Hybrid Temporal Situation Calculus Vitaliy Batusov 1 Giuseppe De - - PowerPoint PPT Presentation

Hybrid Temporal Situation Calculus

Vitaliy Batusov 1 Giuseppe De Giacomo 2 Mikhail Soutchanski 3

1York University, Toronto 2Sapienza Universit`

a di Roma, Rome

3Ryerson University, Toronto

May 30, 2019

Batusov et al. (YorkU / Sapienza / Ryerson) Hybrid Situation Calculus May 30, 2019 1 / 23

Overview

1

Background Dynamical Systems Formalisms Situation Calculus

2

Hybrid Situation Calculus General Considerations Methodology Hybrid Action Theories Features

3

Conclusions

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Dynamical Systems

Dynamical systems: systems that undergo change of some kind Examples: pendulum, animal populations, digital circuits Also: A model which represents a part of the world relevant to the questions we want to answer Have states and transitions between them

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Kinds of change

Continuous (macroscopic physical world) Discrete (quantum world; approximations of macroscopic world) In modelling, a mix of the two is often necessary ⇒ Hybrid Example: computer-controlled machinery

self-driving cars chemical plant high-speed train system

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Formalisms

Need a language to describe and analyse dynamical systems Various formalisms for different applications are available Hybrid Automata: disrete automata with differential equations for each state Come from physics and engineering. Good for continuous dynamics, can’t express rich relational states Situation Calculus: “system as a logical theory” Comes form KRR. Good for relational states with internal structure, limited in continuous dynamics PDDL+: a planning formalism Comes from automated planning. Offers a better balance between discrete and continuous, but with limitations on both

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Our goal

Propose an extension to Raymond Reiter’s version of situation calculus which would organically combine discrete change with true continuous change over real-valued time without compromising too much.

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Situation Calculus

System is described axiomatically in first-order logic Theory has three sorts: action, situation, domain object actions: symbols which trigger change situations: sequences of actions (world histories) domain objects: everything else (cats, cars, numbers, etc.) Predicates and functions describe properties of objects: Cat(John) Actions are used to construct situations: feed(John) executed in the initial situation S0 yields a new situation do(feed(John), S0) Predicates/functions whose last argument is a situation are called fluents: Happy(x, s) — “x is happy in situation s” Fluents are what changes from one situation to another. Thus, situations are a frame of reference, but fluents are the state.

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Modelling Systems in Situation Calculus

Basic Action Theories — Reiter (2001) To describe dynamics of fluent F, begin with effect axioms φ+(¯ x, a, s) → F(¯ x, do(a, s)) (positive) φ−(¯ x, a, s) → ¬F(¯ x, do(a, s)) (negative) Example: Cat(x) ∧ ¬Happy(x, s) → Happy(x, do(feed(x), s)) Cat(x) → ¬Happy(x, do(bathe(x), s))

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Frame problem

Want the theory to unambiguously describe what happens when an action is executed Things that change — effect axioms Things that simply carry over — ? (too many to list)

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Reiter’s solution

Causal completeness assumption: there is no other source of change to fluent F other than what is asserted in the effect axioms Formally: F(¯ x, s) ∧ ¬F(¯ x, do(a, s)) → φ−(¯ x, a, s) ¬F(¯ x, s) ∧ F(¯ x, do(a, s)) → φ+(¯ x, a, s). Assuming that φ+ and φ− can never happen simultaneously, Reiter derives the successor state axiom F(¯ x, do(a, s)) ↔ φ+(¯ x, a, s) ∨ F(¯ x, s) ∧ ¬φ−(¯ x, a, s) which is logically equivalent to the conjunction of the above axioms.

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Successor State Axioms

Example: Happy(x, do(a, s)) ↔ a=feed(x) ∧ Cat(x) ∧ ¬Happy(x, s) ∨ Happy(x, s) ∧ ¬[a=bathe(x) ∧ Cat(x)] This concise form is the key feature of BATs Makes regression possible: given a query about a far-away situation, can transform it to equivalent query about S0

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General considerations

Goal: allow for true continuous change while providing the same kind

• f systematic modelling methodology

Two good ideas:

Autonomous processes: a modelling pattern where the scope of continuous behaviours is limited to a set of well-defined relational states (processes) started and ended by actions. Appears in Pinto (1994), PDDL+. HA too inexpressive to raise this issue. Temporal fluents: fluents whose values are defined for the entire real-valued timeline, and not just the time points of actions. PDDL+ authors Fox & Long (2003): When determining how to achieve a goal a planner must be able to access the values of these continuous quantities at arbitrary points on the time-line of the plan. Example: Reiter’s BATs can’t handle queries about arbitrary timepoints: velocity(Cat, τ, σ) > 10

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Methodology

Add time arguments to all actions: feed(x) ⇒ feed(x, t) (Reiter) Add temporal fluents — functional fluents with arguments t, s velocity(Cat, t, s) Describe them using temporal change axioms of the form γ(¯ x, s) ∧ δ(¯ x, y, t, s) → f (¯ x, t, s)=y, where γ(¯ x, s) is a discrete “context” and δ(¯ x, y, t, s) encodes a function defining y. Modeller must ensure integrity γ(¯ x, s) → ∃y δ(¯ x, y, t, s)

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Example

A reaction in beaker b yields a product at rate R(b) whenever b contains a mix of reagents in situation s (Mix(b, s)). Adding one of the catalysts c into b (in(c, b, s)) multiplies the rate by m(c). Pouring the products from two beakers into an empty one creates a new mix and starts another reaction. Amount of product in beaker b at time t in situation s: Mix(b, s) ∧ ¬∃c[in(b, c, s)]

• context γ

∧ y =prod(b, start(s), s) + R(b) × (t − start(s))

• function δ

→ prod(b, t, s)=y The behaviour of atemporal fluents Mix and in is described in the usual way using SSA.

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Continuous Frame Problem

Must solve a continuous version of the frame problem Given a set of k temporal change axioms for fluent f γ1(¯ x, s) ∧ δ1(¯ x, y, t, s) → f (¯ x, t, s)=y . . . γk(¯ x, s) ∧ δk(¯ x, y, t, s) → f (¯ x, t, s)=y Equivalently express as a single normal-form axiom Φ(¯ x, y, t, s) → f (¯ x, t, s)=y, Assert an explanation closure axiom f (¯ x, t, s) = f (¯ x, start(s), s) → ∃y Φ(¯ x, y, t, s).

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State Evolution Axioms

Assuming the integrity and consistency of the TCA in normal form,

• btain an equivalent axiom called the state evolution axiom

f (¯ x, t, s)=y ↔ [Φ(¯ x, y, t, s) ∨ y =f (¯ x, start(s), s) ∧ ¬

• 1≤i≤k

γi(¯ x, s)] Compare to SSA: F(¯ x, do(a, s)) ↔ [φ+(¯ x, a, s) ∨ F(¯ x, s) ∧ ¬φ−(¯ x, a, s)] This is the main result of the paper. Note that the SEA talks about a single situation, unlike SSA

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Example (continued)

Modeller provides temporal change axioms only for non-trivial change Methodology yields a SEA that covers all circumstances: prod(b, t, s)=y ↔ ∃τ.τ = (t − start(s)) ∧

• ∃c[Mix(b, s) ∧ in(b, c, s)
• there is a catalyst

∧y =prod(b, start(s), s) + m(c)R(b)τ] ∨ Mix(b, s) ∧ ¬∃c[in(b, c, s)]

• no catalyst

∧y =prod(b, start(s), s) + R(b)τ ∨ ¬Mix(b, s)

• frame context

∧ y =prod(b, start(s), s)

• trivial evolution
• The frame context is derived from given and covers all other

possibilities Exactly one of the contexts of the SEA holds for every situation

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Hybrid BATs and Properties

Can formally define a Hybrid BAT Consists of a Reiter’s BAT, a set of SEA, and some consistency conditions to ensure the interactions between the discrete and the continuous sides are well-defined when an action occurs Stratified HBATs: useful fragment with no cyclic dependence between

• temp. fluents by virtue of SEA

If stratified, can easily define regression If stratified, can prove a relative satisfiability property: the question of SAT of the theory is reduced to SAT of the initial state We thus preserve the most important properties of BATs

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Compared to Hybrid Automata

Recall example. In hybrid BATs, any system with k beakers and n catalysts can be captured concisely by the same theory (save for the initial state, which grows as k + n). An equivalent⋆ hybrid automaton grows at least linearly with k × n in the number of states and at least quadratically in the number of transitions A system with incomplete information further blows up the HA state

• space. This is hard to quantify because it depends on the kind of

incompleteness. A SC action may introduce new objects (beakers, catalysts), dynamically growing the transition graph, which cannot be expressed in HA HBAT offers symbolic reasoning about the future

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Compared to BATs

In HBATs, we can query fluents at arbitrary time points and so can pose queries on the continuous state directly: ∃t, b1, b2(t > T ∧ prod(b1, t, s)−prod(b2, t, s)<2) To achieve similar behaviour, Reiter’s SC, in place of the temporal fluents above, must express their values in terms of values at start(s), taking continuous evolution into account—a sinister modelling challenge

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Conclusions

We can show (see paper) that a hybrid automaton can be captured by a hybrid BAT Informed by HA and PDDL+, hybrid situation calculus brings modern treatment of continuous time and a modelling methodology for mixed domains to SC Also brings complex agent actions, unbounded rich relational states with uncertainty, and symbolic reasoning about states to the domain

• f hybrid automata

Also allows to replace complex meta-theoretical semantics of PDDL+ by a logical theory (to appear in ICAPS-2019)

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Rough map of expressive power

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Questions?

1

Background Dynamical Systems Formalisms Situation Calculus

2

Hybrid Situation Calculus General Considerations Methodology Hybrid Action Theories Features

3

Conclusions

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