Regular separability of languages of well-structured transition - - PowerPoint PPT Presentation
Regular separability of languages of well-structured transition systems Roland Mayer Wojciech Czerwi ski Sebastian Muskalla S awomir Lasota K Narayan Kumar Prakash Saivasan University of Warsaw CMI Chennai TU Braunschweig Infinity
Wojciech Czerwiński Sławomir Lasota
languages of
University of Warsaw
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TU Braunschweig
Roland Mayer Sebastian Muskalla Prakash Saivasan
CMI Chennai
K Narayan Kumar
Wojciech Czerwiński Sławomir Lasota
languages of
University of Warsaw
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TU Braunschweig
Roland Mayer Sebastian Muskalla Prakash Saivasan
CMI Chennai
K Narayan Kumar
[Mukund, Kumar, Radhakrishnan, Sohoni ’98]
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Fix a class of languages C
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Input: two (disjoint) languages L, K from C Fix a class of languages C L K
R
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Input: two (disjoint) languages L, K from C Question: are these two languages separated by a regular language? Fix a class of languages C L K
R
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Input: two (disjoint) languages L, K from C Question: are these two languages separated by a regular language? I.e., is there a regular language R with L ⊆ R and R ∩ K = ∅? Fix a class of languages C L K
R
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Input: two (disjoint) languages L, K from C Question: are these two languages separated by a regular language? I.e., is there a regular language R with L ⊆ R and R ∩ K = ∅? Fix a class of languages C L K Symmetric in L, K
R
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Input: two (disjoint) languages L, K from C Question: are these two languages separated by a regular language? I.e., is there a regular language R with L ⊆ R and R ∩ K = ∅? Fix a class of languages C L K Symmetric in L, K
Parametric in C
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K R L classify a word from L ∪ K into L or K separator as a classifier:
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K R L classify a word from L ∪ K into L or K separator as a classifier:
Bad
R
System
separator proves absence of undesirable behavior language-theoretic verification:
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K R L classify a word from L ∪ K into L or K separator as a classifier:
Bad
R
System
separator proves absence of undesirable behavior language-theoretic verification: K L separator as a recognizer: recognize L inside K
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K R L classify a word from L ∪ K into L or K separator as a classifier:
Bad
R
System
separator proves absence of undesirable behavior language-theoretic verification: K L separator as a recognizer: R recognize L inside K
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R
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L K
R
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L K
R
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L K
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for regular-separability L K
R
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for regular-separability
L K
R
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for regular-separability
L K
R
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WSTS language WSTS language
R
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WSTS language WSTS language
Theorem: Every two disjoint WSTS languages are regular-separable,
R
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WSTS language WSTS language
Theorem: Every two disjoint WSTS languages are regular-separable, under some mild assumptions.
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U/ DWSTS: well-structured transition system
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U/ DWSTS: well-structured transition system
transition system
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U/ DWSTS: well-structured transition system
transition system
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U/ DWSTS: well-structured transition system
transition system
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U/ DWSTS: well-structured transition system
transition system
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U/ DWSTS: well-structured transition system
transition system
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U/ DWSTS: well-structured transition system
transition system
the language
a
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U/ DWSTS: well-structured transition system
transition system
the language
a
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U/ DWSTS: well-structured transition system
transition system
well-structured
the language
a
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U/ DWSTS: well-structured transition system
transition system
well-structured
the language
a
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U/ DWSTS: well-structured transition system
transition system
well-structured
upward-compatible
the language
a
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U/ DWSTS: well-structured transition system
transition system
well-structured
upward-compatible
the language
a
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U/ DWSTS: well-structured transition system
transition system
well-structured
upward-compatible
the language
a
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U/ DWSTS: well-structured transition system
transition system
well-structured
upward-compatible
the language
a
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U/ DWSTS: well-structured transition system
transition system
well-structured
upward-compatible
the language
≼ a ∀
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U/ DWSTS: well-structured transition system
transition system
well-structured
upward-compatible
the language
≼ a ≼
∀ ∃
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U/ DWSTS: well-structured transition system
transition system
well-structured
upward-compatible
downward-compatible
the language
≼ a ≼
∀ ∃
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Def: a quasi order is a WQO if it has:
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Examples:
Def: a quasi order is a WQO if it has:
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Examples:
Def: a quasi order is a WQO if it has:
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Examples:
Def: a quasi order is a WQO if it has:
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Examples:
Def: a quasi order is a WQO if it has:
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Examples:
Def: a quasi order is a WQO if it has:
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Examples:
Def: a quasi order is a WQO if it has: Def: a quasi order is an 𝜕2-WQO if its downward closed subsets (ordered by inclusion) are a WQO
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9
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≼ a ≼
∃ upward compatibility:
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≼ a ≼
∃ upward compatibility:
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UWSTS language
DWSTS languages
UWSTS language
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UWSTS language
DWSTS languages
Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is finitely-branching.
UWSTS language
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UWSTS language
DWSTS languages
Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is finitely-branching. Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic.
UWSTS language
R
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UWSTS language
DWSTS languages
Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is finitely-branching. Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic.
every state has finitely many a-successors UWSTS language
R
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UWSTS language
DWSTS languages
Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is finitely-branching. Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic.
every state has finitely many a-successors every state has exactly one a-successor UWSTS language
R
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UWSTS language
DWSTS languages
Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is finitely-branching. Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic.
every state has finitely many a-successors every state has exactly one a-successor UWSTS language
deterministic.
R
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UWSTS language
DWSTS languages
Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is finitely-branching. Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic.
every state has finitely many a-successors every state has exactly one a-successor UWSTS language
Corollary: Every two disjoint 𝜕2-UWSTS or 𝜕2-DWSTS languages are regular-separable. deterministic.
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Corollary: Every two disjoint 𝜕2-UWSTS or 𝜕2-DWSTS languages are regular-separable.
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Corollary: Every two disjoint 𝜕2-UWSTS or 𝜕2-DWSTS languages are regular-separable. Corollary: Every two disjoint languages of
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Corollary: Every two disjoint 𝜕2-UWSTS or 𝜕2-DWSTS languages are regular-separable. Corollary: Every two disjoint languages of
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Corollary: Every two disjoint 𝜕2-UWSTS or 𝜕2-DWSTS languages are regular-separable. Corollary: Every two disjoint languages of
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Corollary: Every two disjoint 𝜕2-UWSTS or 𝜕2-DWSTS languages are regular-separable. Corollary: Every two disjoint languages of
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Corollary: Every two disjoint 𝜕2-UWSTS or 𝜕2-DWSTS languages are regular-separable. Corollary: Every two disjoint languages of
are regular-separable. Alike for gainy FIFO/counter automata.
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Corollary: Every two disjoint 𝜕2-UWSTS or 𝜕2-DWSTS languages are regular-separable. Corollary: Every two disjoint languages of
are regular-separable. Alike for gainy FIFO/counter automata. Corollary: No subclass of
U/ DWSTS languages closed under complement
beyond regular languages.
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Proof: Main ingredients Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is deterministic. Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic. R
UWSTS language UWSTS language
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Proof: Main ingredients
U/ DWSTS
Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is deterministic. Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic. R
UWSTS language UWSTS language
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Proof: Main ingredients
U/ DWSTS
Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is deterministic. Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic. R
UWSTS language UWSTS language
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Def: An inductive invariant in a UTS is a subset X ⊆ S of states s.t.
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Def: An inductive invariant in a UTS is a subset X ⊆ S of states s.t.
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Def: An inductive invariant in a UTS is a subset X ⊆ S of states s.t.
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Def: An inductive invariant in a UTS is a subset X ⊆ S of states s.t.
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Def: An inductive invariant in a UTS is a subset X ⊆ S of states s.t.
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Def: An inductive invariant in a UTS is a subset X ⊆ S of states s.t. Fact: Every empty-language UTS admits an inductive invariant, e.g.,
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Def: An inductive invariant in a UTS is a subset X ⊆ S of states s.t.
Fact: Every empty-language UTS admits an inductive invariant, e.g.,
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Def: An inductive invariant in a UTS is a subset X ⊆ S of states s.t.
Fact: Every empty-language UTS admits an inductive invariant, e.g., In particular, the synchronized product of two disjoint UTS admits one.
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Def: An inductive invariant in a UTS is a subset X ⊆ S of states s.t.
Fact: Every empty-language UTS admits an inductive invariant, e.g., In particular, the synchronized product of two disjoint UTS admits one. We will need finitary inductive invariants Q↓, namely Q finite.
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Key Lemma: If the synchronized product W×V of two UTS, V deterministic, admits an inductive invariant Q↓, then W and V are separated by an automaton with state space Q.
Proof: We define automaton A to overapproximate W×V wrt ≼.
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Key Lemma: If the synchronized product W×V of two UTS, V deterministic, admits an inductive invariant Q↓, then W and V are separated by an automaton with state space Q.
I ⊆ Q↓
≼
Proof: We define automaton A to overapproximate W×V wrt ≼. Final states of A: the W-component is final in W.
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Key Lemma: If the synchronized product W×V of two UTS, V deterministic, admits an inductive invariant Q↓, then W and V are separated by an automaton with state space Q.
I ⊆ Q↓
≼
Proof: We define automaton A to overapproximate W×V wrt ≼. Final states of A: the W-component is final in W. Thus L(W) ⊆ L(A).
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Key Lemma: If the synchronized product W×V of two UTS, V deterministic, admits an inductive invariant Q↓, then W and V are separated by an automaton with state space Q.
I ⊆ Q↓
≼
Proof: We define automaton A to overapproximate W×V wrt ≼. Final states of A: the W-component is final in W. Thus L(W) ⊆ L(A). Using determinacy of V, the V-component of every state reached by A along some word
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Key Lemma: If the synchronized product W×V of two UTS, V deterministic, admits an inductive invariant Q↓, then W and V are separated by an automaton with state space Q.
I ⊆ Q↓
≼
Proof: We define automaton A to overapproximate W×V wrt ≼. Final states of A: the W-component is final in W. Thus L(W) ⊆ L(A). Using determinacy of V, the V-component of every state reached by A along some word ≼-dominates the unique state reached by V along this word.
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Key Lemma: If the synchronized product W×V of two UTS, V deterministic, admits an inductive invariant Q↓, then W and V are separated by an automaton with state space Q.
I ⊆ Q↓
≼
Proof: We define automaton A to overapproximate W×V wrt ≼. Final states of A: the W-component is final in W. Thus L(W) ⊆ L(A). Using determinacy of V, the V-component of every state reached by A along some word ≼-dominates the unique state reached by V along this word. Thus L(A) ∩ L(V) = ∅. ☐
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Key Lemma: If the synchronized product W×V of two UTS, V deterministic, admits an inductive invariant Q↓, then W and V are separated by an automaton with state space Q.
I ⊆ Q↓
≼
Proof: We define automaton A to overapproximate W×V wrt ≼. Final states of A: the W-component is final in W. Thus L(W) ⊆ L(A). Using determinacy of V, the V-component of every state reached by A along some word ≼-dominates the unique state reached by V along this word. Thus L(A) ∩ L(V) = ∅. ☐
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Key Lemma: If the synchronized product W×V of two UTS, V deterministic, admits an inductive invariant Q↓, then W and V are separated by an automaton with state space Q.
It remains to demonstrate existence of a finite Q.
I ⊆ Q↓
≼
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Theorem: Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic.
Key Lemma: If the synchronized product W×V of two UTS, V deterministic, admits an inductive invariant Q↓, then W and V are separated by an automaton with state space Q.
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Theorem: Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic.
Proof: Apply Key Lemma to inverses of DWSTS which are UTS. Key Lemma: If the synchronized product W×V of two UTS, V deterministic, admits an inductive invariant Q↓, then W and V are separated by an automaton with state space Q.
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Theorem: Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic.
Proof: Apply Key Lemma to inverses of DWSTS which are UTS. Finite min of upward closed set inverses to finite max of downward closed sets. ☐ Key Lemma: If the synchronized product W×V of two UTS, V deterministic, admits an inductive invariant Q↓, then W and V are separated by an automaton with state space Q.
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Recall: We need a finitary inductive invariant Q↓, for Q finite.
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Def: An ideal in a quasi-order is any downward closed (3, 𝜕, 4) directed subset thereof. Recall: We need a finitary inductive invariant Q↓, for Q finite.
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Def: An ideal in a quasi-order is any downward closed (3, 𝜕, 4) directed subset thereof. Finite ideal decomposition: Every downward closed subset of a WQO is a finite union of ideals. Recall: We need a finitary inductive invariant Q↓, for Q finite.
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Def: An ideal in a quasi-order is any downward closed (3, 𝜕, 4) directed subset thereof. Finite ideal decomposition: Every downward closed subset of a WQO is a finite union of ideals. Ideal completion: extend quasi-order by all its ideals. Recall: We need a finitary inductive invariant Q↓, for Q finite.
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Def: An ideal in a quasi-order is any downward closed (3, 𝜕, 4) directed subset thereof. Finite ideal decomposition: Every downward closed subset of a WQO is a finite union of ideals. Ideal completion: extend quasi-order by all its ideals. Fact 1: Ideal completion of a (deterministic) UWSTS is a language-equivalent (deterministic) UTS. Recall: We need a finitary inductive invariant Q↓, for Q finite.
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Def: An ideal in a quasi-order is any downward closed (3, 𝜕, 4) directed subset thereof. Finite ideal decomposition: Every downward closed subset of a WQO is a finite union of ideals. Ideal completion: extend quasi-order by all its ideals. Fact 1: Ideal completion of a (deterministic) UWSTS is a language-equivalent (deterministic) UTS. Recall: We need a finitary inductive invariant Q↓, for Q finite. Fact 2: Ideal completion commutes with synchronized product.
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Def: An ideal in a quasi-order is any downward closed (3, 𝜕, 4) directed subset thereof. Finite ideal decomposition: Every downward closed subset of a WQO is a finite union of ideals. Ideal completion: extend quasi-order by all its ideals. Fact 1: Ideal completion of a (deterministic) UWSTS is a language-equivalent (deterministic) UTS. Recall: We need a finitary inductive invariant Q↓, for Q finite. Fact 3: For every inductive invariant in a UWSTS, its finite ideal decomposition is a finitary inductive invariant in the ideal completion of this UWSTS. Fact 2: Ideal completion commutes with synchronized product.
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Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is deterministic.
Key Lemma: If the synchronized product W×V of two UTS, V deterministic, admits an inductive invariant Q↓, then W and V are separated by an automaton with state space Q.
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Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is deterministic.
Proof: Apply Key Lemma to the ideal completions of the UWSTS. Key Lemma: If the synchronized product W×V of two UTS, V deterministic, admits an inductive invariant Q↓, then W and V are separated by an automaton with state space Q.
Fact 1: Ideal completion of a (deterministic) UWSTS is a language-equivalent (deterministic) UTS.
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Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is deterministic.
Proof: Apply Key Lemma to the ideal completions of the UWSTS. Synchronized product of idea completions, isomorphic to Key Lemma: If the synchronized product W×V of two UTS, V deterministic, admits an inductive invariant Q↓, then W and V are separated by an automaton with state space Q.
Fact 1: Ideal completion of a (deterministic) UWSTS is a language-equivalent (deterministic) UTS.
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Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is deterministic.
Proof: Apply Key Lemma to the ideal completions of the UWSTS. Synchronized product of idea completions, isomorphic to ideal completion of synchronized product, Key Lemma: If the synchronized product W×V of two UTS, V deterministic, admits an inductive invariant Q↓, then W and V are separated by an automaton with state space Q.
Fact 1: Ideal completion of a (deterministic) UWSTS is a language-equivalent (deterministic) UTS. Fact 2: Ideal completion commutes with synchronized product.
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Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is deterministic.
Proof: Apply Key Lemma to the ideal completions of the UWSTS. Synchronized product of idea completions, isomorphic to ideal completion of synchronized product, admits a finitary inductive invariant. ☐ Key Lemma: If the synchronized product W×V of two UTS, V deterministic, admits an inductive invariant Q↓, then W and V are separated by an automaton with state space Q.
Fact 1: Ideal completion of a (deterministic) UWSTS is a language-equivalent (deterministic) UTS. Fact 2: Ideal completion commutes with synchronized product. Fact 3: For every inductive invariant in a UWSTS, its finite ideal decomposition is a finitary inductive invariant in the ideal completion of this UWSTS.
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DWSTS
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DWSTS
Theorem: The following relations between the language classes:
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DWSTS
Theorem: The following relations between the language classes:
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DWSTS
Theorem: The following relations between the language classes:
rev det. DWSTS
rev det. UWSTS
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Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is finitely-branching. Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic.
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Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is finitely-branching. Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic.
Can these assumptions be dropped?
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Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is finitely-branching. Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic.
Can these assumptions be dropped?
Theorem: The following relations between the language classes:
rev det. DWSTS
rev det. UWSTS
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Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is finitely-branching. Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic.
Can these assumptions be dropped?
Theorem: The following relations between the language classes:
rev det. DWSTS
rev det. UWSTS
Are the inclusions strict?
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Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is finitely-branching. Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic.
Can these assumptions be dropped?
Theorem: The following relations between the language classes:
rev det. DWSTS
rev det. UWSTS
Are the inclusions strict?
Obvious generalizations:
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Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is finitely-branching. Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic.
Can these assumptions be dropped?
Theorem: The following relations between the language classes:
rev det. DWSTS
rev det. UWSTS
Are the inclusions strict?
Obvious generalizations:
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Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is finitely-branching. Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic.
Can these assumptions be dropped?
Theorem: The following relations between the language classes:
rev det. DWSTS
rev det. UWSTS
Are the inclusions strict?
Obvious generalizations:
20
Theorem: Every two disjoint UWSTS are regular-separable, whenever one of them is finitely-branching. Every two disjoint DWSTS are regular-separable, whenever one of them is deterministic.
Can these assumptions be dropped?
Theorem: The following relations between the language classes:
rev det. DWSTS
rev det. UWSTS
Are the inclusions strict?
Obvious generalizations:
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undecidable decidable
r e g u l a r s e p a r a b i l i t y
21
[Hunt ’82]
undecidable decidable
r e g u l a r s e p a r a b i l i t y
21
[Hunt ’82]
[Czerwiński, L. ’17]
undecidable decidable
r e g u l a r s e p a r a b i l i t y
21
[Hunt ’82]
[Czerwiński, L. ’17]
[Clemente, Czerwiński, L., Paperman ’17]
undecidable decidable
r e g u l a r s e p a r a b i l i t y
21
[Hunt ’82]
[Czerwiński, L. ’17] commutative closures of VASS [Clemente, Czerwiński, L., Paperman ’17]
[Clemente, Czerwiński, L., Paperman ’17]
undecidable decidable
r e g u l a r s e p a r a b i l i t y
21
[Hunt ’82]
[Czerwiński, L. ’17] commutative closures of VASS [Clemente, Czerwiński, L., Paperman ’17]
[Clemente, Czerwiński, L., Paperman ’17]
undecidable decidable
coverability VASS
r e g u l a r s e p a r a b i l i t y
21
[Hunt ’82]
[Czerwiński, L. ’17] commutative closures of VASS [Clemente, Czerwiński, L., Paperman ’17]
[Clemente, Czerwiński, L., Paperman ’17]
undecidable decidable
coverability VASS
r e g u l a r s e p a r a b i l i t y
21
[Hunt ’82]
[Czerwiński, L. ’17] commutative closures of VASS [Clemente, Czerwiński, L., Paperman ’17]
[Clemente, Czerwiński, L., Paperman ’17]
regularity
undecidable decidable undecidable decidable
[Valk, Vidal-Naquet ’81] [Cadilhac, Finkel, McKenzie ’11]
coverability VASS
[Worrell ’17]
r e g u l a r s e p a r a b i l i t y
21
[Hunt ’82]
[Czerwiński, L. ’17] commutative closures of VASS [Clemente, Czerwiński, L., Paperman ’17]
[Clemente, Czerwiński, L., Paperman ’17]
[Czerwiński, L. ’17]
regularity
undecidable decidable undecidable decidable
[Valiant ’75] [Valk, Vidal-Naquet ’81] [Cadilhac, Finkel, McKenzie ’11]
coverability VASS
[Worrell ’17]
[Cadilhac, Finkel, McKenzie ’11]
r e g u l a r s e p a r a b i l i t y
21
[Hunt ’82]
[Czerwiński, L. ’17] commutative closures of VASS [Clemente, Czerwiński, L., Paperman ’17]
[Clemente, Czerwiński, L., Paperman ’17]
[Czerwiński, L. ’17]
regularity
undecidable decidable undecidable decidable
[Valiant ’75] [Valk, Vidal-Naquet ’81] [Cadilhac, Finkel, McKenzie ’11]
coverability VASS
[Worrell ’17]
[Cadilhac, Finkel, McKenzie ’11]
r e g u l a r s e p a r a b i l i t y