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Abstract State Machines, and lessons of an ASM-based project at Microsoft Yuri Gurevich ( Erdos #2 ) Microsoft Research

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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? 2

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Turing’s analysis of computation Great Yet limited 3

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Improving on Turing’s analysis Emile Post Andrei Kolmogorov “Algorithms compute in steps of bounded complexity.” Pointer machines Robin Gandy 4

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Another line of analysis Recursive functions Skolem to Gödel Lambda calculus Church’s thesis Comparing the two lines 5

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6 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? 7

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8 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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9 Postulate 2: Abstract State The states are structures in the sense of mathematical logic. 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. 10

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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? 11

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12 Locations and updates Locations = (f,(a 1,..,a j )) Content( ) = f(a 1,..,a j ) Updates (,v) The update set of state X is (X) = { (,v) : v = Content( ) in Next(X) Content( ) in X }

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13 Postulate 3: Bounded Exploration There is a finite set t 1,..,t n of critical terms such that (X) = (Y) if every Val X (t i ) = Val Y (t i ).

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14 Definition A sequential algorithm is an abstract-state bounded-exploration transition system.

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15 Sequential ASMs SyntaxSemantics = ? f(t 1,..,t j ):= t 0 {(,a 0 )} where = (f,(a 1,..,a j )) and each a i = Val(t i ) do in parallel R 1 … R k (R 1 ) … (R k ) if t then R 1 else R 2 if Val(t) = true then (R 1 ) else (R 2 )

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16 Example if b = 0 then d := a else [do in-parallel] a := b b := a mod b Nullary dynamic functions:a, b, d Static functions: =, 0, mod

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17 Example (cont.) if a(s)=0 then d(s) := b(s) s := s+1 else a(s) := b(s) mod a(s) b(s) := a(s)

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18 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 Next A (X) = Next B (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. 19

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20 In-place one-swap-a-time sorting var A as Seq of Integer = [3,1,2] Swap() choose i,j in Indices(A) where i A(j) A(i) := A(j) A(j) := A(i) Sort() step until fixpoint Swap() A = [2,3,1] A = [1,3,2] A = [1,2,3] A = [2,1,3] Nondeterminsm Parallelism

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21 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 22

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23 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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24 Early ASM engines ASM Workbench Uni Paderborn, Siemens ASM Gopher Uni Ulm, Siemens XASM Uni Berlin, Kestrel

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25 AsmL creators In the hiring order: Wolfram Schulte, Margus Veanes, Colin Campbell, Lev Nachmanson, Mike Barnett, Wolfgang Grieskamp, Nikolai Tillmann

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26 ModelingValidation Refinement Verification AsmL Model Implementation C, C++, C#,... Product Idea / Informal Spec Are you building the product right ? Are you building the right product? What product are you building? FSE propaganda example

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27 Spec ValidateEnforce Comprehend Play scenarios Test Model check Prove properties Generate test suites Lockstep runtime verification On-the-fly testing

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28 Conformance testing I AsmL model Test harness I Implementation under test Discrepancies flagged Any client I

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Spec Explorer Original purpose Model based testing Why model-based testing? Arguably the largest model-based-testing operation anywhere. Success of sorts 29

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Probability of success Coburn: (pain of crisis) divided by (pain of adoption) where pain means perceived pain. 30

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Love triangle 31

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