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 Everything you need to know about Lossy Counter Machines – p. 1

Lossy Counter Machines (LCM) in a Nutshell 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 about Lossy Counter Machines – p. 2

Lossy Counter Machines (LCM) in a Nutshell 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 about Lossy Counter Machines – p. 2

Lossy Counter Machines (LCM) in a Nutshell 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 about Lossy Counter Machines – p. 2

Lossy Counter Machines (LCM) in a Nutshell 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. Reachability and termination are decidable for LCMs but not with primitive-recursive complexity. 2. Finiteness and liveness are undecidable ( Σ 0 1 -complete). Everything you need to know about Lossy Counter Machines – p. 2

Basic notions Everything you need to know about Lossy Counter Machines – p. 3

Counters and Counter Machines Counters, aka registers, store values that are positive integers. Minsky (Counter) Machines = finite-state control + finite number of counters + increments and decrements + zero-tests . c 1 1 c 1 ++ c 2 >0? c 2 -- c 3 =0? c 2 ℓ 0 ℓ 1 ℓ 2 ℓ 3 4 c 3 0 A configuration: e.g., s = ( ℓ 0 , 1 , 4 , 0) . All configurations: S def = Loc × N C = { s, t, . . . } . Operational semantics: s − → c t , e.g., ( ℓ 0 , 1 , 4 , 0) − → c ( ℓ 1 , 2 , 4 , 0) . Useful notations: Pre ( t ) def → t } , Pre ( X ) (for X ⊆ S ), Post ( X ) , Pre ∗ ( X ) , Post + ( X ) , . . . = { s | s − Recall: Minsky Machines are Turing-powerful. Everything you need to know about Lossy Counter Machines – p. 4

Lossy Counter Machines, aka LCM’s 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 about Lossy Counter Machines – p. 5

Lossy Counter Machines, aka LCM’s 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: → c s ′ def (Minsky) s − ⇔ . . . → l s ′ def → c t ′ ≥ s ′ for some t, t ′ s − ⇔ s ≥ 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 about Lossy Counter Machines – p. 5

Lossy Counter Machines, aka LCM’s 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: → c s ′ def (Minsky) s − ⇔ . . . → l s ′ def → c t ′ ≥ s ′ for some t, t ′ (LCM) s − ⇔ s ≥ t − Implies monotony : if s + → l s ′ then t ′ + → l s ′ for any t ≥ s and s + → l t ′ for any t ′ ≤ s ′ . − − − → l s ′ def → c s ′ ∨ s ′ < s Alternative definitions: s − ⇔ s − ( s ′ = t, or → l s ′ def op or: s − ⇔ s → c t and − op = c i ++ ∧ s ′ = t − c i . → l s ′ def or: s − ⇔ . . . Everything you need to know about Lossy Counter Machines – p. 5

A closer look at comparing configurations ( ℓ, a 1 , . . . , a m ) ≤ ( ℓ ′ , b 1 , . . . , b m ) def ⇔ ℓ = ℓ ′ ∧ a 1 ≤ b 1 ∧ · · · ∧ a m ≤ b m . ( S, ≤ ) is a well-quasi-ordering: in any infinite sequence s 0 , s 1 , s 2 , . . . there is an increasing subsequence s i 0 ≤ s i 1 ≤ s i 2 ≤ . . . (Dickson’s Lemma). Coro. Any subset of S has finitely many minimal elements. Coro. 1. Any upward-closed subset of S is semilinear (or Presburger). 2. Idem for downward-closed subsets of S . Everything you need to know about Lossy Counter Machines – p. 6

A closer look at comparing configurations ( ℓ, a 1 , . . . , a m ) ≤ ( ℓ ′ , b 1 , . . . , b m ) def ⇔ ℓ = ℓ ′ ∧ a 1 ≤ b 1 ∧ · · · ∧ a m ≤ b m . ( S, ≤ ) is a well-quasi-ordering: in any infinite sequence s 0 , s 1 , s 2 , . . . there is an increasing subsequence s i 0 ≤ s i 1 ≤ s i 2 ≤ . . . (Dickson’s Lemma). Coro. Any subset of S has finitely many minimal elements. Coro. 1. Any upward-closed subset of S is semilinear (or Presburger). 2. Idem for downward-closed subsets of S . 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 about Lossy Counter Machines – p. 6

A closer look at comparing configurations ( ℓ, a 1 , . . . , a m ) ≤ ( ℓ ′ , b 1 , . . . , b m ) def ⇔ ℓ = ℓ ′ ∧ a 1 ≤ b 1 ∧ · · · ∧ a m ≤ b m . ( S, ≤ ) is a well-quasi-ordering: in any infinite sequence s 0 , s 1 , s 2 , . . . there is an increasing subsequence s i 0 ≤ s i 1 ≤ s i 2 ≤ . . . (Dickson’s Lemma). Coro. Any subset of S has finitely many minimal elements. Coro. 1. Any upward-closed subset of S is semilinear (or Presburger). 2. Idem for downward-closed subsets of S . For LCM’s: monotony implies that Post + ( X ) is downward-closed and Pre + ( X ) is upward-closed. Hence both are semilinear (or Presburger). Note. All one needs for decidability of LCM’s is on this slide. Everything you need to know about Lossy Counter Machines – p. 6

A quick survey Everything you need to know about Lossy Counter Machines – p. 7

Reachability 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 about Lossy Counter Machines – p. 8

Reachability 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 about Lossy Counter Machines – p. 8

Reachability 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 about Lossy Counter Machines – p. 8

Reachability Logic ∃ 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 about Lossy Counter Machines – p. 9

Reachability Logic 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 ∗ Post ∗ ( X ) ⊇ Y − → t ∃ t ∈ Y ∀ s ∈ X : s ∗ − → t Everything you need to know about Lossy Counter Machines – p. 9