Semantical Characterization of unbounded-nondeterministic ASMs
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1 Semantical Characterization of unbounded-nondeterministic ASMs Berlin, 26/27 Feb 2007 Andreas Glausch Humboldt-Universität zu Berlin Department of Computer Science
2 Abstract State Machines (ASMs) state next-state Σ-structure ASM program Σ-structure syntax Can we do without syntax? 2
3 Semantical Characterization Gurevich 99 sequential ASMs = sequential algorithms 3
4 Semantical Characterization Gurevich 99 sequential ASMs = sequential algorithms Gurevich, Yavorskaya 06 bounded-nondeterministic ASM = bounded-nondeterministic algorithms k 4
5 This Talk Lipari Guide unbounded-nondeterministic ASMs = unbounded-nondeterministic algorithms new 0 nondeterministic = def unbounded-nondeterministic 5
6 Semantical Characterization of Sequential ASMs by Yuri Gurevich
7 Sequential ASMs A sequential ASM consists of a set of Σ-structures, a sequential ASM program: 1. assignment statements 2. conditional statements 3. block statements a := next(a) if (a b) then par endpar 7
8 Characterization sequential ASMs = sequential algorithms 8
9 Sequential Algorithms A sequential algorithm Α satisfies (S1) sequential-time axiom: Α determines a set of states S and a next-state function τ : S S. 9
10 Sequential Algorithms A sequential algorithm Α satisfies (S1) sequential-time axiom, (S2) state axiom: All states are Σ-structures. 10
11 Sequential Algorithms A sequential algorithm Α satisfies (S1) sequential-time axiom, (S2) state axiom, (S3) universe axiom: S and τ(s) have the same universe. 11
12 Sequential Algorithms A sequential algorithm Α satisfies (S1) sequential-time axiom, (S2) state axiom, (S3) universe axiom, (S4) isomorphism axiom: S is closed under isomorphism. τ preserves isomorphisms. 12
13 Sequential Algorithms A sequential algorithm Α satisfies (S1) sequential-time axiom, (S2) state axiom, (S3) universe axiom, (S4) isomorphism axiom, (S5) bounded-exploration axiom: A finite set T of ground terms characterizes τ: S = T R implies (S, τ(s)) = (R, τ(r)). 13
14 Sequential Algorithms A sequential algorithm Α satisfies (S1) sequential-time axiom, (S2) state axiom, (S3) universe axiom, (S4) isomorphism axiom, (S5) bounded-exploration axiom. 14
15 Nondeterministic ASMs
16 Nondeterministic ASMs A nondeterministic ASM consists of a set of Σ-structures, a nondeterministic ASM program: 1. assignment statements a := next(a) 2. conditional statements if (a=b) then 3. block statements par endpar 4. choice statements choose x do 16
17 Example State a() 1 b() 2 universe: N x next(x) signature Σ: a: N b: N next: N N 17
18 Example Program choose x with x a do b:=x x=3 a() b() 1 3 a() b() 1 2 x=4 x=5 a() b() 1 4 a() 1 b() 5 18
19 Semantical Characterization of nondeterministic ASMs
20 Characterization nondeterministic ASMs = nondeterministic algorithms 20
21 Nondeterministic Algorithms A nondeterministic algorithm Ν satisfies (N1) nondeterministic-time axiom, (N2) state axiom, (N3) universe axiom, (N4) isomorphism axiom, (N5)... 21
22 Nondeterministic Algorithms A nondeterministic algorithm Ν satisfies (N1) nondeterministic-time axiom, (N2) state axiom, (N3) universe axiom, (N4) isomorphism axiom, (N5) bounded-work axiom. 22
23 Nondeterministic Algorithms A nondeterministic algorithm Ν satisfies (N1) nondeterministic-time axiom: Ν consists of a set of states S and a next-state relation S S. 23
24 Nondeterministic Algorithms A nondeterministic algorithm Ν satisfies (N1) nondeterministic-time axiom, (N2) state axiom: All states are Σ-structures. 24
25 Nondeterministic Algorithms A nondeterministic algorithm Ν satisfies (N1) nondeterministic-time axiom, (N2) state axiom, (N3) universe axiom: For S S, S and S have the same universe. 25
26 Nondeterministic Algorithms A nondeterministic algorithm Ν satisfies (N1) nondeterministic-time axiom, (N2) state axiom, (N3) universe axiom, (N4) isomorphism axiom: S is closed under isomorphism. Isomorphic states have isomorphic next-states. 26
27 Nondeterministic Algorithms A nondeterministic algorithm Ν satisfies (N1) nondeterministic-time axiom, (N2) state axiom, (N3) universe axiom, (N4) isomorphism axiom, (N5) bounded-work axiom. 27
28 Bounded-Exploration Axiom A finite set T of ground terms characterizes : S S and S = T R imply R R with (S, S ) = (R, R ). bounded-nondeterministic algorithms S k 28
29 (N5) Bounded-Work Axiom Our aim: unbounded nondeterminism Our solution: bounded-work axiom amount of work is bounded in each step S S S S 29
30 (N5) Bounded-Work Axiom Our aim: unbounded nondeterminism Our solution: bounded-work axiom amount of work is bounded in each step S S S S 30
31 (N5) Bounded-Work Axiom S S 1. Decompose S into molecules 2. Each step involves only boundedly many molecules of S 31
32 Molecule of a State (next, [1], 2) molecule a() 1 x b() 2 next(x) Gurevich: Describe changes of a state by molecules. Here: Describe the complete state by molecules. 32
33 Molecule of a State a() 1 x b() 2 next(x) (a, [], 1) (b, [], 2) (next, [1], 2) (next, [2], 3) (next, [3], 4) (next, [4], 5) 33
34 Bounded-Work S S involves boundedly many molecules of S S S 34
35 Bounded-Work S S involves boundedly many molecules of S S (a, [], 1) (b, [], 2) (next, [1], 2) (next, [2], 3) (next, [3], 4) (next, [4], 5) 35
36 Bounded-Work S S involves boundedly many molecules of S S (a, [], 1) (b, [], 2) (next, [1], 2) (next, [2], 3) (next, [3], 4) (next, [4], 5) S (a, [], 1) (b, [], 3) (next, [1], 1) (next, [2], 3) (next, [3], 4) (next, [4], 5) 36
37 Bounded-Work S S involves boundedly many molecules of S Substep S S (a, [], 1) (b, [], 2) (next, [1], 2) (next, [2], 3) (next, [3], 4) (next, [4], 5) (a, [], 1) (b, [], 3) (next, [1], 1) (next, [2], 3) (next, [3], 4) (next, [4], 5) k S S is caused by the substep. 37
38 (N5) Bounded-Work Axiom A set W of bounded substeps characterizes all steps: S S is a step iff S S is caused by a substep in W. 38
39 Nondeterministic Algorithms Any object satisfying (N1) nondeterministic-time axiom, (N2) state axiom, (N3) universe axiom, (N4) isomorphism axiom, (N5) bounded-work axiom, is a nondeterministic algorithm. 39
40 The Theorem nondeterministic ASMs = nondeterministic algorithms 40
41 Applications of the Characterization
42 Reversibility Nondeterministic algorithms are reversible: For a nondeterministic algorithm Ν, Ν -1 is a nondeterministic algorithm, too. Ν Ν -1 42
43 Reversibility: Proof Ν -1 satisfies the axioms (N1) (N5): (N1) Ν -1 determines states and next-state relation -1. (N2) States are Σ-structures. (N3) -1 preserves Universes. (N4) -1 preserves Isomorphisms. (N5) -1 performs bounded substeps: (a, [], 1) (b, [], 2) (next, [1], 2) (next, [2], 3) Ν Ν -1 (a, [], 1) (b, [], 3) (next, [1], 1) (next, [2], 3) k 43
44 Reversibility of ASMs Nondeterministic algorithms are reversible. nondeterministic ASM Theorem Nondeterministic ASMs are reversible. 44
45 Reversibility: Example a() b() 3 2 a:=b a() b() 4 2 a() b() 2 2 a() b()
46 Reversibility: Example a() b() 3 2 choose x with a=b do a:=x a() b() 4 2 a() b() 2 2 a() b()
47 Linear-Speedup Nondeterministic algorithms are linear-speedup: For a nondeterministic algorithm Ν, Ν 2 is a nondeterministic algorithm, too. Ν Ν 2 47
48 Linear-Speedup: Proof Ν 2 satisfies the axioms (N1) (N5): (N1) Ν 2 determines states and next-state relation 2. (N2) States are Σ-structures. (N3) 2 preserves Universes. (N4) 2 preserves Isomorphisms. 48
49 Linear-Speedup: Proof Ν 2 satisfies the axioms (N1) (N5): (N5) 2 performs bounded substeps: (a, [], 1) (b, [], 2) (next, [1], 2) (next, [2], 3) Ν (a, [], 1) (b, [], 3) (next, [1], 1) (next, [2], 3) Ν (a, [], 1) (b, [], 3) (next, [1], 3) (next, [2], 1) k (a, [], 1) (b, [], 2) (next, [1], 2) (next, [2], 3) Ν 2 (a, [], 1) (b, [], 3) (next, [1], 3) (next, [2], 1) 2k 49
50 Linear-Speedup: Example choose x do next(x):=a speedup choose x,y do par next(x):=a next(y):=a endpar 50
51 Summary nondeterministic algorithms generalize sequential algorithms Idea: replace bounded-exploration by bounded-work nondeterministic algorithms = nondeterministic ASMs nondeterministic ASMs are reversible and linear-speedup 51
52 Future Work Check further closure properties: union and intersection of algorithms, generalize nondeterministic algorithms to distributed algorithms Thank you! 52
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