Everything you need to know about Lossy Counter Machines Ph. - - PowerPoint PPT Presentation
Everything you need to know about Lossy Counter Machines Ph. Schnoebelen http://www.lsv.ens-cachan.fr/ phs Lab. Sp ecification et V erification (LSV) CNRS & ENS de Cachan & INRIA-Saclay Journ ee DOTS / Mar. 18th, 2010
http://www.lsv.ens-cachan.fr/∼phs
ecification et V´ erification (LSV) CNRS & ENS de Cachan & INRIA-Saclay
Journ´ ee DOTS / Mar. 18th, 2010
Everything you need to know aboutLossy Counter Machines – p. 1
LCM’s are a weaker version of Minsky Counter Machines. “Weaker” because counters are not reliable, they may leak. “Weaker” implies “easier to verify”.
Everything you need to know aboutLossy Counter Machines – p. 2
LCM’s are a weaker version of Minsky Counter Machines. “Weaker” because counters are not reliable, they may leak. “Weaker” implies “easier to verify”. “Weaker” also implies “easier to reduce from”.
Everything you need to know aboutLossy Counter Machines – p. 2
LCM’s are a weaker version of Minsky Counter Machines. “Weaker” because counters are not reliable, they may leak. “Weaker” implies “easier to verify”. “Weaker” also implies “easier to reduce from”. At the moment, LCM’s are mostly used in hardness proofs. Recently LCM’s have been faithfully encoded in MITL, in XPath, in LTL+Past with 1 register, in Post’s Embedding Problem, in the product of modal logics, in alternating one-clock timed automata, etc., (wherein classic Minsky machines cannot be encoded).
Everything you need to know aboutLossy Counter Machines – p. 2
LCM’s are a weaker version of Minsky Counter Machines. “Weaker” because counters are not reliable, they may leak. “Weaker” implies “easier to verify”. “Weaker” also implies “easier to reduce from”. At the moment, LCM’s are mostly used in hardness proofs. Recently LCM’s have been faithfully encoded in MITL, in XPath, in LTL+Past with 1 register, in Post’s Embedding Problem, in the product of modal logics, in alternating one-clock timed automata, etc., (wherein classic Minsky machines cannot be encoded). Mostly use two results (by Abdulla, Jonsson, Mayr, Schnoebelen, . . . ):
1-complete).
Everything you need to know aboutLossy Counter Machines – p. 2
Everything you need to know aboutLossy Counter Machines – p. 3
Counters, aka registers, store values that are positive integers. Minsky (Counter) Machines = finite-state control + finite number of counters + increments and decrements + zero-tests. ℓ0 ℓ1 ℓ2 ℓ3 c1++ c2>0? c2-- c3=0? 1 4 c1 c2 c3 A configuration: e.g., s = (ℓ0, 1, 4, 0). All configurations: S def = Loc × NC = {s, t, . . .}. Operational semantics: s − →c t, e.g., (ℓ0, 1, 4, 0) − →c (ℓ1, 2, 4, 0). Useful notations: Pre(t) def = {s | s − → t}, Pre(X) (for X ⊆ S), Post(X), Pre∗(X), Post+(X), . . . Recall: Minsky Machines are Turing-powerful.
Everything you need to know aboutLossy Counter Machines – p. 4
Introduced by R. Mayr (2000, TCS 2003). LCM = Minsky machine with lossy counters. NB: this is a limitation, not an extra feature.
Everything you need to know aboutLossy Counter Machines – p. 5
Introduced by R. Mayr (2000, TCS 2003). LCM = Minsky machine with lossy counters. NB: this is a limitation, not an extra feature. Best seen as Minsky machines with modified operational semantics: s − →c s′ def ⇔ . . . (Minsky) s − →l s′ def ⇔ s ≥ t − →c t′ ≥ s′ for some t, t′ (LCM) Implies monotony : if s + − →l s′ then t′ + − →l s′ for any t ≥ s and s + − →l t′ for any t′ ≤ s′.
Everything you need to know aboutLossy Counter Machines – p. 5
Introduced by R. Mayr (2000, TCS 2003). LCM = Minsky machine with lossy counters. NB: this is a limitation, not an extra feature. Best seen as Minsky machines with modified operational semantics: s − →c s′ def ⇔ . . . (Minsky) s − →l s′ def ⇔ s ≥ t − →c t′ ≥ s′ for some t, t′ (LCM) Implies monotony : if s + − →l s′ then t′ + − →l s′ for any t ≥ s and s + − →l t′ for any t′ ≤ s′. Alternative definitions: s − →l s′ def ⇔ s − →c s′ ∨ s′ < s
s − →l s′ def ⇔ s
− →c t and ( s′ = t, or
s − →l s′ def ⇔ . . .
Everything you need to know aboutLossy Counter Machines – p. 5
(ℓ, a1, . . . , am) ≤ (ℓ′, b1, . . . , bm) def ⇔ ℓ = ℓ′ ∧ a1 ≤ b1 ∧ · · · ∧ am ≤ bm. (S, ≤) is a well-quasi-ordering: in any infinite sequence s0, s1, s2, . . . there is an increasing subsequence si0 ≤ si1 ≤ si2 ≤ . . . (Dickson’s Lemma).
Everything you need to know aboutLossy Counter Machines – p. 6
(ℓ, a1, . . . , am) ≤ (ℓ′, b1, . . . , bm) def ⇔ ℓ = ℓ′ ∧ a1 ≤ b1 ∧ · · · ∧ am ≤ bm. (S, ≤) is a well-quasi-ordering: in any infinite sequence s0, s1, s2, . . . there is an increasing subsequence si0 ≤ si1 ≤ si2 ≤ . . . (Dickson’s Lemma).
For LCM’s: monotony implies that Post+(X) is downward-closed and Pre+(X) is upward-closed. Hence both are semilinear (or Presburger).
Everything you need to know aboutLossy Counter Machines – p. 6
(ℓ, a1, . . . , am) ≤ (ℓ′, b1, . . . , bm) def ⇔ ℓ = ℓ′ ∧ a1 ≤ b1 ∧ · · · ∧ am ≤ bm. (S, ≤) is a well-quasi-ordering: in any infinite sequence s0, s1, s2, . . . there is an increasing subsequence si0 ≤ si1 ≤ si2 ≤ . . . (Dickson’s Lemma).
For LCM’s: monotony implies that Post+(X) is downward-closed and Pre+(X) is upward-closed. Hence both are semilinear (or Presburger).
Everything you need to know aboutLossy Counter Machines – p. 6
Everything you need to know aboutLossy Counter Machines – p. 7
Problem statement: does s ∗ − → t ? Does X ∗ − → Y for two (Presburger) sets X, Y ⊆ S? An invariant is some I ⊆ S with Post(I) ⊆ I (equivalently, with Pre(S \ I) ⊆ (S \ I)). Invariants witness non-reachability: X ∗ − → Y iff X ⊆ I and I ∩ Y = ∅ for some invariant I.
Everything you need to know aboutLossy Counter Machines – p. 8
Problem statement: does s ∗ − → t ? Does X ∗ − → Y for two (Presburger) sets X, Y ⊆ S? An invariant is some I ⊆ S with Post(I) ⊆ I (equivalently, with Pre(S \ I) ⊆ (S \ I)). Invariants witness non-reachability: X ∗ − → Y iff X ⊆ I and I ∩ Y = ∅ for some invariant I. For LCM, one can even restrict to downward-closed (hence Presburger) invariants. Such downward-closed invariants can be enumerated and checked effectively. Hence non-reachability is r.e. for LCM’s.
Everything you need to know aboutLossy Counter Machines – p. 8
Problem statement: does s ∗ − → t ? Does X ∗ − → Y for two (Presburger) sets X, Y ⊆ S? An invariant is some I ⊆ S with Post(I) ⊆ I (equivalently, with Pre(S \ I) ⊆ (S \ I)). Invariants witness non-reachability: X ∗ − → Y iff X ⊆ I and I ∩ Y = ∅ for some invariant I. For LCM, one can even restrict to downward-closed (hence Presburger) invariants. Such downward-closed invariants can be enumerated and checked effectively. Hence non-reachability is r.e. for LCM’s. Since reachability is also r.e. very generally, reachability is decidable for LCM’s.
Everything you need to know aboutLossy Counter Machines – p. 8
∃s ∈ X ∃t ∈ Y : s ∗ − → t decidable ∀s ∈ X ∃t ∈ Y : s ∗ − → t ∃s ∈ X ∀t ∈ Y : s ∗ − → t ∀s ∈ X ∀t ∈ Y : s ∗ − → t ∀t ∈ Y ∃s ∈ X : s ∗ − → t ∃t ∈ Y ∀s ∈ X : s ∗ − → t
Everything you need to know aboutLossy Counter Machines – p. 9
X ∗ − → Y ∃s ∈ X ∃t ∈ Y : s ∗ − → t ≡ ¬[Pre∗(Y ) ⊆ (S \ X)] decidable ≡ ¬[Post∗(X) ⊆ (S \ Y )] ∀s ∈ X ∃t ∈ Y : s ∗ − → t X ⊆ Pre∗(Y ) ∃s ∈ X ∀t ∈ Y : s ∗ − → t ∀s ∈ X ∀t ∈ Y : s ∗ − → t ∀t ∈ Y ∃s ∈ X : s ∗ − → t Post∗(X) ⊇ Y ∃t ∈ Y ∀s ∈ X : s ∗ − → t
Everything you need to know aboutLossy Counter Machines – p. 9
X ∗ − → Y ∃s ∈ X ∃t ∈ Y : s ∗ − → t ≡ ¬[Pre∗(Y ) ⊆ (S \ X)] decidable ≡ ¬[Post∗(X) ⊆ (S \ Y )] ∀s ∈ X ∃t ∈ Y : s ∗ − → t X ⊆ Pre∗(Y ) decidable!! ∃s ∈ X ∀t ∈ Y : s ∗ − → t undecidable, Π0
1-hard
∀s ∈ X ∀t ∈ Y : s ∗ − → t undecidable, Π0
1-complete
∀t ∈ Y ∃s ∈ X : s ∗ − → t Post∗(X) ⊇ Y undecidable, Π0
1-complete
∃t ∈ Y ∀s ∈ X : s ∗ − → t decidable!!
Everything you need to know aboutLossy Counter Machines – p. 9
X ∗ − → Y ∃s ∈ X ∃t ∈ Y : s ∗ − → t ≡ ¬[Pre∗(Y ) ⊆ (S \ X)] decidable ≡ ¬[Post∗(X) ⊆ (S \ Y )] ∀s ∈ X ∃t ∈ Y : s ∗ − → t X ⊆ Pre∗(Y ) decidable!! ∃s ∈ X ∀t ∈ Y : s ∗ − → t undecidable, Π0
1-hard
∀s ∈ X ∀t ∈ Y : s ∗ − → t undecidable, Π0
1-complete
∀t ∈ Y ∃s ∈ X : s ∗ − → t Post∗(X) ⊇ Y undecidable, Π0
1-complete
∃t ∈ Y ∀s ∈ X : s ∗ − → t decidable!!
Everything you need to know aboutLossy Counter Machines – p. 9
X ∗ − → Y ∃s ∈ X ∃t ∈ Y : s ∗ − → t ≡ ¬[Pre∗(Y ) ⊆ (S \ X)] decidable ≡ ¬[Post∗(X) ⊆ (S \ Y )] ∀s ∈ X ∃t ∈ Y : s ∗ − → t X ⊆ Pre∗(Y ) decidable!! ∃s ∈ X ∀t ∈ Y : s ∗ − → t undecidable, Π0
1-hard
∀s ∈ X ∀t ∈ Y : s ∗ − → t undecidable, Π0
1-complete
∀t ∈ Y ∃s ∈ X : s ∗ − → t Post∗(X) ⊇ Y undecidable, Π0
1-complete
∃t ∈ Y ∀s ∈ X : s ∗ − → t decidable!!
specifications [Kuˇ cera+S.2006].
Everything you need to know aboutLossy Counter Machines – p. 9
“Non-termination”: There is an infinite run sinit − → s1 − → s2 − → · · · sn − → · · · “Looping”: There is a looping run sinit − → s1 − → s2 − → · · · sn − → · · · sn+m = sn
Everything you need to know aboutLossy Counter Machines – p. 10
“Non-termination”: There is an infinite run sinit − → s1 − → s2 − → · · · sn − → · · · “Looping”: There is a looping run sinit − → s1 − → s2 − → · · · sn − → · · · sn+m = sn Looping entails non-termination. Looping is r.e. (“always”, i.e. for all sensible operational models). Non-termination is co-r.e. (“always”).
Everything you need to know aboutLossy Counter Machines – p. 10
“Non-termination”: There is an infinite run sinit − → s1 − → s2 − → · · · sn − → · · · “Looping”: There is a looping run sinit − → s1 − → s2 − → · · · sn − → · · · sn+m = sn Looping entails non-termination. Looping is r.e. (“always”, i.e. for all sensible operational models). Non-termination is co-r.e. (“always”). For LCM’s, non-termination entails looping!! The two notions coincide.
Everything you need to know aboutLossy Counter Machines – p. 10
“Non-termination”: There is an infinite run sinit − → s1 − → s2 − → · · · sn − → · · · “Looping”: There is a looping run sinit − → s1 − → s2 − → · · · sn − → · · · sn+m = sn Looping entails non-termination. Looping is r.e. (“always”, i.e. for all sensible operational models). Non-termination is co-r.e. (“always”). For LCM’s, non-termination entails looping!! The two notions coincide.
Everything you need to know aboutLossy Counter Machines – p. 10
“B¨ uchi property”: There is an infinite run sinit − → s1 − → s2 − → · · · sn − → · · · that visits ℓ infinitely many times. “Looping on ℓ”: There is a looping run sinit − → s1 − → s2 − → · · · sn = (ℓ, . . .) − → · · · sn+m = sn that visits ℓ infinitely many times.
Everything you need to know aboutLossy Counter Machines – p. 11
“B¨ uchi property”: There is an infinite run sinit − → s1 − → s2 − → · · · sn − → · · · that visits ℓ infinitely many times. “Looping on ℓ”: There is a looping run sinit − → s1 − → s2 − → · · · sn = (ℓ, . . .) − → · · · sn+m = sn that visits ℓ infinitely many times. Looping on ℓ entails the B¨ uchi property. Looping on ℓ is r.e. (“always”). For LCM’s, the B¨ uchi property entails looping on ℓ!! The two notions coincide.
Everything you need to know aboutLossy Counter Machines – p. 11
“B¨ uchi property”: There is an infinite run sinit − → s1 − → s2 − → · · · sn − → · · · that visits ℓ infinitely many times. “Looping on ℓ”: There is a looping run sinit − → s1 − → s2 − → · · · sn = (ℓ, . . .) − → · · · sn+m = sn that visits ℓ infinitely many times. Looping on ℓ entails the B¨ uchi property. Looping on ℓ is r.e. (“always”). For LCM’s, the B¨ uchi property entails looping on ℓ!! The two notions coincide. For Minsky machines, the B¨ uchi property is not co-r.e. (it is Σ1
1-complete).
Everything you need to know aboutLossy Counter Machines – p. 11
“B¨ uchi property”: There is an infinite run sinit − → s1 − → s2 − → · · · sn − → · · · that visits ℓ infinitely many times. “Looping on ℓ”: There is a looping run sinit − → s1 − → s2 − → · · · sn = (ℓ, . . .) − → · · · sn+m = sn that visits ℓ infinitely many times. Looping on ℓ entails the B¨ uchi property. Looping on ℓ is r.e. (“always”). For LCM’s, the B¨ uchi property entails looping on ℓ!! The two notions coincide. For Minsky machines, the B¨ uchi property is not co-r.e. (it is Σ1
1-complete).
Liveness properties are undecidable for LCM’s. The B¨ uchi property is Σ0
1-complete for LCM’s. s |
= EGFX is undecidable. s | = AXUY is decidable, but one cannot compute Sat(AXUY ), or model-check TL(AF, . . .).
Everything you need to know aboutLossy Counter Machines – p. 11
Games: For LCM’s, game-theoretical questions are not very interesting. But see [Baier+Bertrand+S.2006].
Everything you need to know aboutLossy Counter Machines – p. 12
Games: For LCM’s, game-theoretical questions are not very interesting. But see [Baier+Bertrand+S.2006]. Finiteness Finiteness (whether Post∗(s0) is finite) and regularity (whether Traces(s0) is regular) are undecidable.
Everything you need to know aboutLossy Counter Machines – p. 12
Games: For LCM’s, game-theoretical questions are not very interesting. But see [Baier+Bertrand+S.2006]. Finiteness Finiteness (whether Post∗(s0) is finite) and regularity (whether Traces(s0) is regular) are undecidable. Equivalences All sensible comparison relations are undecidable between LCM’s, already for lossy VASS’s (S. 2001, after Janˇ car). (Recall that many behavioural relations are decidable between LCM’s and finite systems.)
Everything you need to know aboutLossy Counter Machines – p. 12
Everything you need to know aboutLossy Counter Machines – p. 13
ℓ0 ℓ1 ℓ2 ℓ3 c1++ c2>0? c2-- c3=0? ℓbankrupt B>0? B-- B++ B=0? 1 4 95 c1 c2 c3 B
Everything you need to know aboutLossy Counter Machines – p. 14
ℓ0 ℓ1 ℓ2 ℓ3 c1++ c2>0? c2-- c3=0? ℓbankrupt B>0? B-- B++ B=0? 1 4 95 c1 c2 c3 B
Sum of counters remain stable (in classic, non-lossy, semantics. It can decrease in lossy semantics).
Everything you need to know aboutLossy Counter Machines – p. 14
ℓ0 ℓ1 ℓ2 ℓ3 c1++ c2>0? c2-- c3=0? ℓbankrupt B>0? B-- B++ B=0? 1 4 95 c1 c2 c3 B
Sum of counters remain stable (in classic, non-lossy, semantics. It can decrease in lossy semantics). Machine becomes finite-state (for given budget) but standard behaviour is preserved with “large enough” budget.
Everything you need to know aboutLossy Counter Machines – p. 14
ℓ0
ℓ1
B++ c1 c2 c3 B “reset counters” B ← c1 + · · · + cm, c1 ← 0, . . . , cm ← 0
Everything you need to know aboutLossy Counter Machines – p. 15
ℓ0
ℓ1
B++ c1 c2 c3 B “reset counters” B ← c1 + · · · + cm, c1 ← 0, . . . , cm ← 0
M exits (classically)
Everything you need to know aboutLossy Counter Machines – p. 15
ℓ0
ℓ1
B++ c1 c2 c3 B “reset counters” B ← c1 + · · · + cm, c1 ← 0, . . . , cm ← 0
M exits (classically) iff Mon budget exits for some budget (classically)
Everything you need to know aboutLossy Counter Machines – p. 15
ℓ0
ℓ1
B++ c1 c2 c3 B “reset counters” B ← c1 + · · · + cm, c1 ← 0, . . . , cm ← 0
M exits (classically) iff Mon budget exits for some budget (classically) iff M ′ has a loop on ℓ1 (under lossy semantics).
Everything you need to know aboutLossy Counter Machines – p. 15
ℓ0
ℓ1
B++ c1 c2 c3 B “reset counters” B ← c1 + · · · + cm, c1 ← 0, . . . , cm ← 0
M exits (classically) iff Mon budget exits for some budget (classically) iff M ′ has a loop on ℓ1 (under lossy semantics).
uchi property is undecidable (Σ0
1-complete).
Everything you need to know aboutLossy Counter Machines – p. 15
Assume M is deterministic (classically).
ℓ0
ℓ1
ℓbankrupt B++ c1 c2 c3 B “reset counters” B ← c1 + · · · + cm, c1 ← 0, . . . , cm ← 0
Everything you need to know aboutLossy Counter Machines – p. 16
Assume M is deterministic (classically).
ℓ0
ℓ1
ℓbankrupt B++ c1 c2 c3 B “reset counters” B ← c1 + · · · + cm, c1 ← 0, . . . , cm ← 0
M is unbounded (classically)
Everything you need to know aboutLossy Counter Machines – p. 16
Assume M is deterministic (classically).
ℓ0
ℓ1
ℓbankrupt B++ c1 c2 c3 B “reset counters” B ← c1 + · · · + cm, c1 ← 0, . . . , cm ← 0
M is unbounded (classically) iff Mon budget bankrupts on all budgets (classically)
Everything you need to know aboutLossy Counter Machines – p. 16
Assume M is deterministic (classically).
ℓ0
ℓ1
ℓbankrupt B++ c1 c2 c3 B “reset counters” B ← c1 + · · · + cm, c1 ← 0, . . . , cm ← 0
M is unbounded (classically) iff Mon budget bankrupts on all budgets (classically) iff M ′ is unbounded (under lossy semantics).
Everything you need to know aboutLossy Counter Machines – p. 16
Assume M is deterministic (classically).
ℓ0
ℓ1
ℓbankrupt B++ c1 c2 c3 B “reset counters” B ← c1 + · · · + cm, c1 ← 0, . . . , cm ← 0
M is unbounded (classically) iff Mon budget bankrupts on all budgets (classically) iff M ′ is unbounded (under lossy semantics).
1-complete).
Everything you need to know aboutLossy Counter Machines – p. 16
Assume M is deterministic (classically).
ℓ0
ℓ1
ℓbankrupt B++ c1 c2 c3 B “reset counters” B ← c1 + · · · + cm, c1 ← 0, . . . , cm ← 0
M is unbounded (classically) iff Mon budget bankrupts on all budgets (classically) iff M ′ is unbounded (under lossy semantics).
1-complete).
Everything you need to know aboutLossy Counter Machines – p. 16
Everything you need to know aboutLossy Counter Machines – p. 17
For m ∈ N, Fm : N → N is defined by: F0(n) def = n + 1, (D1) Fm+1(n) def = F n+1
m
(n) =
n+1 times
z }| { Fm(Fm(. . . Fm(n) . . .)), (D2)
Everything you need to know aboutLossy Counter Machines – p. 18
For m ∈ N, Fm : N → N is defined by: F0(n) def = n + 1, (D1) Fm+1(n) def = F n+1
m
(n) =
n+1 times
z }| { Fm(Fm(. . . Fm(n) . . .)), (D2) Yields F1(n) = 2n + 1 F2(n) = (n + 1)2n+1 − 1 and F3(n) > 22 . . .
2
) n times. Further ensures Fm(n) > n, Fm(n + 1) > Fm(n) and Fm+1(n) ≥ Fm(n).
Everything you need to know aboutLossy Counter Machines – p. 18
For m ∈ N, Fm : N → N is defined by: F0(n) def = n + 1, (D1) Fm+1(n) def = F n+1
m
(n) =
n+1 times
z }| { Fm(Fm(. . . Fm(n) . . .)), (D2) Yields F1(n) = 2n + 1 F2(n) = (n + 1)2n+1 − 1 and F3(n) > 22 . . .
2
) n times. Further ensures Fm(n) > n, Fm(n + 1) > Fm(n) and Fm+1(n) ≥ Fm(n). Every Fm is primitive-recursive. Every primitive-recursive function is dominated by some Fm. Fω(n) def = Fn(n), a variant of Ackermann’s function, is not primitive-recursive.
Everything you need to know aboutLossy Counter Machines – p. 18
For m ∈ N, Fm : N → N is defined by: F0(n) def = n + 1, (D1) Fm+1(n) def = F n+1
m
(n) =
n+1 times
z }| { Fm(Fm(. . . Fm(n) . . .)), (D2) Yields F1(n) = 2n + 1 F2(n) = (n + 1)2n+1 − 1 and F3(n) > 22 . . .
2
) n times. Further ensures Fm(n) > n, Fm(n + 1) > Fm(n) and Fm+1(n) ≥ Fm(n). Every Fm is primitive-recursive. Every primitive-recursive function is dominated by some Fm. Fω(n) def = Fn(n), a variant of Ackermann’s function, is not primitive-recursive. LCM’s with m + 2 counters can compute Fm exactly. Hence verifying LCM’s cannot be bounded by a primitive-recursive function.
Everything you need to know aboutLossy Counter Machines – p. 18
For a = (am, . . . , a0) ∈ Nm+1, let Fa def = F am
m
0 .
In particular Fm = F(1,0,0,...,0).
Everything you need to know aboutLossy Counter Machines – p. 19
For a = (am, . . . , a0) ∈ Nm+1, let Fa def = F am
m
0 .
In particular Fm = F(1,0,0,...,0). F(am,...,a0)(n) = F(am,...,a0−1)(n + 1) if a0 > 0 (D1) F(am,...,ak,0, . . . , 0 | {z }
k zeroes
)(n) = F(am,...,ak−1,n+1,0,...,0)(n)
if k > 0 and ak > 0 (D2) F(0,...,0)(n) = n (D0)
Everything you need to know aboutLossy Counter Machines – p. 19
For a = (am, . . . , a0) ∈ Nm+1, let Fa def = F am
m
0 .
In particular Fm = F(1,0,0,...,0). F(am,...,a0)(n) = F(am,...,a0−1)(n + 1) if a0 > 0 (D1) F(am,...,ak,0, . . . , 0 | {z }
k zeroes
)(n) = F(am,...,ak−1,n+1,0,...,0)(n)
if k > 0 and ak > 0 (D2) F(0,...,0)(n) = n (D0) This version of the Fm’s can be implemented as a while loop, with m + 2 integer variables.
Everything you need to know aboutLossy Counter Machines – p. 19
ℓ0
/* eval Fa one step */ if a0> 0 then a0--; n++; elif a1> 0 then a1--; a0:=n+1; . . . elif am> 0 then am--; am−1:=n+1; else /* a == 0: do nothing */ . . . n a0 a1 am F(am,...,a0)(n) = F(am,...,a0−1)(n + 1) if a0 > 0 (D1) F(am,...,ak,0, . . . , 0 | {z }
k zeroes
)(n) = F(am,...,ak−1,n+1,0,...,0)(n)
if k > 0 and ak > 0 (D2)
Everything you need to know aboutLossy Counter Machines – p. 20
ℓ0
/* eval Fa one step */ if a0> 0 then a0--; n++; elif a1> 0 then a1--; a0:=n+1; . . . elif am> 0 then am--; am−1:=n+1; else /* a == 0: do nothing */ . . . n a0 a1 am Classic step: (ℓ0, n, a) eval − − →c (ℓ0, n′, a′) implies Fa(n) = Fa′(n′) and a >lexico a′. Reliable evaluation: (ℓ0, n, a) eval − − →c (ℓ0, n′, 0) iff n′ = Fa(n). Lossy semantics: (ℓ0, n, a) eval − − →l (ℓ0, n′, 0) iff n′ ≤ Fa(n).
Everything you need to know aboutLossy Counter Machines – p. 20
. . . n a0 a1 am ℓ0 n>0? n--;a0++ F(am,...,a0)(n) = F(am,...,a0−1)(n + 1) if a0 > 0 (D1) F(am,...,ak,0, . . . , 0 | {z }
k zeroes
)(n) = F(am,...,ak−1,n+1,0,...,0)(n)
if k > 0 and ak > 0 (D2)
Everything you need to know aboutLossy Counter Machines – p. 21
. . . n a0 a1 am ℓ0 n>0? n--;a0++ a0=n + 1? a0:=0;a1++ F(am,...,a0)(n) = F(am,...,a0−1)(n + 1) if a0 > 0 (D1) F(am,...,ak,0, . . . , 0 | {z }
k zeroes
)(n) = F(am,...,ak−1,n+1,0,...,0)(n)
if k > 0 and ak > 0 (D2)
Everything you need to know aboutLossy Counter Machines – p. 21
. . . n a0 a1 am ℓ0 n>0? n--;a0++ a0=n + 1? a0:=0;a1++ (a0=0 ∧ a1=n + 1)? a1:=0;a2++ F(am,...,a0)(n) = F(am,...,a0−1)(n + 1) if a0 > 0 (D1) F(am,...,ak,0, . . . , 0 | {z }
k zeroes
)(n) = F(am,...,ak−1,n+1,0,...,0)(n)
if k > 0 and ak > 0 (D2)
Everything you need to know aboutLossy Counter Machines – p. 21
. . . n a0 a1 am ℓ0 n>0? n--;a0++ a0=n + 1? a0:=0;a1++ (a0=0 ∧ a1=n + 1)? a1:=0;a2++ “a0=a1= · · · =am−2 = 0 ∧ am−1=n + 1 ” ? am−1:=0;am++ F(am,...,a0)(n) = F(am,...,a0−1)(n + 1) if a0 > 0 (D1) F(am,...,ak,0, . . . , 0 | {z }
k zeroes
)(n) = F(am,...,ak−1,n+1,0,...,0)(n)
if k > 0 and ak > 0 (D2)
Everything you need to know aboutLossy Counter Machines – p. 21
. . . n a0 a1 am ℓ0 n>0? n--;a0++ a0=n + 1? a0:=0;a1++ (a0=0 ∧ a1=n + 1)? a1:=0;a2++ “a0=a1= · · · =am−2 = 0 ∧ am−1=n + 1 ” ? am−1:=0;am++ Classic step: (ℓ0, n, a) back − − →c (ℓ0, n′, a′) iff (ℓ0, n′, a′) eval − − →c (ℓ0, n, a) Reliable evaluation: (ℓ0, n, a) eval+back − − − − − →c (ℓ0, n′, a′) iff Fa(n) = Fa′(n′). Lossy semantics: (ℓ0, n, a) eval+back − − − − − →l (ℓ0, n′, a′) iff Fa(n) ≥ Fa′(n′).
Everything you need to know aboutLossy Counter Machines – p. 21
ℓi ℓf . . . 1 n a0 a1 am B c1 c2 c3 ℓ0 eval+back
Everything you need to know aboutLossy Counter Machines – p. 22
ℓi ℓf . . . 1 n a0 a1 am B c1 c2 c3 ℓ0 eval+back ℓ′ eval+back
Everything you need to know aboutLossy Counter Machines – p. 22
ℓi ℓf . . . 1 n a0 a1 am B c1 c2 c3 ℓ0 eval+back ℓ′ eval+back
(ℓ0, 0, . . . , 0, 1, 0, 0, 0)
∗
− →l (ℓ′
0, 0, . . . , 0, 1, 0, 0, 0)
in M Ack ⊗ M on budget (lossy) iff (ℓ0, 0, . . . , 0, 1, 0, 0, 0)
∗
− →c (ℓ0, 0, . . . , 0, 1, 0, 0, 0) in M Ack ⊗ M on budget (classic) iff (ℓi, Fm(0), 0, 0, 0)
∗
− →c (ℓf, Fm(0), 0, 0, 0) in M on budget (classic)
Everything you need to know aboutLossy Counter Machines – p. 22
ℓi ℓf . . . 1 n a0 a1 am B c1 c2 c3 ℓ0 eval+back ℓ′ eval+back
(ℓ0, 0, . . . , 0, 1, 0, 0, 0)
∗
− →l (ℓ′
0, 0, . . . , 0, 1, 0, 0, 0)
in M Ack ⊗ M on budget (lossy) iff (ℓ0, 0, . . . , 0, 1, 0, 0, 0)
∗
− →c (ℓ0, 0, . . . , 0, 1, 0, 0, 0) in M Ack ⊗ M on budget (classic) iff (ℓi, Fm(0), 0, 0, 0)
∗
− →c (ℓf, Fm(0), 0, 0, 0) in M on budget (classic)
Everything you need to know aboutLossy Counter Machines – p. 22
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