A Formal Basis for a Theory of Composability

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Spring 2003 Simulation Interoperability
Workshop 03S-SIW-054 A Formal Basis for a Theory
of Composability Mikel D. Petty, Ph.D. and Eric
W. Weisel Virginia Modeling, Analysis
Simulation Center Old Dominion University mpetty_at_v
masc.odu.edu 757-686-6210
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Acknowledgements ? Project sponsors - Office of
Naval Research gt Harold Hawkins,
Ph.D. - Defense Modeling and Simulation
Office gt Philomena M. Zimmerman Douglas L.
Clark ? Collaborators - ODU/VMASC gt Eric W.
Weisel Roland R. Mielke, Ph.D. ? Composability
complexity paper Page,1999 - MITRE gt Ernest
H. Page, Ph.D Jeffrey M. Opper
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Outline ? Introduction ? Definitions ? Formal
reasoning and semantics ? Project
goals ? Recent progress ? Summary
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Introduction
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Composability Composability is the capability to
select and assemble simulation components in
various combinations into simulation systemsto
satisfy specific user requirements.
Component Repository
Simulation B
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Simulation A
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1
2
8
3
3
22
. . .
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N
N
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Syntactic and semantic composability ? Syntactic
composability - aka engineering
composability Pratt,1999 Ceranowicz,2002 -
Implementing composable components - Concerned
with component connectability gt e.g., data
interfaces gt e.g., invocation mechanisms - Can
the components be combined so as to execute
together and exchange data?
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? Semantic composability - aka modeling
composability Pratt,1999 Ceranowicz,2002 -
Ensuring composable models - Concerned with
model compatibility gt e.g., domains of
validity gt e.g., consistent assumptions - Is
the combined computation of the composed models
meaningful? - e.g., subsonic flight dynamics
model composed with supersonic jet engine model
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Composability research ? Much work on syntactic
composability - Dynamically loadable modules
Franceschini,1999 - Autonomous behaviors von
der Lippe,2002 Peters,2002 - OneSAF models
Henderson,2002 ? Seek to address semantic
composability - Develop formal theory of
composability - Some previous work
Overstreet,1982 Page,1999 - Basis for formal
theory ? Project underway, current status
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Basis for a formal theory ? Formal theory
requires formal basis ? Basis of a formal theory,
in general - Objects things or ideas that are
theorys subject formally defined - Axioms
statements about objects accepted as true
without proof - Rules of inference methods of
reasoning that can be applied to axioms and
theorems to generate new theorems - Purpose
reason for theory development
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? Basis of a formal theory of composability - Obj
ects model, simulation, composition, composabil
ity, validity formally defined - Axioms
definitions of objects - Rules of inference
normal rules of mathematical reasoning - Purpos
e reason formally about validity and
complexity of compositions of models ? Supporting
theories - Computability theory Davis,1994
Sommerhalder,1988 - Computational complexity
theory Garey,1979 - Model theory Hodges,1993
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Definitions
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Model ? Official definition of model - Model a
physical, mathematical, or otherwise logical
representation of a system, entity, phenomenon,
or process DOD,1996 DOD,1998 - Informal - A
ssumes validity by definition?
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? Definition of model for theory - Model a
computable function - Function relates each
input to unique output - Computable calculable
by finite procedure - No assumption of
validity - Computable functions may be
composite - Domain, codomain vectors of
integers
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Simulation ? Official definition of
simulation - Simulation a method for
implementing a model over time DOD,1996
DOD,1998 - Informal - Inherits validity from
model by definition? - Assumes time-based models
by definition?
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? Definition of simulation for theory - Simulatio
n a sequence of executions of a model - No
assumption of validity - (mn, on) f(mn-1, in-1)
Halbwachs,1991
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Composition and composability ? Definitions for
theory - Composition composition of functions
- Composability a set of models is
composable if and only if their composition is
valid - But valid not yet defined
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Validity ? Informal definition of
validity - Model typically intended to represent
system, i.e., modeled system - Valid
models behavior is close enough to that of
the modeled system
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? Definition of validity for theory - Models are
computable functions gt Input, output vectors of
integers gt Input represents before state of
modeled system gt Output represents after
state of modeled system - Perfect model
notional model that computes, given input
vector representing before state of modeled
system, output vector that exactly matches
after state of modeled system - Perfect
validity metric of closeness between model and
perfect model - v(f, x) f(x) - f(x)
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? Observational validity - Perfect models not
ordinarily available - Observational model set
of vector pairs representing before and
after observations of modeled
system - Observational validity metric of
closeness between model and observational
model - v(f, x) y - f(x) iff (x, y) ?
O
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Formal reasoning and semantics
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Linking formal reasoning to semantics ? Formal
reasoning is devoid of meaning - Axioms, rules,
theorems used in formal, i.e., syntactic,
manner - Interpretations do not affect
reasoning ? Semantics in formal composability
theory? - Semantics (i.e., meaning)
enters composability theory via
validity - Model meaning depends on model
validity - Reasoning about validity is reasoning
about semantics
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Reasoning about validity ? When is a composite
model valid? - e.g., when valid ranges and
domains intersect
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Project goals
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Project goals ? Composition validity
goals - Necessary and sufficient conditions
for preserving validity in compositions - Class
es of models that remain valid under
composition - Algorithm to determine validity of
a composition of models
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? Model selection goals - Necessary and
sufficient conditions for meeting requirements
with composition - Algorithm to select set of
models to meet requirements - Computational
complexity of model selection process
Page,1999 ? Composition complexity
goals - Computational complexity of compositions
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Recent progress
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Recent progress ? Definitions - Labeled
transition systems (LTSs) gt LTSs have states
and labeled transitions gt Use LTS to trace
model computation in simulation gt Compare LTS
trajectories using bisimulation gt Define
validity as metric-based bisimulation
relation? Theorems - Tentatively identified
class of models valid under composition
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Summary
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Summary ? Composability - Ability to combine
simulation components - Syntactic, semantic
composability different ? Formal theory of
composability - Based on formal definitions
of model, simulation, composability,
validity - Formally reason about semantic
composability ? Validity links formal theory to
semantics
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Answers
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Integers ? Objection - Using only integers in
model domains and codomains is too
restrictive. ? Response - Only integers exist
in digital computers - Computability theory
defined for integers i.e., all computable
functions are functions of type N ? N
Davis,1994 Sommerhalder,1988 - Vectors of
integers can be mapped to/from integers via
Cantors method
More?
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Functions ? Objection - Simulation systems are
more than functions. ? Response - Functions gt
Rules that relate input to output gt May be
expressed by arithmetic expressions, enumeratio
n, if-then rules, Hein,2002 - Computability
theory argument gt Anything that runs on a
digital computer is a computable function
Davis,1994 Sommerhalder,1988 gt Simulation
systems run on digital computers
More?
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- Practical argument gt Input to function is
state of computer before step gt Output of
function is state after step gt Step is
instruction cycle or main loop or
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Compositions of functions ? Objection - Simulat
ion systems are built using methods that go
beyond function composition. ? Response - Comput
ability theory argument gt Anything that runs on
a digital computer is a computable function
Davis,1994 Sommerhalder,1988 gt Simulation
systems run on digital computers gt Computable
functions closed under composition Davis,1994
Sommerhalder,1988
More?
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- General form of composition h(X) f(g1(X),
g2(X), , gn(X)) gt f, gi, h are functions gt X
is any number of arguments gt Covers any
combination of functions Hein,2002
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Perfect models ? Objection - Perfect models
dont exist. ? Response - Perfect model allows
formal and quantifiable definition of
validity - Conditions that preserve validity
w.r.t. perfect model of interest, even if
notional - Validity special case of
consistency - Observational model allows
approximation of validity consistent with
formal definition
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End of presentation