step Run Valid Step Distance Valid Run Denotational Outputs: Let Ω be the set of valid runs R. Let Outs Ω D N N be function which given a valid run of a Chu space will return a tuple consisting of a constraint representing a store, a number of resources and an amount of time (the outputs of a process).

53 Denotational Semantics The observables of a process then, can be recovered as follows: Denotational Observables: Let R P,i be the set of valid runs of P i. The observations of a process P is the set of chains of its inputs and outputs, that is Obs(P) = {(α, α ) α(i) = Ins(i) and α (i) = Outs(R P,i )}

54 Denotational Semantics The observables of a process then, can be recovered as follows: Denotational Observables: Let R P,i be the set of valid runs of P i. The observations of a process P is the set of chains of its inputs and outputs, that is Obs(P) = {(α, α ) α(i) = Ins(i) and α (i) = Outs(R P,i )} Full Abstraction: For two processes P and Q, Obs(P) = Obs(Q) iff io(p) = io(q).

55 Real-time Logics We define a Real-time logic for a simplify transition system (without resources): We assume Maximal Parallelism Based on RTTL. An explicit clock variable T. Syntax: A, B,... = c π A A A x.a A A A

56 Real-time Logics Semantics: Definition A timed observation sequence ρ = σ, τ is a pair consisting of an infinite sequence σ of states, and a monotonic function τ N I that maps every c i σ to a time interval. A timed observation sequence is defined as c 1, τ 1 c 2, τ 2 c 3, τ 3...

57 Real-time Logics tell(c) c if ρ, 1 T + Φ T (c, σ(1)) r(τ 1 ) i I when c i do P i i I P i A i (c i A i ) ( c i ) i I i I if ρ, 1 T + Φ A (c i, σ(1)) r(τ 1 ) P A Q B P Q A B P A local x in P x A P A unless c next P c A if ρ, 1 T + Φ A(c, σ(1)) r(τ 1 ) P A Q B catch c in P finally Q ( c A) (c B) if ρ, 1 T + Φ A (c, σ(1)) r(τ 1 ) P A delay P until δ A if ρ, 1 T l(τ 1) + δ P A next P A P A!P A P A P A

58 Concluding Remarks A survey of formal models for music. We review some of the formal models that have been proposed for expressing a variety of musical applications from a computation perspective. Development of rtcc. We develop a new calculus justified in the lack of a formalism capable of expressing both reactive and real-time systems for music. Applications. We illustrate the potential of the new calculus by formalizing a well-known musical tool and a musical improvisation process.

59 Future Work Real-time logic with resources or Girard s Linear Logic. An interpreter. Modelling of complex musical systems. Expressivity proofs Native notion of repair processes

60 Thank you!