Classes of Rational Graphs Classes of Rational Graphs Christophe Morvan Irisa Journ´ ees Montoises 2006 1/25
Classes of Rational Graphs Plan 1 Definitions Notations Rational Graphs 2 Structure defined families Deterministic Rational Graphs Rational Trees Rational DAG 3 Automata defined families Automatic Graphs Monotonous Rational Graphs Commutative Rational Graphs 4 Conclusion and future works 2/25
Classes of Rational Graphs Plan 1 Definitions Notations Rational Graphs 2 Structure defined families Deterministic Rational Graphs Rational Trees Rational DAG 3 Automata defined families Automatic Graphs Monotonous Rational Graphs Commutative Rational Graphs 4 Conclusion and future works 2/25
Classes of Rational Graphs Plan 1 Definitions Notations Rational Graphs 2 Structure defined families Deterministic Rational Graphs Rational Trees Rational DAG 3 Automata defined families Automatic Graphs Monotonous Rational Graphs Commutative Rational Graphs 4 Conclusion and future works 2/25
Classes of Rational Graphs Plan 1 Definitions Notations Rational Graphs 2 Structure defined families Deterministic Rational Graphs Rational Trees Rational DAG 3 Automata defined families Automatic Graphs Monotonous Rational Graphs Commutative Rational Graphs 4 Conclusion and future works 2/25
Classes of Rational Graphs Definitions Plan 1 Definitions Notations Rational Graphs 2 Structure defined families Deterministic Rational Graphs Rational Trees Rational DAG 3 Automata defined families Automatic Graphs Monotonous Rational Graphs Commutative Rational Graphs 4 Conclusion and future works 3/25
Classes of Rational Graphs Definitions Notations Graphs Graphs A graphe G is a subset of V × A × V • V is an arbitrary set of vertices • A is a finite set of labels Conventions a ( u , a , v ) is denoted by u − → v For rational graphs V := X ∗ ( X some finite alphabet) 4/25
Classes of Rational Graphs Definitions Notations Graphs Graphs A graphe G is a subset of V × A × V • V is an arbitrary set of vertices • A is a finite set of labels Conventions a ( u , a , v ) is denoted by u − → v For rational graphs V := X ∗ ( X some finite alphabet) 4/25
Classes of Rational Graphs Definitions Notations Graphs Graphs A graphe G is a subset of V × A × V • V is an arbitrary set of vertices • A is a finite set of labels Conventions a ( u , a , v ) is denoted by u − → v For rational graphs V := X ∗ ( X some finite alphabet) 4/25
Classes of Rational Graphs Definitions Notations Graphs Graphs A graphe G is a subset of V × A × V • V is an arbitrary set of vertices • A is a finite set of labels Conventions a ( u , a , v ) is denoted by u − → v For rational graphs V := X ∗ ( X some finite alphabet) 4/25
Classes of Rational Graphs Definitions Notations Graphs Graphs A graphe G is a subset of V × A × V • V is an arbitrary set of vertices • A is a finite set of labels Conventions a ( u , a , v ) is denoted by u − → v For rational graphs V := X ∗ ( X some finite alphabet) 4/25
Classes of Rational Graphs Definitions Rational Graphs Rational graphs Definition A rational graph is defined by a rational transducer 0 / 0 ε/ 0 000 a 0 / 0 r 1 a 00 b a q 1 a 0 / 0 0 / 1 1 / 1 ε 0 001 b 0 / 1 01 b b p r 2 b 1 / A A 1 011 b c 1 / 1 c 11 1 /ε b c q 2 r 3 111 c 1 / 1 5/25
Classes of Rational Graphs Definitions Rational Graphs Rational graphs Definition A rational graph is defined by a rational transducer 0 / 0 ε/ 0 000 a 0 / 0 r 1 a 00 b a q 1 a 0 / 0 0 / 1 1 / 1 ε 0 001 b 0 / 1 01 b b p r 2 b 1 / A A 1 011 b c 1 / 1 c 11 1 /ε b c q 2 r 3 111 c 1 / 1 5/25
Classes of Rational Graphs Definitions Rational Graphs Rational graphs Definition A rational graph is defined by a rational transducer 0 / 0 ε/ 0 000 a 0 / 0 r 1 a 00 b a q 1 a 0 / 0 0 / 1 1 / 1 ε 0 001 b 0 / 1 01 b b p r 2 b 1 / A A 1 011 b c 1 / 1 c 11 1 /ε b c q 2 r 3 111 c 1 / 1 5/25
Classes of Rational Graphs Definitions Rational Graphs Rational graphs Definition A rational graph is defined by a rational transducer 0 / 0 ε/ 0 000 a 0 / 0 r 1 a 00 b a q 1 a 0 / 0 0 / 1 1 / 1 ε 0 001 b 0 / 1 01 b b p r 2 b 1 / A A 1 011 b c 1 / 1 c 11 1 /ε b c q 2 r 3 111 c 1 / 1 5/25
Classes of Rational Graphs Definitions Rational Graphs Rational graphs Definition A rational graph is defined by a rational transducer 0 / 0 ε/ 0 000 a 0 / 0 r 1 a 00 b a q 1 a 0 / 0 0 / 1 1 / 1 ε 0 001 b 0 / 1 01 b b p r 2 b 1 / A A 1 011 b c 1 / 1 c 11 1 /ε b c q 2 r 3 111 c 1 / 1 5/25
Classes of Rational Graphs Definitions Rational Graphs Classical questions Proposition Following properties are undecidable for rational graphs: (i) Accessibility (between two vertices); (ii) Connectedness (of the whole graph); (iii) Isomorphism (of two graphs); (iv) First order theory (of a rational graph). 6/25
Classes of Rational Graphs Definitions Rational Graphs Traces Example (traces) 000 a 00 b a a ε 0 b 001 b 01 b A 1 b 011 c c 11 b c 111 Theorem (Morvan Stirling 01) The traces of rational graphs between two vertices coincide with the context sensitive languages. 7/25
Classes of Rational Graphs Definitions Rational Graphs Traces Example (traces) 000 a The trace of this graph between ε and A 00 b a is { a n b n c n | n ∈ N } a ε 0 b 001 b 01 b Tr ( G , ε, A ) = { a n b n c n | n ∈ N } A 1 b 011 c c 11 b c 111 Theorem (Morvan Stirling 01) The traces of rational graphs between two vertices coincide with the context sensitive languages. 7/25
Classes of Rational Graphs Definitions Rational Graphs Traces Example (traces) 000 a The trace of this graph between ε and A 00 b a is { a n b n c n | n ∈ N } a ε 0 b 001 b 01 b Tr ( G , ε, A ) = { a n b n c n | n ∈ N } A 1 b 011 c c 11 b c 111 Theorem (Morvan Stirling 01) The traces of rational graphs between two vertices coincide with the context sensitive languages. 7/25
Classes of Rational Graphs Structure defined families Plan 1 Definitions Notations Rational Graphs 2 Structure defined families Deterministic Rational Graphs Rational Trees Rational DAG 3 Automata defined families Automatic Graphs Monotonous Rational Graphs Commutative Rational Graphs 4 Conclusion and future works 8/25
Classes of Rational Graphs Structure defined families Deterministic Rational Graphs Deterministic Rational Graphs Definition A rational graph is deterministic if each vertex is the source of at most on arc for each label Proposition Determinism is decidable for rational graphs Proposition First order theory and accessibility are undecidable for deterministic rational graphs 9/25
Classes of Rational Graphs Structure defined families Deterministic Rational Graphs Deterministic Rational Graphs Definition A rational graph is deterministic if each vertex is the source of at most on arc for each label Proposition Determinism is decidable for rational graphs Proposition First order theory and accessibility are undecidable for deterministic rational graphs 9/25
Classes of Rational Graphs Structure defined families Deterministic Rational Graphs Deterministic Rational Graphs Definition A rational graph is deterministic if each vertex is the source of at most on arc for each label Proposition Determinism is decidable for rational graphs Proposition First order theory and accessibility are undecidable for deterministic rational graphs 9/25
Classes of Rational Graphs Structure defined families Deterministic Rational Graphs Traces Proposition The traces of deterministic rational graphs form a boolean algebra of deterministic context-sensitive languages containing non context-free languages. Question Are there context-free languages not contained in the traces of deterministic rational graphs? 10/25
Classes of Rational Graphs Structure defined families Deterministic Rational Graphs Traces Proposition The traces of deterministic rational graphs form a boolean algebra of deterministic context-sensitive languages containing non context-free languages. Question Are there context-free languages not contained in the traces of deterministic rational graphs? 10/25
Classes of Rational Graphs Structure defined families Deterministic Rational Graphs Traces Proposition The traces of deterministic rational graphs form a boolean algebra of deterministic context-sensitive languages containing non context-free languages. Question Are there context-free languages not contained in the traces of deterministic rational graphs? 10/25
Classes of Rational Graphs Structure defined families Rational Trees Rational Trees Definition (Tree) • at most one ancestor per vertex decidable • a single root decidable • connected undecidable 11/25
Classes of Rational Graphs Structure defined families Rational Trees Rational Trees Definition (Tree) • at most one ancestor per vertex decidable • a single root decidable • connected undecidable 11/25
Classes of Rational Graphs Structure defined families Rational Trees Rational Trees Definition (Tree) • at most one ancestor per vertex decidable • a single root decidable • connected undecidable 11/25
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