Composition of ConGolog Programs

Transcription

1 Composition of ConGolog Programs Sebastian Sardina 1 Giuseppe De Giacomo 2 1 Department of Computer Science and Information Technology RMIT University, Melbourne, AUSTRALIA 2 Dipartimento di Informatica e Sistemistica Antonio Ruberti Sapienza Universita di Roma, Rome, ITALY 1 / 18

2 Behavior Composition: The Basic Idea... Environment (description of actions; prec. & effects) Available Behaviors (description of the behavior of available agents/devices) 2 / 18

3 Behavior Composition: The Basic Idea... Environment (description of actions; prec. & effects) Target Behavior (desired behavior) Available Behaviors (description of the behavior of available agents/devices) 2 / 18

4 Behavior Composition: The Basic Idea... Environment (description of actions; prec. & effects) Controller Target Behavior (desired behavior) Available Behaviors (description of the behavior of available agents/devices) 2 / 18

5 Behavior Composition: The Basic Idea... Environment (description of actions; prec. & effects) Controller Target Behavior (desired behavior) Available Behaviors (description of the behavior of available agents/devices) 2 / 18

6 Behavior Composition: The Basic Idea... Environment (description of actions; prec. & effects) Now with unbounded data! Controller Target Behavior (desired behavior) Available Behaviors (description of the behavior of available agents/devices) 2 / 18

7 The ConGolog Composition Problem Given: 1 An action theory D; 2 n available programs δ 1,..., δ n ; 3 a target program δ t. 3 / 18

8 Given: 1 An action theory D; The ConGolog Composition Problem 2 n available programs δ 1,..., δ n ; 3 a target program δ t. Task: find an orchestrator/delegator that coordinates the concurrent execution of the available programs so as to mimic/realize the target program. 3 / 18

9 Given: 1 An action theory D; The ConGolog Composition Problem 2 n available programs δ 1,..., δ n ; 3 a target program δ t. Task: find an orchestrator/delegator that coordinates the concurrent execution of the available programs so as to mimic/realize the target program. agent/plan coordination, virtual agents; web-service composition; composition of business processes 3 / 18

10 Given: 1 An action theory D; The ConGolog Composition Problem 2 n available programs δ 1,..., δ n ; 3 a target program δ t. Task: find an orchestrator/delegator that coordinates the concurrent execution of the available programs so as to mimic/realize the target program. agent/plan coordination, virtual agents; web-service composition; composition of business processes Notable features: Programs may include non-deterministic points & may not terminate. Domain may be infinite. Programs may go over infinite states. 3 / 18

11 Controller for a Music Jukebox while True do { if ( Playing ( song)pending(song)) then (π song, disk).{ (Pending(song) InDisk(song, disk))?; select(song); load(disk); play(song) } else wait } 4 / 18

12 Controller for a Music Jukebox while True do { if ( Playing ( song)pending(song)) then (π song, disk).{ (Pending(song) InDisk(song, disk))?; select(song); load(disk); play(song) } else wait } Domain tests relative to an action theory. 4 / 18

13 Controller for a Music Jukebox while True do { if ( Playing ( song)pending(song)) then (π song, disk).{ (Pending(song) InDisk(song, disk))?; select(song); load(disk); play(song) } else wait } Domain tests relative to an action theory. Domain actions. 4 / 18

14 Controller for a Music Jukebox while True do { if ( Playing ( song)pending(song)) then (π song, disk).{ (Pending(song) InDisk(song, disk))?; select(song); load(disk); play(song) } else wait } Domain tests relative to an action theory. Domain actions. Nondeterministic features. 4 / 18

15 Semantics for High-Level Programs In terms of two predicates: 1 Trans(δ, s, δ, s ): program δ can evolve one step from situation s to situation s with remaining program δ. Trans(δ 1 ; δ 2, s, δ, s ) Trans(δ 1, s, δ 1, s ) δ = δ 1 ; δ 2 Final(δ 1, s) Trans(δ 2, s, δ, s ). 2 Final(δ, s): program δ may terminate successfully in s. Final(δ 1 ; δ 2, s) Final(δ 1, s) Final(δ 2, s) 5 / 18

16 Formalizing the Composition Problem: Simulation Informally: System S simulates system T if S can match all T s moves, forever. 6 / 18

17 Formalizing the Composition Problem: Simulation Informally: System S simulates system T if S can match all T s moves, forever. Formally [Milner IJCAI 71]: Given two labelled TSs S = (Σ S, A S, S ) and T = (Σ T, A T, T ), the simulation is the largest relation Sim Σ S Σ T such that: if Sim(s, t) holds (state s simulates state t), then: if t α T t, then there exists s α S s and Sim(s, t ). 6 / 18

18 The Composition Problem: Simulation Sim(δ t, δ 1,..., δ n, s): available programs can simulate the target program in s. where Sim(δ t, δ 1,..., δ n, s) S. ( S(δ t, δ 1,..., δ n, s) δ t, δ 1,..., δ n, s.θ[s](δ t, δ 1,..., δ n, s) ), Θ[S](δ t, δ 1,..., δ n, s) = def S(δ ( t, δ 1,..., δ n, s) Final(δt, s) i=1,...,n Final(δ i, s) ) ( δ t, s Trans(δ t, s, δ t, s ) i=1,...,n δ i.trans(δ i, s, δ i, s ) S(δ t, δ 1,..., δ i,..., δ n, s ) ). If the simulation holds then one can build an orchestrator generator based on it. 7 / 18

19 The Composition Problem: Simulation Sim(δ t, δ 1,..., δ n, s): available programs can simulate the target program in s. where Sim(δ t, δ 1,..., δ n, s) S. ( S(δ t, δ 1,..., δ n, s) δ t, δ 1,..., δ n, s.θ[s](δ t, δ 1,..., δ n, s) ), Second Θ[S](δ t, δ 1,..., δ n, s) = def order! S(δ ( t, δ 1,..., δ n, s) Final(δt, s) i=1,...,n Final(δ i, s) ) ( δ t, s Trans(δ t, s, δ t, s ) i=1,...,n δ i.trans(δ i, s, δ i, s ) S(δ t, δ 1,..., δ i,..., δ n, s ) ). If the simulation holds then one can build an orchestrator generator based on it. 7 / 18

20 The Technique Relies on the following tools /notions: 1 Simulation approximates: [Tarski 55] Check simulation in a finite way. 2 Regression mechanism: [Reiter 91; Pirri & Reiter 99] Reason on formulas after action performance. 3 Characteristic graphs: [Classen&Lakemeyer KR 08] Abstract (infinite) program states into a finite graph. 8 / 18

21 The Technique Relies on the following tools /notions: 1 Simulation approximates: [Tarski 55] Check simulation in a finite way. 2 Regression mechanism: [Reiter 91; Pirri & Reiter 99] Reason on formulas after action performance. 3 Characteristic graphs: [Classen&Lakemeyer KR 08] Abstract (infinite) program states into a finite graph. 9 / 18

22 Simulation Approximates Sim k (δ t, δ 1,..., δ n, s): the available programs can simulate k steps of the target program in s. Sim 0(δ t, δ 1,..., δ n, s) (Final(δ t, s) V i=1,...,n Final(δ i, s)). Sim k+1 (δ t, δ 1,..., δ n, s) Sim k (δ t, δ 1,..., δ n, s) ` δ t, s.trans(δ t, s, δ t, s ) W i=1,...,n δ i.trans(δ i, s, δ i, s ) Sim k (δ t, δ 1,..., δ i,..., δ n, s ). 10 / 18

23 Simulation Approximates Sim k (δ t, δ 1,..., δ n, s): the available programs can simulate k steps of the target program in s. Sim 0(δ t, δ 1,..., δ n, s) (Final(δ t, s) V i=1,...,n Final(δ i, s)). Sim k+1 (δ t, δ 1,..., δ n, s) Sim k (δ t, δ 1,..., δ n, s) ` δ t, s.trans(δ t, s, δ t, s ) W i=1,...,n δ i.trans(δ i, s, δ i, s ) Sim k (δ t, δ 1,..., δ i,..., δ n, s ). Proposition For every k 0, if Sim k (δ t, δ 1,..., δ n, s) Sim k+1 (δ t, δ 1,..., δ n, s), then Sim k (δ t, δ 1,..., δ n, s) Sim(δ t, δ 1,..., δ n, s). 10 / 18

24 The Technique Relies on the following tools /notions: 1 Simulation approximates: [Tarski 55] Check simulation in a finite way. 2 Regression mechanism: [Reiter 91; Pirri & Reiter 99] Reason on formulas after action performance. computes what has to be true in situation s so that φ is true after doing action α in s. R[φ(do(α, s))] = φ (s) action α has been eliminated! 3 Characteristic graphs: [Classen&Lakemeyer KR 08] Abstract (infinite) program states into a finite graph. 11 / 18

25 The Technique Relies on the following tools /notions: 1 Simulation approximates: [Tarski 55] Check simulation in a finite way. 2 Regression mechanism: [Reiter 91; Pirri & Reiter 99] Reason on formulas after action performance. 3 Characteristic graphs: [Classen&Lakemeyer KR 08] Abstract (infinite) program states into a finite graph. 12 / 18

26 Characteristic Graph for δ music πsong, disk : select(song), v 0 Pending(song) InDisk(song, disk) Playing v 1 wait, Playing song.pending(song) play(song), True load(disk), True v 2 δ music. = while True do { if ( Playing ( song)pending(song)) then πsong, disk.{ (Pending(song) InDisk(song, disk))?; select(song); load(disk); play(song) } else wait } v 0 = δ music, False v 1 = load(disk); play(song), False v 2 = play(song), False 13 / 18

27 The Composition Problem: Simulation Sim(δ t, δ 1,..., δ n, s): available programs can simulate the target program in s. where Sim(δ t, δ 1,..., δ n, s) S. ( S(δ t, δ 1,..., δ n, s) δ t, δ 1,..., δ n, s.θ[s](δ t, δ 1,..., δ n, s) ), Θ[S](δ t, δ 1,..., δ n, s) = def S(δ ( t, δ 1,..., δ n, s) Final(δt, s) i=1,...,n Final(δ i, s) ) ( δ t, s Trans(δ t, s, δ t, s ) i=1,...,n δ i.trans(δ i, s, δ i, s ) S(δ t, δ 1,..., δ i,..., δ n, s ) ). If the simulation holds then one can build an orchestrator generator based on it. 14 / 18

28 Algorithm SymSim(δ 0 t, δ 0 1,..., δ0 n) Computes relation X containing tuples of the form v t, v 1,..., v n, φ : v t, v 1,..., v n are nodes in the characteristic graphs. FO formula φ: the target program in v t is simulated by the available programs in v 1,..., v n. 15 / 18

29 Algorithm SymSim(δ 0 t, δ 0 1,..., δ0 n) Computes relation X containing tuples of the form v t, v 1,..., v n, φ : v t, v 1,..., v n are nodes in the characteristic graphs. FO formula φ: the target program in v t is simulated by the available programs in v 1,..., v n. 1 X 0 := { (δ t, γ t ), (δ 1, γ 1 ),..., (δ n, γ n ), γ t n i=1 γ i (δ j, γ j ) in G δ 0 j } 0-step simulation: check for termination mimicking. 15 / 18

30 Algorithm SymSim(δ 0 t, δ 0 1,..., δ0 n) Computes relation X containing tuples of the form v t, v 1,..., v n, φ : v t, v 1,..., v n are nodes in the characteristic graphs. FO formula φ: the target program in v t is simulated by the available programs in v 1,..., v n. 1 X 0 := { (δ t, γ t ), (δ 1, γ 1 ),..., (δ n, γ n ), γ t n i=1 γ i (δ j, γ j ) in G δ 0 j } 0-step simulation: check for termination mimicking. 2 At every step, compute Next[X ]: one step refinement of the simulation: Next[X ] = { v t, v 1,..., v n, φ old φ new v t, v 1,..., v n, φ old X }. φ new : we can safely mimic (any) single action from the target. 15 / 18

31 Algorithm SymSim(δ 0 t, δ 0 1,..., δ0 n) Computes relation X containing tuples of the form v t, v 1,..., v n, φ : v t, v 1,..., v n are nodes in the characteristic graphs. FO formula φ: the target program in v t is simulated by the available programs in v 1,..., v n. 1 X 0 := { (δ t, γ t ), (δ 1, γ 1 ),..., (δ n, γ n ), γ t n i=1 γ i (δ j, γ j ) in G δ 0 j } 0-step simulation: check for termination mimicking. 2 At every step, compute Next[X ]: one step refinement of the simulation: Next[X ] = { v t, v 1,..., v n, φ old φ new v t, v 1,..., v n, φ old X }. φ new : we can safely mimic (any) single action from the target. 3 X represents the approximates of the simulation, refined at each iteration. 15 / 18

32 Algorithm SymSim(δ 0 t, δ 0 1,..., δ0 n) Computes relation X containing tuples of the form v t, v 1,..., v n, φ : v t, v 1,..., v n are nodes in the characteristic graphs. FO formula φ: the target program in v t is simulated by the available programs in v 1,..., v n. 1 X 0 := { (δ t, γ t ), (δ 1, γ 1 ),..., (δ n, γ n ), γ t n i=1 γ i (δ j, γ j ) in G δ 0 j } 0-step simulation: check for termination mimicking. 2 At every step, compute Next[X ]: one step refinement of the simulation: Next[X ] = { v t, v 1,..., v n, φ old φ new v t, v 1,..., v n, φ old X }. φ new : we can safely mimic (any) single action from the target. 3 X represents the approximates of the simulation, refined at each iteration. 4 Stop when X = Next[X ]. 15 / 18

33 Next[X ]: One-step refinement of the simulation Next[X ] = { v t, v 1,..., v n, φ old φ new v t, v 1,..., v n, φ old X }, φ new = π x αt v t v t ( Et ψ t x.ψ t [s] Poss(α t, s) n i=1 π y.α v i i v ψi i E i v t,v1,...,v i,...,vn,φ i X ) y.α t = α i ψ i [s] R[φ i (do(α i, s))]. For every potential target evolution from v t to v t via action α t,... if it can be done in the program (ψ t holds) and the action α t is possible,... then some available prog. δ i can evolve from v i to v i via action α i such that: 1 the action α i can be matched to α t ; 2 the program can indeed do the step; 3 after doing the step, we are still in simulation. 16 / 18

34 Next[X ]: One-step refinement of the simulation Next[X ] = { v t, v 1,..., v n, φ old φ new v t, v 1,..., v n, φ old X }, φ new = π x αt v t v t ( Et ψ t x.ψ t [s] Poss(α t, s) n i=1 π y.α v i i v ψi i E i v t,v1,...,v i,...,vn,φ i X ) y.α t = α i ψ i [s] R[φ i (do(α i, s))]. For every potential target evolution from v t to v t via action α t,... if it can be done in the program (ψ t holds) and the action α t is possible,... then some available prog. δ i can evolve from v i to v i via action α i such that: 1 the action α i can be matched to α t ; 2 the program can indeed do the step; 3 after doing the step, we are still in simulation. 16 / 18

35 Next[X ]: One-step refinement of the simulation Next[X ] = { v t, v 1,..., v n, φ old φ new v t, v 1,..., v n, φ old X }, φ new = π x αt v t v t ( Et ψ t x.ψ t [s] Poss(α t, s) n i=1 π y.α v i i v ψi i E i v t,v1,...,v i,...,vn,φ i X ) y.α t = α i ψ i [s] R[φ i (do(α i, s))]. For every potential target evolution from v t to v t via action α t,... if it can be done in the program (ψ t holds) and the action α t is possible,... then some available prog. δ i can evolve from v i to v i via action α i such that: 1 the action α i can be matched to α t ; 2 the program can indeed do the step; 3 after doing the step, we are still in simulation. 16 / 18

36 Next[X ]: One-step refinement of the simulation Next[X ] = { v t, v 1,..., v n, φ old φ new v t, v 1,..., v n, φ old X }, φ new = π x αt v t v t ( Et ψ t x.ψ t [s] Poss(α t, s) n i=1 π y.α v i i v ψi i E i v t,v1,...,v i,...,vn,φ i X ) y.α t = α i ψ i [s] R[φ i (do(α i, s))]. For every potential target evolution from v t to v t via action α t,... if it can be done in the program (ψ t holds) and the action α t is possible,... then some available prog. δ i can evolve from v i to v i via action α i such that: 1 the action α i can be matched to α t ; 2 the program can indeed do the step; 3 after doing the step, we are still in simulation. 16 / 18

37 Next[X ]: One-step refinement of the simulation Next[X ] = { v t, v 1,..., v n, φ old φ new v t, v 1,..., v n, φ old X }, φ new = π x αt v t v t ( Et ψ t x.ψ t [s] Poss(α t, s) n i=1 π y.α v i i v ψi i E i v t,v1,...,v i,...,vn,φ i X ) y.α t = α i ψ i [s] R[φ i (do(α i, s))]. For every potential target evolution from v t to v t via action α t,... if it can be done in the program (ψ t holds) and the action α t is possible,... then some available prog. δ i can evolve from v i to v i via action α i such that: 1 the action α i can be matched to α t ; 2 the program can indeed do the step; 3 after doing the step, we are still in simulation. 16 / 18

38 Next[X ]: One-step refinement of the simulation Next[X ] = { v t, v 1,..., v n, φ old φ new v t, v 1,..., v n, φ old X }, φ new = π x αt v t v t ( Et ψ t x.ψ t [s] Poss(α t, s) n i=1 π y.α v i i v ψi i E i v t,v1,...,v i,...,vn,φ i X ) y.α t = α i ψ i [s] R[φ i (do(α i, s))]. For every potential target evolution from v t to v t via action α t,... if it can be done in the program (ψ t holds) and the action α t is possible,... then some available prog. δ i can evolve from v i to v i via action α i such that: 1 the action α i can be matched to α t ; 2 the program can indeed do the step; 3 after doing the step, we are still in simulation. 16 / 18

39 Next[X ]: One-step refinement of the simulation Next[X ] = { v t, v 1,..., v n, φ old φ new v t, v 1,..., v n, φ old X }, φ new = π x αt v t v t ( Et ψ t x.ψ t [s] Poss(α t, s) n i=1 π y.α v i i v ψi i E i v t,v1,...,v i,...,vn,φ i X ) y.α t = α i ψ i [s] R[φ i (do(α i, s))]. For every potential target evolution from v t to v t via action α t,... if it can be done in the program (ψ t holds) and the action α t is possible,... then some available prog. δ i can evolve from v i to v i via action α i such that: 1 the action α i can be matched to α t ; 2 the program can indeed do the step; 3 after doing the step, we are still in simulation. 16 / 18

40 Technical Results Theorem If algorithm SymSim(δ 0 t, δ 0 1,..., δ0 n) terminates returning the set X. Then, Axioms = Sim(δ t, δ 1,..., δ n, s) φ[s], where (δ t, γ t ), (δ 1, γ 1 ),..., (δ n, γ n ), φ X. 17 / 18

41 Technical Results Theorem If algorithm SymSim(δ 0 t, δ 0 1,..., δ0 n) terminates returning the set X. Then, Axioms = Sim(δ t, δ 1,..., δ n, s) φ[s], where (δ t, γ t ), (δ 1, γ 1 ),..., (δ n, γ n ), φ X. Moreover, we can construct a delegator controller to realize the composition on-the-fly using FO entailment only (on D S0 ). Idea: after a request, jump to a configuration v t, v 1,..., v n, φ X for which φ holds. 17 / 18

42 Conclusions Reasoning on unbounded data and processes is a challenge for CS since it leads infinite state systems. Standard approach: abstract to finite systems (see literature on Verification). Here instead we are proposing an alternative approach rooted in KR and AI. 18 / 18

43 Conclusions Reasoning on unbounded data and processes is a challenge for CS since it leads infinite state systems. Standard approach: abstract to finite systems (see literature on Verification). Here instead we are proposing an alternative approach rooted in KR and AI. Specifically: Based on transforming the second-order formula for checking the dynamic property of simulation into a first-order one talking only about the static properties of the initial situation/db. 18 / 18

44 Conclusions Reasoning on unbounded data and processes is a challenge for CS since it leads infinite state systems. Standard approach: abstract to finite systems (see literature on Verification). Here instead we are proposing an alternative approach rooted in KR and AI. Specifically: Based on transforming the second-order formula for checking the dynamic property of simulation into a first-order one talking only about the static properties of the initial situation/db. Main research direction for future work: 1 incomplete information about the initial situation. offline vs online interpreters (see literature on high-level programs in AI). 2 identify cases in which the technique becomes sound & complete. 18 / 18