Hybrid Temporal Situation Calculus for Planning with Continuous Processes
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This tutorial focuses on the representation and semantics of change. The basic
language is the situation calculus. This tutorial covers both the wellknown
logical foundations and the recent developments in the situation calculus.
Specifically, the initial part of the tutorial includes logical characterizations
of qualitative change within the situation calculus, and the middle part presents
a recent approach that extends the previous version of the situation calculus
with new representations for reasoning about quantitative change over time.
It turns out that this new version of temporal situation calculus can serve
as a basis for the development of a new lifted heuristic planner that can solve
the planning problems in the mixed discretecontinuous domains. It is “lifted”
in a sense that it works with action schemata, but not with instantiated actions.
These hybrid domains include not only the usual discrete actions and qualitative
properties, but also continuous processes that can be possibly initiated or
terminated by actions.
The final part of the tutorial includes an overview of the experimental results
collected from running a tentative implementation of a new planner on a selected
benchmarks designed by the planning community for the hybrid domains. Since
this tutorial focuses on logical representations and semantics, there will be
no discussion of the algorithmic or implementation details related to planning.
No previous background in the situation calculus or planning is required.
However, it is expected that the attendees have basic understanding of knowledge
representation and working knowledge of classical first order logic.
The situation calculus (SC) is a logical approach to representation and
reasoning about actions and their effects. It was proposed in [65,66]
to capture common sense reasoning patterns of how the actions and events change
properties of the world and mental states of the agents. It was inspired by Isaac
Newton's ideas [72], and therefore it described change in terms
of fluents and inertia [25]. In particular, it was
observed that only the specified fluents change their values from one situation
to a new situation, but by default most common sense properties are subject to
inertia and remain the same in the new situation. The problem of finding a
parsimonious characterization of the inertial properties became known as
the frame problem by analogy with the fixed coordinate frame.
In contrast to Newton, who described change and
inertia in terms of quantitative fluents and their fluxions (derivatives),
the original SC approached change in purely qualitative terms [66].
This emphasis on the study of the qualitative aspects of change within SC
continued for years, e.g. see [47,42],
despite a very productive role that the study of continuous change played in
the history of science [15,91].
The research papers about a possible integration within SC of reasoning about
qualitative and continuous change started to appear only in the middle of
the 1990s, e.g., see [24,70] and the references below.
From the very beginning of Artificial Intelligence (AI), it was recognized that
the planning problems at the common sense level can be conveniently formulated
in SC as the entailment problem in predicate logic.
Cordell Green proposed to solve the (planning) problems
using resolutionbased theorem proving in situation calculus
[43,44,45].
Since his program did not work well [81,16],
the subsequent planning research switched to more specialized representations
such as STRIPS developed by Richard Fikes and Nils Nilsson
[32], and ADL, developed by Edwin Pednault
[73]. Accordingly,
the semantics of planning has changed. Namely, the planning problem was often
reformulated as either the satisfiability problem in propositional logic, or
as the reachability problem in an instantiated  propositional level 
transition system, where both the initial state and the goal state
are specified using fluents with explicitly named constants from
the planning instance. In 1999, this historical shift was summarized in
[96] as follows:
“Despite the early formulation of planning as
theorem proving [45], most researchers have long assumed that
specialpurpose planning algorithms are necessary for practical performance".
However, the knowledge representation research of SC continued, since
SC is formulated in a well understood manysorted first order logic (FOL)
with the standard Tarskian semantics [82]. This is in contrast to
alternative less expressive approaches to reasoning about action that rely on
a nonstandard syntax and semantics. In particular, Ray Reiter solved the frame
problem on a FOL level in SC [82],
and in [83] he provided logical
foundations for SC and logical theories for reasoning about actions, events
and their effects. He proposed Basic Action Theories (BATs) that include
precondition axioms, successor state axioms, initial state axioms, unique name
axioms, and the foundational axioms for situations. The latter characterize
situations as a (finitely) branching tree with the root in the initial
situation [78]. Subsequently, Ray Reiter, Hector Levesque
[53], their collaborators and other researchers developed several
extensions of SC, including sequential temporal SC
[92,75,84,51,52,79,34],
SC for reasoning about concurrent actions, reasoning about knowledge
[88],
a systematic axiomatization of direct and indirect effects of actions
[54,59,69,61],
reasoning about noisy sensors and effects in SC [4,11],
decisiontheoretic planning in SC [80,14],
and the high level robot control
[2,46,90,33].
The foundations of this wellestablished KR&R
research are summarized in the Reiter's book [85,77].
It is explained there that the planning problem can be understood as a special
case of the entailment problem in SC within the standard FOL semantics
[35,85,16].
In Reiter's SC [85], the change is represented using atemporal
fluents, i.e., predicates or functions with situational argument, but
without explicit time argument. This
is somewhat counterintuitive, since a large body of knowledge about dynamical
systems in science and engineering is formulated in terms of mathematical functions
that depend on time explicitly and vary over continuous time. Therefore, it is
natural to think about a new temporal SC that can represent changes in hybrid
systems, where the actions/events switch between situations, but within each
situations the specified quantities can change over time. In hybrid automata
[71,1,49,3,27],
actions or events are responsible for discrete transitions between finitely
many atomic states, while states may include processes with continuous evolution
over time. However, in some practical applications, e.g., a traffic domain
representing a large city with flows of cars moving between intersections
[94], or the network of electric power generating stations that
feed electricity to large regions [74], a
hybrid system is more general than a hybrid automaton. These large scale hybrid
systems have relational structure, since their states are no longer atomic,
but they can be represented using fluents with object parameters. The Hybrid
Temporal SC [8,9], in addition to atemporal
fluents that are used to represent qualitatively different contexts, introduces
temporal fluents that have an explicit time argument. The latter are suitable to
represent continuously varying parameterized processes within a state (context).
These new representations facilitate reasoning about the hybrid
systems with relational structure. There is an ICAPS2019 paper
[10] demonstrating that this new
Hybrid Temporal SC can be used to provide a declarative semantics for PDDL+.
PDDL+ is the variant of the Planning Domain Definition Language (PDDL),
a standardized language developed to address some concerns related to modelling
numeric fluents and durative (continuous) actions in PDDL2.1
[6,36,40,68,89]. Maria Fox and Derek
Long mentioned in [37] that it is desirable to have a purely
declarative semantics for PDDL+, and the Hybrid Temporal SC addresses this need.
It turns out that one can develop an efficient lifted heuristic
planner for PDDL+ domains using the Hybrid Temporal SC. The shift towards general domainindependent planning algorithms
based on heuristic search happened in the end of 1990s, e.g., see
[67,12,50,17,48,41].
At the same time and more recently, the researchers gradually realized
the importance of lifted planning, i.e., that AI planners can efficiently work
with action schemata without constructing beforehand a propositional level
transition system, e.g., see
[22,26,30,31,93,63,64,19,38,7,39,23,97,13] and other references mentioned there.
Moreover, Fangzhen Lin developed and implemented a situation calculus based
planner R [55,56] that is a variant of the original
STRIPS planning algorithm. His planner R did not use heuristics, but it
participated in the international competitions on planning at the commonsense
level [56,5]. However, to the best of
our knowledge, SC was never taken as a foundation for designing a lifted heuristic
domainindependent planner.
Our provisional
implementation has been tested on several benchmarks developed to compare
the performance of the state of the art PDDL+ planners such as DiNo
[76],
SMTPlan [20,21],
ENHSP [87,86].
The tutorial will briefly review experimental data collected from our tentative
implementation, and comparison with the stateofthe art in PDDL+ planning.
The tutorial has a duration of 1/2 day.
This tutorial focuses on the semantics and logical representation for actions
and temporally continuous processes in SC. It is expected that the attendees
are not familiar with SC and planning, but they have a basic knowledge of first
order logic [29].
All algorithmic and implementation issues will be discussed at another venue.
The tutorial will include the following topics.
 Introduction to reasoning about action.
Intuitive ontology for the situation calculus.
Deterministic, primitive, atemporal actions without sideeffects
[16,53,85,57].
 Frame axioms (axioms about lack of effects for actions),
the frame problem: Reiter's solution.
Effect axioms, normal form for effect axioms. Transforming effect axioms for a given fluent
into a single positive effect axiom and a single negative effect axiom for the fluent.
Explanation closure, causal completeness, successor state axioms.
[82,85]
 Foundational axioms for the situation calculus, the tree of situations.
Uniform formulas, regressable formulas [78].
 Basic Action Theories (BATs):
precondition axioms, successor state axioms, initial theory,
foundational axioms, unique name axioms (UNA).
[78,85].
Closed world assumption (CWA), open world assumption (OWA).
Domain closure assumption (DCA) vs open domains
with possibly unspecified objects [35].
State constraints (derived predicates): [54,59].
A compilation approach for acyclic constraints: [69].
 The projection and executability problems. A brief review of two techniques
for solving these problems: regression (reasoning backwards) [95]
and progression (reasoning forward).
 The regression operator [95]. The relative satisfiability theorem.
The theorem about reducing the projection problem to entailment of a regressed
formula from an initial theory (together with UNA)
[82,78].
 Reasoning forward in the situation calculus. Forgetting about a single
ground atom and about multiple ground atoms [58].
Progression in the situation calculus [60].
Computing progression efficiently in localeffect basic action theories
[62,31].
 Time and concurrency [85].
Instantaneous actions, processes extended in time.
Approaches to axiomatizing concurrency.
The sequential, temporal situation calculus. We discuss
(1) what new representation has to be introduced, (2) how the foundational
axioms have to be amended, (3) whether we need new axioms,
(4) what has to change in the precondition and successor state axioms.
 A recent extension: Hybrid Temporal Situation Calculus (HTSC)
[8,9].
Atemporal fluents vs temporal fluents.
Continuous processes initiated or terminated by the last action (event).
Temporal change axioms (TCA) to represent how temporal numerical fluents
continuously change over time in each context.
Deriving state evolution axioms (SEA).
 Temporal Basic Action Theories in HTSC for sequential actions
[8,28,18].
Examples.
Relation with PDDL+ [10].
 A lifted heuristic planner based on HTSC.
Experimental assessment, and comparison with the other state of the art
PDDL+ planners on the challenging benchmarks.
 Discussion of possible future research directions.
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Hybrid Temporal Situation Calculus for Planning with Continuous Processes
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