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Foundations for Restraining Bolts: Reinforcement Learning with LTLf/LDLf Restraining Specifications Giuseppe De Giacomo Actions@KR18 Oct. 29, 2018 Joint work with Marco Favorito, Luca Iocchi, & Fabio Patrizi Restraining Bolts

  1. Foundations for Restraining Bolts: Reinforcement Learning with LTLf/LDLf Restraining Specifications Giuseppe De Giacomo Actions@KR18 – Oct. 29, 2018 Joint work with Marco Favorito, Luca Iocchi, & Fabio Patrizi

  2. Restraining Bolts https://www.starwars.com/databank/restraining-bolt Giuseppe De Giacomo (Sapienza) Foundations for Restraining Bolts Actions@KR18 – Oct. 29, 2018 1 / 14

  3. Restraining Bolts Restraining bolts cannot rely on the internals of the agent they control. The controlled agent is not built to be controlled by the restraining bolt . Giuseppe De Giacomo (Sapienza) Foundations for Restraining Bolts Actions@KR18 – Oct. 29, 2018 2 / 14

  4. Restraining Bolts Two distinct representations of the world : ◮ one for the agent , by the designer of the agent ◮ one for the restraining bolt , by the authority imposing the bolt Are these to representations related to each other? ◮ NO: the agent designer and the authority imposing the bolt are not aligned (why should they!) ◮ YES: the agent and the bolt act in the real world. But can restraining bolt exist at all? ◮ YES: for example based on Reinforcement Learning ! Giuseppe De Giacomo (Sapienza) Foundations for Restraining Bolts Actions@KR18 – Oct. 29, 2018 3 / 14

  5. RL with ltl f / ldl f restraining bolt Two distinct representations of the world W : A learning agent represented by an MDP with LA-accessible features S , and reward R ltl f / ldl f rewards { ( ϕ i , r i ) m i =1 } over a set of RB-accessible features L Solution : a non-Markovian policy ρ : S ∗ → A that is optimal wrt rewards r i and R . Observe L not used in ρ ! LA s s Features Extractor a LEARNING AGENT r RB l RESTRAINING Features BOLT Extractor R w WORLD Giuseppe De Giacomo (Sapienza) Foundations for Restraining Bolts Actions@KR18 – Oct. 29, 2018 4 / 14

  6. RL with ltl f / ldl f restraining bolt Formally: Problem definition: RL with ltl f / ldl f restraining specifications Given a learning agent M = � S, A, Tr ag , R ag � with Tr ag and R ag unknown, and a restraining bolt RB = �L , { ( ϕ i , r i ) } m i =1 � formed by a set of ltl f / ldl f formulas ϕ i over L with associated rewards r i . learn a non-Markovian policy ρ : S ∗ → A that maximizes the expected cumulative reward. Giuseppe De Giacomo (Sapienza) Foundations for Restraining Bolts Actions@KR18 – Oct. 29, 2018 5 / 14

  7. Example: Breakout + remove column left to right Learning Agent ◮ LA features : paddle position, ball speed/position ◮ LA actions : move the paddle ◮ LA rewards : reward when a brick is hit Restraining Bolt ◮ RB features : bricks status (broken/not broken) ◮ RB ltl f / ldl f restraining specification : all the bricks in column i must be removed before completing any other column j > i ( l i means: the i th column of bricks has been removed): � ( ¬ l 0 ∧ ¬ l 1 ∧ . . . ∧ ¬ l n ) ∗ ; ( l 0 ∧ ¬ l 1 ∧ . . . ∧ ¬ l n ); ( l 0 ∧ ¬ l 1 ∧ . . . ∧ ¬ l n ) ∗ ; . . . ; ( l 0 ∧ l 1 ∧ . . . ∧ l n ) � tt Giuseppe De Giacomo (Sapienza) Foundations for Restraining Bolts Actions@KR18 – Oct. 29, 2018 6 / 14

  8. Example: Sapientino + pair colors in a given order Learning Agent ◮ LA features: robot position ( x, y ) and facing θ ◮ LA actions: forward, backward, turn left, turn right, beep ◮ LA reward: negative rewards are given when the agent exits the board. Restraining Bolt ◮ RB features: color of current cell, just beeped ◮ RB ltl f / ldl f restraining specification: visit (just beeped) at least two cells of the same color for each color, in a given order among the colors Giuseppe De Giacomo (Sapienza) Foundations for Restraining Bolts Actions@KR18 – Oct. 29, 2018 7 / 14

  9. Example: CocktailParty Robot + don’t serve twice & no alcohol to minors Learning Agent ◮ LA features: robot’s pose, location of objects (drinks and snacks), and location of people ◮ LA actions: move in the environment, can grasp and deliver items to people ◮ LA reward: rewards when a deliver task is completed. Restraining Bolt ◮ RB features: identity, age and received items (in practice, tools like Microsoft Cognitive Services Face API can be integrated into the bolt to provide this information.) ◮ RB ltl f / ldl f restraining specification: serve exactly one drink and one snack to every person, but do not serve alcoholic drinks to minors Giuseppe De Giacomo (Sapienza) Foundations for Restraining Bolts Actions@KR18 – Oct. 29, 2018 8 / 14

  10. Building blocks Classic Reinforcement Learning: ◮ An agent interacts with an environment by taking actions so to maximize rewards ; ◮ No knowledge about the transition model, but assume Markov property (history does not matter): Markov Decision Process (MDP) ◮ Solution: Markovian policy ρ : S → A Temporal logic on finite traces (De Giacomo, Vardi 2013) : ◮ Linear-time Temporal Logic on Finite Traces ltl f ◮ Linear-time Dynamic Logic on Finite Traces ldl f ◮ Reasoning: transform formulas ϕ into NFA/DFA A ϕ s.t. for every trace π and ltl f / ldl f formula ϕ : π | = ϕ ⇐ ⇒ π ∈ L ( A ϕ ) RL for Non-Markovian Decision Process with ltl f / ldl f rewards (Brafman, De Giacomo, Patrizi 2018) : ◮ Rewards depend from history , not just the last transition; ◮ Specify proper behaviours by using ltl f / ldl f formulas; ◮ Solution: Non-Markovian policy ρ : S ∗ → A ◮ Reduce the problem to MDP (with extended state space) Giuseppe De Giacomo (Sapienza) Foundations for Restraining Bolts Actions@KR18 – Oct. 29, 2018 9 / 14

  11. RL for Non-Markovian Decision Process with ltl f / ldl f reward (Brafman, De Giacomo, Patrizi 2018) Lemma (BDP18): Every non-Markovian policy for N is equivalent to a Markovian policy for M which guarantees the same expected reward, and viceversa. Theorem (BDP18): One can find optimal non-Markovian policies solving the N by searching for optimal Markovian policies for M . Corollary: We can reduce non-Markovian RL for N to standard RL for M Giuseppe De Giacomo (Sapienza) Foundations for Restraining Bolts Actions@KR18 – Oct. 29, 2018 10 / 14

  12. RL with ltl f / ldl f restraining specifications (De Giacomo, Favorito, Iocchi, Patrizi 2018) Problem definition: RL with ltl f / ldl f restraining specifications Given a learning agent M = � S, A, Tr ag , R ag � with Tr ag and R ag unknown, and a restraining bolt RB = �L , { ( ϕ i , r i ) } m i =1 � formed by a set of ltl f / ldl f formulas ϕ i over L with associated rewards r i . learn a non-Markovian policy ρ : S ∗ → A that maximizes the expected cumulative reward. Theorem ( De Giacomo, Favorito, Iocchi, Patrizi 2018 ) RL with ltl f / ldl f restraining specifications for learning agent M = � S, A, Tr ag , R ag � and restraining bolt RB = �L , { ( ϕ i , r i ) } m i =1 � can be reduced to classical RL over the MDP M ′ = � Q 1 × · · · × Q m × S, A, Tr ′ ag , R ′ ag � ag learned for M ′ corresponds to an optimal policy of the original i.e., the optimal policy ρ ′ problem. R ′ ag ( q 1 , . . . , q m , s, a, q ′ 1 , . . . , q ′ m , s ′ )= � r i + R ag ( s, a, s ′ ) i : q ′ i ∈ Fi We can rely on off-the-shelf RL algorithms (Q-Learning, Sarsa, ...)! Giuseppe De Giacomo (Sapienza) Foundations for Restraining Bolts Actions@KR18 – Oct. 29, 2018 11 / 14

  13. RL with ltl f / ldl f restraining specifications (De Giacomo, Favorito, Iocchi, Patrizi 2018) Our approach: Transform each ϕ i into dfa A ϕ i Do RL over an MDP M ′ with a transformed state space: S ′ = Q 1 × · · · × Q m × S LA s s Features Extractor q a LEARNING AGENT RB l r RESTRAINING Features BOLT Extractor R w WORLD Notice: the agent ignores RB features L ! RL relies on standard algorithms (e.g. Sarsa( λ )) Giuseppe De Giacomo (Sapienza) Foundations for Restraining Bolts Actions@KR18 – Oct. 29, 2018 12 / 14

  14. Relationship between the LA and RB representations Question 1: What is the relationship between S and L that needs to hold, in order to allow the agent to learn an optimal policy for the RB restraining specification? Answer: None! The LA will learn anyway to comply as much as possible to the RB restraining specifications. Note that from a KR viewpoint being able to synthesize policies by merging two formally unrelated representations S for LA and L for RB is unexpected, and speaks loudly about certain possibilities of RL vs. reasoning/planning. Question 2: Will LA policies surely satisfy RB restraining specification? Answer: Not necessarily! “ You can’t teach pigs to fly! ” But if it does not then anyway no policy are possible! If we want to check formally that the optimal policy satisfies the RB restraining specification, we first need to model how LA actions affects RB L ( the glue ) and then we can use e.g., model checking Question 3: Is the policy computed the same as if we did not make distinction between the features? Answer: No! We learn optimal non-Markovian policies of the form S ∗ → A not of the form ( S ∪ L ) ∗ → A Giuseppe De Giacomo (Sapienza) Foundations for Restraining Bolts Actions@KR18 – Oct. 29, 2018 13 / 14

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