Logical Characterization of Weighted Pebble Walking Automata

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1 Logical Characterization of Weighted Pebble Walking Automata Benjamin Monmege Université libre de Bruxelles, Belgium Benedikt Bollig and Paul Gastin (LSV, ENS Cachan, France) Marc Zeitoun (LaBRI, Bordeaux University, France) CSL-LICS 2014 Vienna - July 15, 2014

2 Benjamin Monmege, ULB, Belgium 2/17 Equivalence between automata and logic Well-known and studied model of computation: NFA over words Existing extensions over trees, grids, graphs... Robustness of automata intrinsically linked to logical characterization

3 Benjamin Monmege, ULB, Belgium 2/17 Equivalence between automata and logic Well-known and studied model of computation: NFA over words Existing extensions over trees, grids, graphs... Robustness of automata intrinsically linked to logical characterization Büchi-Elgot-Trakhtenbrot: NFA vs MSO Engelfriet-Hoogeboom: pebble walking automata vs FOposTC Droste-Gastin: weighted automata vs restricted weighted MSO

4 Benjamin Monmege, ULB, Belgium 2/17 Equivalence between automata and logic Well-known and studied model of computation: NFA over words Existing extensions over trees, grids, graphs... Robustness of automata intrinsically linked to logical characterization Büchi-Elgot-Trakhtenbrot: NFA vs MSO Engelfriet-Hoogeboom: pebble walking automata vs FOposTC Droste-Gastin: weighted automata vs restricted weighted MSO Aim: extend Engelfriet-Hoogeboom result to the quantitative setting, relating weighted pebble walking automata with weighted FOposTC

5 Benjamin Monmege, ULB, Belgium 3/17 Graphs as a general model Words: D = { } computations of sequential programs a a b a a a a b a a b a b b programs

6 Benjamin Monmege, ULB, Belgium 3/17 Graphs as a general model Words: D = { } computations of sequential programs a a b a a a a b a a b a b b programs Nested words: D = {, } computations of recursive programs, XML documents a a b a a a a b a a b a b b programs!

7 Benjamin Monmege, ULB, Belgium 3/17 Graphs as a general model Words: D = { } computations of sequential programs a a b a a a a b a a b a b b programs Nested words: D = {, } computations of recursive programs, XML documents a a b a a a a b a a b a b b programs! Ranked trees: D = { 1, 2 } expressions, formulae, parse trees p q r

8 Benjamin Monmege, ULB, Belgium 3/17 Definition: directed graphs G = (V, (E d ) d D, λ, ι) where Graphs as a general model V is a nonempty and finite set of vertices; for all edge label d D, E d V V is an irreflexive relation, describing the d-edges of the graph, which is deterministic and codeterministic; λ: V A is a vertex-labeling function; ι V is an initial vertex. For all edge label d, we consider its reverse d 1 letting E d 1 = (E d ) 1

9 Benjamin Monmege, ULB, Belgium 3/17 Definition: directed graphs G = (V, (E d ) d D, λ, ι) where Graphs as a general model V is a nonempty and finite set of vertices; for all edge label d D, E d V V is an irreflexive relation, describing the d-edges of the graph, which is deterministic and codeterministic; P λ: V A is a vertex-labeling function; ι V is an initial vertex. For all edge label d, we consider its reverse d 1 letting E d 1 = (E d ) 1 Grids: D = {, } pictures

10 Benjamin Monmege, ULB, Belgium 4/17 Weighted pebble walking automata Definition: Finite number of states, initial and final states Ability to navigate in the graph (using the deterministic edge labels) Bounded supply of pebbles able to mark temporarily a position Pebbles are treated with a stack policy: first pebble to lift is the last dropped pebble Transitions equipped with weights in a complete semiring (S, +,, 0, 1) c,drop x a, b, 7 x?, init?,lift Examples of complete semirings: ({0, 1},,, 0, 1) (R + {+ }, +,, 0, 1) (Z {+, }, min, +, +, 0) (Z {+, }, max, +,, 0) ([0, 1], min, max, 1, 0) (2 Σ,,,, {ε}) x?, 2 3

11 Benjamin Monmege, ULB, Belgium 4/17 Weighted pebble walking automata a, b, 7 c,drop x x?, init?,lift x?, 2 3 Definition: Semantics over (S, +,, 0, 1) Configurations over a graph G: (G, q, π, v) with state q, stack π of pebble positions and current vertex v Weight of a run: multiplication of the weights of transitions Semantics [[A]](G): sum of weights of accepting runs over G

12 Benjamin Monmege, ULB, Belgium 5/17 Example of weighted pebble walking automaton drop x lift drop y b, x?,lift b, b, drop y b, x?,b,lift b, x?, w, 1 b, x y computes the biggest size of frames (empty black square) (N { }, max, +,, 0)

13 Benjamin Monmege, ULB, Belgium 5/17 Example of weighted pebble walking automaton drop x lift drop y b, x?,lift b, b, drop y b, x?,b,lift b, x?, w, 1 b, x y computes the biggest size of frames (empty black square) (N { }, max, +,, 0)

14 Benjamin Monmege, ULB, Belgium 5/17 Example of weighted pebble walking automaton drop x lift drop y b, x?,lift b, b, drop y b, x?,b,lift b, x?, w, 1 b, x y computes the biggest size of frames (empty black square) (N { }, max, +,, 0)

15 Benjamin Monmege, ULB, Belgium 5/17 Example of weighted pebble walking automaton drop x lift drop y b, x?,lift b, b, drop y b, x?,b,lift b, x?, w, 1 b, x y computes the biggest size of frames (empty black square) (N { }, max, +,, 0)

16 Benjamin Monmege, ULB, Belgium 6/17 Logical characterization Classical weighted automata are one-way (sometimes branching) and without pebbles Logical characterization for them in terms of a restricted weighted MSO logic: over words [Droste and Gastin, 2009] over trees [Droste and Vogler, 2006] over grids [Fichtner, 2011] over nested words [Mathissen, 2010]...

17 Benjamin Monmege, ULB, Belgium 6/17 Logical characterization Classical weighted automata are one-way (sometimes branching) and without pebbles Logical characterization for them in terms of a restricted weighted MSO logic: over words [Droste and Gastin, 2009] over trees [Droste and Vogler, 2006] over grids [Fichtner, 2011] over nested words [Mathissen, 2010]... Restricted weighted MSO does not even contain full weighted FO a priori Theorem: Our contribution Weighted pebble walking automata over graphs (wpwa) = wfotc

18 Benjamin Monmege, ULB, Belgium 7/17 Weighted first-order logic Definition: Classical first-order logic ϕ ::= (x = y) init(x) P a (x) R d (x, y) R + d (x, y) ϕ ϕ ϕ x ϕ

19 Benjamin Monmege, ULB, Belgium 7/17 Weighted first-order logic Definition: Classical first-order logic ϕ ::= (x = y) init(x) P a (x) R d (x, y) R + d (x, y) ϕ ϕ ϕ x ϕ Weighted first-order logic over graphs with weights in a semiring (S, +,, 0, 1) Φ ::= s ϕ? Φ : Φ Φ Φ Φ Φ x Φ x Φ

20 Benjamin Monmege, ULB, Belgium 7/17 Weighted first-order logic Definition: Classical first-order logic ϕ ::= (x = y) init(x) P a (x) R d (x, y) R + d (x, y) ϕ ϕ ϕ x ϕ Weighted first-order logic over graphs with weights in a semiring (S, +,, 0, 1) Φ ::= s ϕ? Φ : Φ Φ Φ Φ Φ x Φ x Φ Semantics over a graph G and a valuation σ of free variables { [[Φ 1 ]](G, σ) if G, σ = ϕ [[ϕ? Φ 1 : Φ 2 ]](G, σ) = [[Φ 2 ]](G, σ) otherwise [[Φ 1 Φ 2 ]](G, σ) = [[Φ 1 ]](G, σ) +[[Φ 2 ]](G, σ) [[ x Φ]](G, σ) = v V[[Φ]](G, σ[x v]) [[Φ 1 Φ 2 ]](G, σ) = [[Φ 1 ]](G, σ) [[Φ 2 ]]G, σ) [[ x Φ]](G, σ) = [[Φ]](G, σ[x v]) v V

21 Weighted first-order logic Definition: Classical first-order logic ϕ ::= (x = y) init(x) P a (x) R d (x, y) R + d (x, y) ϕ ϕ ϕ x ϕ Weighted first-order logic over graphs with weights in a semiring (S, +,, 0, 1) Φ ::= s ϕ? Φ : Φ Φ Φ Φ Φ x Φ x Φ Semantics over a graph G and a valuation σ of free variables { [[Φ 1 ]](G, σ) if G, σ = ϕ [[ϕ? Φ 1 : Φ 2 ]](G, σ) = [[Φ 2 ]](G, σ) otherwise [[Φ 1 Φ 2 ]](G, σ) = [[Φ 1 ]](G, σ) +[[Φ 2 ]](G, σ) [[ x Φ]](G, σ) = v V[[Φ]](G, σ[x v]) [[Φ 1 Φ 2 ]](G, σ) = [[Φ 1 ]](G, σ) [[Φ 2 ]]G, σ) [[ x Φ]](G, σ) = [[Φ]](G, σ[x v]) v V Examples in (N { }, max, +,, 0) Φ b = x P b(x)? 1 : 0 Φ w = x P w(x)? 1 : 0 Φ b Φ w Benjamin Monmege, ULB, Belgium 7/17

22 Benjamin Monmege, ULB, Belgium 8/17 Transitive closure in graphs Binary predicate R (x, y) = z[r (x, z) R (z, y)] Transitive Closure TC x,y R (x, y) test if square (not doable in FO)

23 Benjamin Monmege, ULB, Belgium 8/17 Transitive closure in graphs Binary predicate R (x, y) = z[r (x, z) R (z, y)] Transitive Closure TC x,y R (x, y) test if square (not doable in FO) Weighted transitive closure: semiring (N { }, max, +,, 0) TC x,y [R (x, y)? 1 : ] verifies that it is a square and computes the length of its diagonal

24 Benjamin Monmege, ULB, Belgium 8/17 Transitive closure in graphs Binary predicate R (x, y) = z[r (x, z) R (z, y)] Transitive Closure TC x,y R (x, y) test if square (not doable in FO) Weighted transitive closure: semiring (N { }, max, +,, 0) TC x,y [R (x, y)? 1 : ] verifies that it is a square and computes the length of its diagonal Semantics given in a complete semiring (S, +,, 0, 1) [[[TC x,y Φ](x, y )]](G, σ) = [[Φ]](G, σ[x v k, y v k+1 ]) sum along sequences of stop-vertices v 0,v 1,...,v m (m>0) 0 k m 1 σ(x )=v 0,σ(y )=v m multiplication along the sequence

25 Benjamin Monmege, ULB, Belgium 9/17 Bounding the Transitive Closure A necessary restriction to obtain a fragment of logic expressively equivalent to wpwa But not so restrictive in most of the cases! TC N x,y Φ(x, y) = TC x,y [dist(x, y) N? Φ(x, y) : 0]

26 Benjamin Monmege, ULB, Belgium 9/17 Bounding the Transitive Closure A necessary restriction to obtain a fragment of logic expressively equivalent to wpwa But not so restrictive in most of the cases! TC N x,y Φ(x, y) = TC x,y [dist(x, y) N? Φ(x, y) : 0] Previous example: TC x,y [R (x, y)? 1 : ] = TC 2 x,y [R (x, y)? 1 : ]

27 Benjamin Monmege, ULB, Belgium 9/17 Bounding the Transitive Closure A necessary restriction to obtain a fragment of logic expressively equivalent to wpwa But not so restrictive in most of the cases! TC N x,y Φ(x, y) = TC x,y [dist(x, y) N? Φ(x, y) : 0] Previous example: TC x,y [R (x, y)? 1 : ] = TC 2 x,y [R (x, y)? 1 : ] Definition: Logic wfotc Φ ::= s ϕ? Φ : Φ Φ Φ Φ Φ x Φ x Φ TCN x,y Φ with s S, ϕ FO, x, y Var and N N \ {0}. Comparison with restricted wmso: unrestricted use of x and, presence of TC x,y, absence of X

28 Benjamin Monmege, ULB, Belgium 10/17 Contribution Theorem: over searchable graphs: wfotc weighted pebble walking automata over zonable graphs: weighted pebble walking automata wfotc

29 Benjamin Monmege, ULB, Belgium 10/17 Contribution Theorem: over searchable graphs: wfotc weighted pebble walking automata over zonable graphs: weighted pebble walking automata wfotc = (un)decidability and complexity results over automata transfer to wfotc

30 Benjamin Monmege, ULB, Belgium 11/17 Translation of wfotc in wpwa Definition: Hypothesis: searchable graphs linear order on vertices with ι (initial vertex) as minimal element existence of a guide: walking automaton A G computing All previously classes of graphs are searchable

31 Benjamin Monmege, ULB, Belgium 11/17 Translation of wfotc in wpwa Definition: Hypothesis: searchable graphs linear order on vertices with ι (initial vertex) as minimal element existence of a guide: walking automaton A G computing All previously classes of graphs are searchable Inductive translation: Φ Ψ disjoint union of automata reset to ι Φ Ψ A Φ A Ψ

32 Benjamin Monmege, ULB, Belgium 11/17 Translation of wfotc in wpwa Definition: Hypothesis: searchable graphs linear order on vertices with ι (initial vertex) as minimal element existence of a guide: walking automaton A G computing All previously classes of graphs are searchable init?next next x Φ drop x drop x A Φ lift

33 Benjamin Monmege, ULB, Belgium 11/17 Translation of wfotc in wpwa Definition: Hypothesis: searchable graphs linear order on vertices with ι (initial vertex) as minimal element existence of a guide: walking automaton A G computing All previously classes of graphs are searchable init? next init? x Φ drop x A Φ lift

34 Benjamin Monmege, ULB, Belgium 11/17 Translation of wfotc in wpwa Definition: Hypothesis: searchable graphs linear order on vertices with ι (initial vertex) as minimal element existence of a guide: walking automaton A G computing All previously classes of graphs are searchable ϕ? Φ : Ψ B ϕ ϕ ϕ reset to ι reset to ι A Φ A Ψ Boolean fragment: linear size automata (pebble and navigation)

35 Translation of wfotc in wpwa Definition: Hypothesis: searchable graphs linear order on vertices with ι (initial vertex) as minimal element existence of a guide: walking automaton A G computing All previously classes of graphs are searchable ϕ? Φ : Ψ Boolean fragment: B ϕ ϕ ϕ disjunction ξ = ϕ ψ B ϕ ϕ ϕ reset to ι reset to ι reset to ι A Φ A Ψ linear size automata (pebble and navigation) B ψ ψ ψ ξ ξ existential quantification ξ = x ϕ drop x init? ϕ B ϕ ϕ Benjamin Monmege, ULB, Belgium 11/17 next lift init? lift ξ ξ

36 Benjamin Monmege, ULB, Belgium 12/17 Translation of wfotc in wpwa Case of a formula [TC N x,y Φ(x, y)] (x, y ) with A a wpwa for Φ: construction of }{{} fresh free variables a wpwa A with two more layers of pebbles that does the following

37 Benjamin Monmege, ULB, Belgium 12/17 Translation of wfotc in wpwa Case of a formula [TC N x,y Φ(x, y)] (x, y ) with A a wpwa for Φ: construction of }{{} fresh free variables a wpwa A with two more layers of pebbles that does the following 1. search free variable x, and drop pebble x 2. guess a sequence of moves of length N, follow it, and drop pebble y (then flush the sequence to save memory) 3. reset to ι and simulate A 4. search pebble y 5. guess sequence π of moves of length N, follow it, check that it holds x 6. lift pebbles y and x (hence returning to the vertex of x) 7. follow π R to reach back the vertex that held y, and drop pebble x 8. if y is held by the current vertex, enter a final state 9. in every case, go back to step 2

38 Benjamin Monmege, ULB, Belgium 12/17 Translation of wfotc in wpwa Case of a formula [TC N x,y Φ(x, y)] (x, y ) with A a wpwa for Φ: construction of }{{} fresh free variables a wpwa A with two more layers of pebbles that does the following 1. search free variable x, and drop pebble x 2. guess a sequence π of moves of length N, follow it, and drop pebble y (then flush the sequence to save memory) test that π is minimal amongst all sequences going from x to y 3. reset to ι and simulate A 4. search pebble y 5. guess sequence π of moves of length N, follow it, check that it holds x test that π is minimal amongst all sequences going q from y to x 6. lift pebbles y and x (hence returning to the vertex of x) 7. follow π R to reach back the vertex that held y, and drop pebble x 8. if y is held by the current vertex, enter a final state 9. in every case, go back to step 2

39 Benjamin Monmege, ULB, Belgium 13/17 Translation of wpwa in wfotc Theorem: Let G be a zonable class of graphs. Then, for every wpwa A, we can construct a formula Φ of wfotc such that for every graph G G, and valuation σ of free variables, [[A]](G, σ) = [[Φ]](G, σ). for a zonable class of graphs G Automaton A Formula Φ Translation depends on the class G

40 Benjamin Monmege, ULB, Belgium 13/17 Translation of wpwa in wfotc Theorem: Let G be a zonable class of graphs. Then, for every wpwa A, we can construct a formula Φ of wfotc such that for every graph G G, and valuation σ of free variables, [[A]](G, σ) = [[Φ]](G, σ). for a zonable class of graphs G Automaton A Formula Φ Translation depends on the class G Proof in two steps: For the considered class of graphs, prove the zonability; Generic translation of automata into formulae for zonable class of graphs

41 Zonable classes of graphs A zoning of a graph G with parameter N: I an equivalence relation, decomposing a graph into zones of diameter bounded by a constant M; I set W of wires = (directed) edges relating different zones; I an injective encoding function encchapter : W {0,..6.., NLOGICAL 1} V SPECIFICATIO re 6.6: Zone partitioning of a graph: zones are related by wires depicted with dash. The encoding of wire (v, v Õ ), for every integer n œ [0.. N 1] is depicted by a red a edbenjamin to the vertex v Õ, or containing vertex v Õ. Monmege, ULB, Belgium 14/17

42 Zonable classes of graphs A zoning of a graph G with parameter N: I an equivalence relation, decomposing a graph into zones of diameter bounded by a constant M; I set W of wires = (directed) edges relating different zones; I an injective encoding function encchapter : W {0,..6.., NLOGICAL 1} V SPECIFICATIO re 6.6: Zone partitioning of a graph: zones are related by wires0 depicted with dash and and enc must be expressible by some formulae zone(z, z ) and. The encoding of wire (v, v Õ ), for every integer n œ [0.. N 1] is depicted by a red a encn (z, z 0, x) (for n {0,..., N 1}) inõ wfotc Õ edbenjamin to the vertex v, or containing vertex v. Monmege, ULB, Belgium 14/17

43 Examples 6.5. FROM AUTOMATA TO LOGICS N (k 1)N 2kN 1 2kN 2KN 2(k+1)N! "# w $ N 1 w 2KN <3N 1 Figure 6.7: Zone partitioning of a word and description of the encoding function f then we can consider that there is a single zone containing all the vertices, and hence no wires. Therefore, in the examples we give below, we only consider graphs having at least M + 1 vertices. It has to be noticed that wires relate two distinct zones of the graph. Hence, they can be characterized by the formula wire(z, z Õ ) = m m x nœ[0.. N 1] (f(z, z Õ, n) = x) which is an unambiguous formula as f is an injection. Before stating the translation theorems, we give a non-exhaustive list of zonable classes of graphs. Words Zones of words will be subwords of length 2N (see Figure 6.7), except the last zone that Benjamin Monmege, ULB, Belgium 15/17

44 Examples 6.5. FROM AUTOMATA TO LOGICS N (k 1)N 2kN 1 2kN 2KN 2(k+1)N! "# w $ N 1 w 2KN <3N 1 CHAPTER 6. LOGICAL SPECIFICATIONS Figure 6.7: Zone partitioning of a word and description of the encoding function f then we can consider that there is a single zone containing all the vertices, and hence no wires. Therefore, in the examples we give below, we only consider graphs having at least M + 1 vertices. It has to be noticed that wires relate two distinct zones of the graph. Hence, they can be characterized by the formula wire(z, z Õ ) = m m x nœ[0.. N 1] (f(z, z Õ, n) = x) which is an unambiguous formula as f is an injection. Before stating the translation theorems, we give a non-exhaustive list of zonable classes of graphs. Words Zones of Figure words6.8:will subwords length 2N (see Figure 6.7), except the last zone that Zonebe partitioning of aof picture Benjamin Monmege, ULB, Belgium 15/17

45 Examples 6.5. FROM AUTOMATA TO LOGICS N kN 1 2kN 2(k 1)N 2(k+1)N CHAPTER 6. LOGICAL SPECIFICATIONS 2KN! "# w $ N 1 w 2KN <3N 1 Figure 6.7: Zone partitioning of a word and description of the encoding function f then we can consider that there is a single zone containing all the vertices, and hence no wires. Therefore, in the examples we give below, we only consider graphs having at least M + 1 vertices. It has to be noticed that wires relate two distinct zones of the graph. Hence, they can be characterized by the formula wire(z, z ) = Õ m m x but also trees, nested words, Õ Mazurkiewicz traces, rings... nœ[0.. N 1] (f(z, z, n) = x) which is an unambiguous formula as f is an injection. Before stating the translation theorems, we give a non-exhaustive list of zonable classes of graphs. Words Zones of Figure words6.8:will subwords length 2N (see Figure 6.7), except the last zone that Zonebe partitioning of aof picture Benjamin Monmege, ULB, Belgium 15/17

46 Translation in a zonable class of graphs I External (bounded) transitive closure jumping from zone to zone: state at the wires encoded using enc; I Internal (bounded) transitive closures to compute the weights of the infinite set of runs restricted to a zone: computation by McNaughton-Yamada HYBRID NAVIGATIONAL algorithm, state directlylogics encoded in the formulae. v (i) x z1 y z2 v (f ) z2 z1 Figure 6.11: Instantiation of formula Benjamin Monmege, ULB, Belgium qi,qf 16/17 1

47 Translation in a zonable class of graphs Weight of the runs from zi in state qi to zf in state qf : L L L 0 0 encq1 (z1, z10, x 0 ) Φqi,q1 (zi, z1 ) [TC3M y1,y2 Ψ](x, y ) x 0,y 0 z1,z10 q1 Q i L L encq2 (z2, z20, y 0 ) trq2,q20 (z2, z20 ) Φq20,qf (z20, zf ) z2,z20 q2,q20 Q with Ψ(y1, y2 ) the formula i L L h (z1, z ) encq (z2, z, y2 ) Φq 0,q (z, z2 ) enc (z, z, y ) tr q 1 1 q,q HYBRID NAVIGATIONAL LOGICS z1,z1, q1,q1, z2,z20 q2 Q v (i) x z1 y z2 z2 v (f ) z1 Benjamin Monmege, ULB, Belgium 16/17

48 Benjamin Monmege, ULB, Belgium 16/17 Translation in a zonable class of graphs Weight of the runs from z i in state q i to z f in state q f : [ enc q1 (z 1, z 1, x ) Φ qi,q 1 (z i, z 1 ) ] [TC 3M y 1,y 2 Ψ](x, y ) x,y z 1,z 1 q 1 Q [ ] encq2 (z 2, z 2, y ) tr q2,q (z q 2,q 2 Q 2 2, z 2) Φ q 2,q f (z 2, z f ) z 2,z 2 with Ψ(y 1, y 2 ) the formula [ ] enc q1 (z 1, z 1, y 1 ) tr q1,q 1 (z 1, z 1) enc q2 (z 2, z 2, y 2 ) Φ q 1,q 2 (z 1, z 2 ) z 1,z 1, z 2,z 2 q 1,q 1, q 2 Q Φ q,q (x, x ) formula computing the weight of the runs from x in q to x in q, staying in the zone containing both x and x built by McNaughton-Yamada algorithm, with cascade of bounded transitive closures (since zones have bounded diameter)

49 Benjamin Monmege, ULB, Belgium 17/17 Conclusion and Perspectives Expressive equivalence between weighted pebble walking automata and weighted first-order logic with bounded transitive closure, over arbitrary complete semirings Additional reasonable requirements on the classes of graphs (searchable and zonable), met by usual examples of graphs (words, nested words, trees, grids, Mazurkiewicz traces, rings...) Interesting special case: a logic for graph-to-word transducers (non-commutative semiring of languages over an alphabet Σ)

50 Benjamin Monmege, ULB, Belgium 17/17 Conclusion and Perspectives Expressive equivalence between weighted pebble walking automata and weighted first-order logic with bounded transitive closure, over arbitrary complete semirings Additional reasonable requirements on the classes of graphs (searchable and zonable), met by usual examples of graphs (words, nested words, trees, grids, Mazurkiewicz traces, rings...) Interesting special case: a logic for graph-to-word transducers (non-commutative semiring of languages over an alphabet Σ) Translation from automata to logic with less transitive closures? as in [Bollig, Gastin, Monmege, and Zeitoun, 2010] for words and the non-looping semantics Case of strong pebbles to deal with unbounded transitive closure? Extension to infinite structures?

51 Benjamin Monmege, ULB, Belgium 17/17 Conclusion and Perspectives Expressive equivalence between weighted pebble walking automata and weighted first-order logic with bounded transitive closure, over arbitrary complete semirings Additional reasonable requirements on the classes of graphs (searchable and zonable), met by usual examples of graphs (words, nested words, trees, grids, Mazurkiewicz traces, rings...) Interesting special case: a logic for graph-to-word transducers (non-commutative semiring of languages over an alphabet Σ) Translation from automata to logic with less transitive closures? as in [Bollig, Gastin, Monmege, and Zeitoun, 2010] for words and the non-looping semantics Case of strong pebbles to deal with unbounded transitive closure? Extension to infinite structures? Thank you!

52 References Benedikt Bollig, Paul Gastin, Benjamin Monmege, and Marc Zeitoun. Pebble weighted automata and transitive closure logics. In Proceedings of ICALP 10, volume 6199 of LNCS, pages Springer, Manfred Droste and Paul Gastin. Weighted automata and weighted logics. EATCS Monographs in TCS, chapter 5, pages Springer, Manfred Droste and Heiko Vogler. Weighted tree automata and weighted logics. Theoretical Computer Science, 366(3): , Ina Fichtner. Weighted picture automata and weighted logics. Theory of Computing Systems, 48(1):48 78, Christian Mathissen. Weighted logics for nested words and algebraic formal power series. Logical Methods in Computer Science, 6(1), Benjamin Monmege. Specification and Verification of Quantitative Properties: Expressions, Logics, and Automata. Phd thesis, ENS de Cachan, 2013.

Weighted Specifications over Nested Words

Weighted Specifications over Nested Words Benedikt Bollig 1, Paul Gastin 1, and Benjamin Monmege 1 LSV, ENS Cachan, CNRS & Inria, France firstname.lastname@lsv.ens-cachan.fr Abstract. This paper studies