# Abstract State Machines: From Foundations to Tools ```Abstract State Machines,
and lessons of
an ASM-based project at Microsoft
Yuri Gurevich (Erdos #2)
Microsoft Research
Modeling
No science without modeling


The virtuous cycle
Maybe even no life without modeling
Physics uses PDEs for modeling.
What are the PDEs of computer science?
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Turing’s analysis of computation
Great
Yet limited
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Improving on Turing’s analysis
Emile Post
Andrei Kolmogorov
“Algorithms compute in steps
of bounded complexity.”
Pointer machines
Robin Gandy
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Another line of analysis
Recursive functions
Skolem to G&ouml;del
Lambda calculus
Church’s thesis
Comparing the two
lines
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A Thought Experiment
A perfect machine model
Step-for-step simulation
of any algorithm
Uses: software specs,
model based testing
What would the model look
like?
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Postulate 1: Sequential Time
An algorithm is a
transition system.
What are states?
What are
transitions?
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States
The state is information that, given
the program, determines the ensuing
computation(s).

More than the values of the variables.
What is the form of states?

Or what is is?
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Postulate 2: Abstract State
The states are structures
in the sense of mathematical logic.
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Same vocabulary
Transitions preserve the state domain.
Everything is preserved under isomorphism.
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What are
transitions?
Deterministic or nondeterministic?
More generally,
interactive or non-interactive?
Let’s consider first the classical case of
non-interactive algorithms.
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What are transitions? (cont.)
How powerful steps are?
Let’s consider first the classical case of
“steps of bounded complexity.”
How to bound the complexity?
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Locations  = (f,(a1,..,aj))
Content() = f(a1,..,aj)
The update set of state X is
(X) =
{ (,v) : v = Content() in Next(X)
 Content() in X
}
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Postulate 3: Bounded Exploration
There is a finite set t1,..,tn
of critical terms such that
(X) = (Y) if every ValX(ti) = ValY(ti).
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Definition
A sequential algorithm is
an abstract-state bounded-exploration
transition system.
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Sequential ASMs
Syntax
f(t1,..,tj):= t0
Semantics
=?
{(,a0)} where
=(f,(a1,..,aj)) and
each ai = Val(ti)
do in parallel
R1 … Rk
(R1)  …  (Rk)
if t then R1
else R2
if Val(t) = true then (R1)
else (R2)
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Example
if b = 0 then d := a
else
[do in-parallel]
a := b
b := a mod b
Nullary dynamic functions:
Static functions:
a, b, d
=, 0, mod
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Example (cont.)
if a(s)=0
d(s) :=
s
:=
else
a(s) :=
b(s) :=
then
b(s)
s+1
b(s) mod a(s)
a(s)
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Seq Characterization Theorem
For any seq algorithm A
there is a seq ASM B such that
states of A are states of B and
every NextA(X) = NextB(X).
#141
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Interaction
The ASM model is relatively
straightforward:

External functions

Choice and import operators
The from-the-first-principles analysis
is not straightforward.
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In-place one-swap-a-time sorting
var A as Seq of Integer = [3,1,2]
Nondeterminsm
Swap()
choose i,j in Indices(A)
where i&lt;j and A(i)&gt;A(j)
A(i) := A(j)
A(j) := A(i)
A = [2,3,1]
A = [1,3,2]
A = [2,1,3]
Parallelism
Sort()
step until fixpoint
Swap()
A = [1,2,3]
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Wide steps
Again, the ASM model is relatively
straightforward

do-for-all
The from-the-first-principles analysis
is not straightforward.
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Topological Sorting Example
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Distributed algorithms
Distributed ASMs were defined long
ago, but the axiomatization problem is
wide (and maybe forever) open.
To simulate, one can interleave (sets
of) actions of the computing agents.
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Early ASM engines
ASM Workbench
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ASM Gopher
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Uni Ulm, Siemens
XASM
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Uni Berlin, Kestrel
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AsmL creators
In the hiring order: Wolfram Schulte, Margus
Veanes, Colin Campbell, Lev Nachmanson, Mike
Barnett, Wolfgang Grieskamp, Nikolai Tillmann
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FSE propaganda example
Product Idea
/ Informal Spec
What product
are you
building?
Modeling
AsmL Model
Refinement
Are you
building the
right product?
Validation
Verification
Implementation
C, C++, C#, ...
Are you building
the product right ?
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Spec
Validate
Enforce
Comprehend
Generate
test suites
Play scenarios
Test
Model check
On-the-fly testing
Lockstep runtime
verification
Prove properties
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Conformance testing
Any
client
Discrepancies
flagged
I
Test harness
I
AsmL
model
I
Implementation
under test
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Spec Explorer
Original purpose
Model based testing
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Why model-based testing?
Arguably the largest model-based-testing
operation anywhere.
Success of sorts
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Probability of
success
Coburn:
(pain of crisis)
divided by