On Data-Structure Rewriting

Transcription

1 On Data-Structure Rewriting Rachid Echahed LIG Lab, Grenoble France June, 2010

2 Rewriting (reminder) A rewrite relation R is a binary relation R A A (u, v) R is read u rewrites into v and written u v

3 Rewriting (reminder) R A A A = set of strings over a vocabulary (V ) A = set of states of the form (variables, valuation) A = set of Turing Machine configurations A = set of lambda-terms A = set of trees (or terms) A = set of clauses A = set of process terms A =...

4 Rewriting R A A How to define a rewrite relation R? How to define a run or the execution of a program? A rewrite derivation : u 0 is the initial call u 0 u 1... u n A narrowing derivation : w 0 is the initial goal (to solve): Where w 0 σ1 w 1... σn w n w i σ w i+1 iff σ(w i ) w i+1 w i is an element of A with partial information σ instantiates w i

5 Motivation : Extension of Term Rewriting ; sharing subterms Function definitions by means of term rewrite rules 0 + x x succ(x) + y succ(x + y) double(x) x + x Very well established domain with several results : Confluence, Termination, Strategies, Proof methods (equational reasoning, induction) etc. double(x) + x double t + t

6 Sharing Subterms (information) and Term Rewriting Consider the following rules: f (a, b) c a b Sharing does not preserve properties of tree (term) rewriting! f (a, a) f (a, b) c f a f b [Plump 99] survey on rewriting with dags.

7 Motivation (continued) Data-structure rewriting including cyclic data-structures with pointers such as circular lists, doubly-linked lists, etc. Data-structures are more complex than terms (Cycles, Sharing) Difficult to encode efficiently using terms Usually described by pointers ( pointer rewriting) Formally described as term-graphs term-graphs = terms with cycles and sharing

8 Term-graphs [Barendregt et al. 87] [Plump 99, survey on acyclic term-graphs] Let Ω be a set of operation symbols. A term-graph t over Ω is defined by: a set of nodes N t, a subset of labeled nodes N Ω t N t, a labeling function L t : N Ω t Ω, a successor function S t : N Ω t N t, 1 1 : f 2 2 : b 3 : g : 5 : h

9 Term-graphs [Barendregt et al. 87] [Plump 99, survey on acyclic term-graphs] Let Ω be a set of operation symbols and F a set of feature symbols. A term-graph t over Ω and F is defined by: a set of nodes N t, a set of edges E t a subset of labeled nodes N Ω t N t, a node labeling function L n t : N Ω t Ω, an edge labeling function L e t : E t F a source function S t : E t N t, a target function T t : E t N t,

10 (Term-)Graph Rewriting Which graphs? (Term-Graphs) Which rules? Which rewrite relation? Two main approaches Algorithmic approaches Algebraic approaches (DPO,SPO,...)

11 Graph Transformation Handbook of Graph Grammars and Computing by Graph Transformation (World Scientific) Vol 1: Foundations, ed. G. Rozenberg, 1997 Vol 2: Applications, Languages and Tools eds. H. Ehrig, G. Engels, H.-J. Kreowski and G. Rozenberg, 1999 Vol 3: Concurrency, Parallelism and Distribution eds. H. Ehrig, H.-J. Kreowski, U. Montanari and G. Rozenberg, 1999 A Monograph in Theoretical Computer Science (An EATCS series) H. Ehrig, K. Ehrig, U. Prange, G. Taentzer: Fundamentals of Algebraic Graph Transformation. Springer-Verlag, 2006

12 Outline Introduction Motivations Termgraph Rewrite Systems Confluence and Rewrite Strategies Narrowing A Modal Logic for Graph Transformation Conclusion

13 Algorithmic approach [Barendregt et al. 87] Shape of a rule: L R where L and R are rooted term-graphs. A rule can be defined as one graph together with two roots (L + R, r 1, r 2 ) where r 1 and r 2 are the roots of L and R respectively Let ρ be the rule (L + R, r 1, r 2 ) We say that G rewrites to H using the rule ρ if L matches a subgraph of G (h : L G n ) (build phase) Construct graph G 1 = G + h(r) (redirection phase) G 2 = [h(r 1 ) h(r 2 )]G 1 (garbage collection phase) H = G 2 root A cumbersome definition, hard to deal with in practice!

14 Rewrite Rules with actions Shape of a rewrite rule : [L C] R L is a term-graph pattern C is a node constraint, n i=1 (α i β i ). R is a sequence of actions a 1 ; a 2 ;... ; a n

15 Actions We consider three kinds of actions : Node definition α:f (α 1,..., α n ) Edge redirection α i β Global redirection α β

16 Application of actions a[t] denotes the application of action(s) a on the term-graph t Let t = n :f (p, q :a) n :f 1 2 p q :a Let t 1 = p :h(p)[t] = n :f (p :h(p), q : a) 1 n :f 1 2 p :h q :a

17 Application of actions a[t] denotes the application of action(s) a on the term-graph t Let t 1 = p :h(p)[t] = n :f (p :h(p), q : a) 1 n :f 1 2 p :h q :a Let t 2 = n 2 p[t 1 ] = n :f (p :h(p), p); q : a 1 p :h n :f 1 2 q :a

18 Application of actions a[t] denotes the application of action(s) a on the term-graph t Let t 2 = n 2 p[t 1 ] = n :f (p :h(p), p); q : a n :f 1 p :h 1 2 Let t 3 = p q[t 2 ] = n :f (q, q); p :h(q) n :f 2 q :a p :h 1 1 q :a

19 Rewrite Step Let t be a term-graph Let ρ be a rewrite rule [L C] R t rewrite to s at node α, t α s iff: m : L t a homomorphism (ρ-matcher) m(root L ) = α α is reachable from root t m(c) holds s = m(r)[t]

20 Term-Graph Rewrite Systems (tgrs) Example Length of a circular list : r : length(p) r : length (p, p) r : length (p 1 : cons(n, p 2 ), p 2 ) r : s(0) [r : length (p 1 : cons(n, p 2 ), p 3 ) p 2 p 3 ] r : s(q); q : length (p 2, p 3 ) Remark: term rewrite systems are tgrs s.

21 Term-Graph Rewrite Systems Example In-situ list reversal : o : reverse(p) o : rev(p, nil) o : rev(p 1 : cons(n, nil), p 2 ) p 1 2 p 2 ; o p 1 o : rev(p 1 : cons(n, p 2 : cons(m, p 3 ), p 4 ) p 1 2 p 4 ; o 1 p 2 ; o 2 p 1 Visual Programming would help!

22 DPO approach of rewrite rules with actions A categorical approach can be found in [TERMGRAPH 06, ENTCS07, RTA07] L l K r R m G d m l D r H Figure: Double pushout: a rewrite step (G H) Redirections of edges (pointers) are handled by K = disconnection(l, E, N) and the morphisms l and r. Remark: Morphisms l and r are not injective! D is not unique!

23 Confluence f (x) x g(x) x The following term-graph n :f q :g rewrites to n :f q :g

24 Confluence α : f (β : c) β : a; α β α : g(β : c) β : b; α β p :f q :g q :c The label of node q may end as q : a or q : b

25 Computing with non-confluent orthogonal Term-graph Rewrite Systems How to evaluate the following term-graph? addlast(length(n : [1, 2]), n) Two normal forms [1, 2, 2] (evaluate addlast after length) [1, 2, 3] (evaluate length after addlast)

26 Term-graphs with Priority [PPDP06][RTA07][RTA08] Endow Term-graphs with priorities (G, < G ) to express which node should be evaluated first m 1 :addlast(m 2 :length(n :[1, 2]), n); m 1 < m 2 Priorities should not be a total order (stay declarative) Which nodes should be ordered? Solution: Order only nodes producing a side-effect

27 Strategies and needed nodes A strategy φ is a partial function which takes a rooted term-graph t and returns a node (position) n and a rule R, φ(t) = (n, R) such that the term-graph t can be reduced at node n using the rule R, t n t

28 Needed Nodes Let φ be a rewrite strategy. Let φ(t) = (p, R). The node p is needed iff for all derivations t β1 t 1 β2... t n 1 βn t n such that t n is a value, there exists i [1..n] s.t. β i = p

29 Inductively sequential Term Rewrite Systems Constitute a subclass of TRSs for which efficient rewrite strategies are available [Antoy 92] Are as expressive as Strongly Sequential TRSs Are the basis of modern functional and logic programming languages. Are defined by means of data-structures called Definitional trees

30 Definitional Trees -case of terms- Let R be the following TRS f(k,nil) R1 f(0,cons(x,l)) R2 f(succ(n),cons(x,l)) R3 A definitional tree of operator f is a hierarchical structure whose leaves are the rules defining f. f(k, l) f(k, nil) R1 f(k, cons (x, u)) f(0, cons (x,u)) R2 f(succ(y), cons (x,u)) R3

31 Definitional trees -case of term-graphsr : length (p 1 : nil, p 2 : ) rhs 1 r : length (p 1 : cons(n :, p 2 : ), p 2 ) rhs 2 [r : length. (p 1 : cons(n :, p 2 : ), p 3 : ) p 2 = p 3 ] rhs 3 A definitional tree T of the operation length is given bellow: r : length (p 1 :, p 2 : ) r : length (p 1 : nil, p 2 : ) rhs 1 r : length (p 1 : cons(n :, p 3 : ), p 2 : ) r : length (p 1 : cons(n :, p 2 : ), p 2 ) rhs 2 [r : length. (p 1 : cons(n :, p 2 : ), p 3 : ) p 2 = p 3 ] rhs 3

32 A Rewrite strategy φ Consider the following definitional tree T of the operation g : r : g(p 1 :, p 2 : ) r : g(p 1 : nil, p 2 : ) rhs 1 r : g(p 1 : cons(n :, p 3 : ), p 2 : ) r : g(p 1 : cons(n :, p 2 : ), p 2 ) rhs 2. [r : g(p 1 : cons(n :, p 2 : ), p 3 : ) p 2 = p 3 ] rhs 3 φ(1 : g (2 : g(3 : g(nil, p), q), 4 : g(nil, o))) = φ(2 : g(3 : g(nil, p), q)) = φ(3 : g(nil, p)) = (3, Rule1)

33 Naive extension of TRS s Contrary to term rewriting, Definitional trees are not enough to ensure the neededness of positions computed by the strategy φ, in the context of term-graph rewriting. Proposition: Let SP = Ω, R be tgrs such that Ω is constructor-based and the rules of every defined operation are stored in a definitional tree. Let t be a rooted term-graph. Then, 1. if φ(t) = (p, R), the node p is not needed in general. 2. if φ(t) is not defined, g can still have a constructor normal form.

34 Counter-examples r : f (p : 0) r p r : h(p : 0, q : succ(n : )) q p r : f (p : succ(p : )) r p Let t = n : succ r : succ p : f q : succ s : h u : 0 φ(t) = (p, r : f (p : succ(p : )) r p). However, the node p is not needed in t.

35 Counter-examples r : g(p : 0) r p r : h(p : 0, q : succ(n : )) q p Let t = n : succ r : succ p : g q : succ s : h u : 0 φ(t) is not defined!. However, the term-graph t rewrites to n : succ(u : 0).

36 Inductively Sequential Term-Graph Rewrite Systems Let SP = Ω, R be a tgrs. SP is called inductively sequential iff The rules of every defined operation can be stored in a definitional tree and for all rules [L C] r in R, for all global (respectively, local) redirections of the form p q (respectively, p i q for some i), occurring in the right-hand side r, p = Root L.

37 Main Properties of Strategy Φ In presence of Inductively Sequential Term-Graph Rewrite Systems The positions computed by Φ are needed Φ is c-normalizing Φ is c-hyper-normalizing Derivations computed by Φ have minimal length

38 Confluence Inductively sequential tgrs are not confluent! f (p :, p) 0 [f (p :, q : ) p q] 1 r : g(q : ) r q Let t = n : f p : g q : 0 There are two different derivations starting from t : t n 1 t p f (q : 0, q) n 0

39 Admissible term-graphs [JICSLP98] Ω is contructor-based, i.e. Ω = D C and D C = D is a set of defined operations C is a set of constructors A term-graph is admissible if none of its cycles includes a defined operation. n : succ(n) is an admissible term-graph n :+(n, n) and n : tail(n) are not admissible

40 Admissible term-graphs The set of admissible term-graphs is not closed under rewriting n :f (m) q :g(n); n m Let Ω = D C with C = {0, succ} and D = {f, g} n 1 :f (m 1 :0) q 1 :g(q 1 )

41 Admissible Inductively sequential Term-Graph Rewrite Systems Let SP = Ω, R be an inductively sequential tgrs. SP is called admissible iff for all rules [π C] r in R the following conditions are satisfied for all global (respectively, local) redirections of the form p q (respectively, p i q for some i), occurring in the right-hand side r, we have p = Root π and q Root π. for all actions of the form α : f (β 1,..., β n ), for all i 1..n, β i Root π the set of actions of the form α : f (β 1,..., β n ), appearing in r, do not construct a cycle including a defined operation.. Constraint C includes disequations of the form p = q where p and q are labeled by constructor symbols.

42 Admissible Inductively sequential Term-Graph Rewrite Systems [ICGT08][JICSLP98] In presence of Admissible Inductively sequential Term-Graph Rewrite Systems The set of admissible term-graphs is closed under the rewrite relation defined by admissible rules. Φ computes needed positions Admissible term-graphs admit unique normal forms

43 Narrowing w i σ w i+1 iff σ(w i ) w i+1 Rewriting = Matching + Transformation Narrowing = Unification + Transformation

44 Narrowing Motivation Automated deduction [Slagle 74] [Fay 79][Hullot 80] Functional and Logic Programming [Goguen and Meseguer 84,...] Security verification [Meadows 89,...] Reachability Analysis [Meseguer and Thati 05,...]...

45 Narrowing Instantiate goal variables and apply a reduction step 0 + X X s(x) + Y s(x + Y ) U + s(0) = s(s(0)) {U s(v )} s(v + s(0)) = s(s(0)) {V 0} s(s(0)) = s(s(0)) Computed answer: {U s(0)}

46 Some Results Needed Term narrowing [POPL04][JACM2000] (main operational semantics of current functionalogic programming languages) Needed Graph Narrowing [JICSLP98] Needed Collapsing Narrowing [Gratra 2000] Narrowing-based algorithm for data-structure rewriting [ICGT06] Goal o : equal(p : length(q), s(s(0))) = true Solution : a circular list of length two [q : cons(n 1, r : cons(n 2, q)) q r]

47 Narrowing: What do we transform? Rule o : f (p : a, q, r) p : b; o 3 q Rewrite Steps o 1, p 1, q 1 and r 1 are constants (names or addresses) o 1 : f (p 1 : a, q 1 : a, r 1 ) o 1 : f (p 1 : b, q 1 : a, q 1 ) o 1 : f (p 1 : a, p 1, r 1 ) o 1 : f (p 1 : b, p 1, p 1 )

48 Narrowing: What do we transform? Rule o : f (p : a, q, r) p : b; o 3 q Rewrite Steps (o 1, p 1, q 1 and r 1 are constants) o 1 : f (p 1 : a, q 1 : a, r 1 ) o 1 : f (p 1 : b, q 1 : a, q 1 ) o 1 : f (p 1 : a, p 1, r 1 ) o 1 : f (p 1 : b, p 1, p 1 ) Narrowing steps (o 2, p 2, q 2 and r 2 are variables) o 2 : f (p 2, q 2 : a, r 2 )? σ labels node p 2 with symbol a. o 2 : f (p 2, q 2 : a, r 2 ) σ o 2 : f (p 2 : b, q 2 : a, q 2 ) p 2 q 2 o 2 : f (p 2, q 2 : a, r 2 ) σ {q2 p 2 } o 2 : f (p 2 : b, p 2, p 2 )

49 Narrowing: What do we transform? Rule Narrowing steps (o 2, p 2, q 2 and r 2 are variables) o : f (p : a, q, r) p : b; o 3 q o 2 : f (p 2, q 2 : a, r 2 ) σ [apply(o 2 : f (p 2 : a, q 2 : a, r 2 ), p 2 : b; o 2 3 q 2 )] [apply(o 2 : f (p 2 : a, q 2 : a, r 2 ), p 2 : b; o 2 3 q 2 ) p 2 q 2 ] [apply(o 2 : f (p 2 : b, q 2 : a, r 2 ), o 2 3 q 2 ) p 2 q 2 ] [apply(o 2 : f (p 2 : a, q 2 : a, q 2 ), ɛ) p 2 q 2 ] [o 2 : f (p 2 : a, q 2 : a, q 2 ) p 2 q 2 ]

50 Narrowing: What do we transform? Rule Narrowing steps o 2, p 2, q 2 and r 2 are variables o : f (p : a, q, r) p : b; o 3 q o 2 : f (p 2, q 2 : a, r 2 ) σ [apply(o 2 : f (p 2 : a, q 2 : a, r 2 ), p 2 : b; o 2 3 q 2 )] {q2 p 2 } [apply(o 2 : f (p 2 : a, p 2, r 2 ), p 2 : b; o 2 3 q 2 )] [apply(o 2 : f (p 2 : b, p 2, r 2 ), o 2 3 q 2 )] [apply(o 2 : f (p 2 : b, p 2, p 2 ), ɛ)] o 2 : f (p 2 : b, p 2, p 2 )

51 g-terms Symbolic handling of actions G is a term-graph φ is a conjunction of disequations τ is a sequence of actions [G φ] [apply(g, τ) φ] Example: [o 2 : f (p 2 : a, q 2 : a, q 2 ) p 2 q 2 ] [apply(o 2 : f (p 2 : b, q 2 : a, r 2 ), o 2 3 q 2 ) p 2 q 2 ]

52 Graph Narrowing Rules Superposition rule (SUP) If: G is a term-graph [G ψ] τ SUP,ρ,θ [H ψ ] σ(τ) ρ is rewrite rule [L φ] R σ is a most general unifier of L and G such that: σ(l) and σ(g) are compatible The root of L unifies with a labeled node in G (non-variable unification) H = apply(σ(g) σ(l), σ(r)) θ = (σ, K ) with K = σ(l)\σ(g) ψ = σ(ψ) σ(φ). p affected p = q. by(σ(τ)),q K Ω

53 Graph Narrowing Rules Action rule: simplify (SIM) [apply(g, ɛ) ψ] τ SIM [G ψ] τ

54 Graph Narrowing Rules Action rule: execute (EXE) [apply(g, α.u) ψ] τ EXE [apply(α[g], u) ψ ] τ.α If: action α is not a node creation and [G ψ] is ready for action α.

55 Graph Narrowing Rules Action rule: new node (NEW) [apply(g, n +.u) ψ] τ NEW,σ [apply(n + [G], σ(u)) ψ ] τ.n+ If: σ = {n n } where n is a fresh effective node ψ = ψ p V N G (p n )

56 Graph Narrowing Rules Isolation rule with equality (EQU) [apply(g, α.u) ψ] τ EQU,σ [apply(σ(g), σ(α).σ(u)) σ(ψ)] σ(τ) If: there exists a node n affected by(α), m is not an α-isolated node in G and σ is a substitution (compatible with G) such that σ(n) = σ(m) o 2 : f (p 2, q 2 : a, r 2 ) σ [apply(o 2 : f (p 2 : a, q 2 : a, r 2 ), p 2 : b; o 2 3 q 2 )] {q2 p 2 } [apply(o 2 : f (p 2 : a, p 2, r 2 ), p 2 : b; o 2 3 q 2 )]

57 Graph Narrowing Rules Isolation rule with disequality (DIS) If: [apply(g, α.u) ψ] τ. DIS [apply(g, α.u) ψ n = m] τ there exists a node n affected by(α), m is not an α-isolated node in G o 2 : f (p 2, q 2 : a, r 2 ) σ [apply(o 2 : f (p 2 : a, q 2 : a, r 2 ), p 2 : b; o 2 3 q 2 )] [apply(o 2 : f (p 2 : a, q 2 : a, r 2 ), p 2 : b; o 2 3 q 2 ) p 2 q 2 ]

58 Computed Solutions [G 0 True] σ1 σn [G n φ] Computed Solution is : (σ 1 σ n, φ) Example: Goal o : equal(p : length(q), s(s(0))) = true Solution [q : cons(n 1, r : cons(n 2, q)) q r]

59 Graph Narrowing: Soundness and Completeness Proposition 1: The proposed narrowing rules are sound. If [G True] σ [H φ] then there exists a ground substitution θ satisfying φ such that: Gσθ Hθ Proposition 2: The proposed narrowing rules are complete. If Gσ H, σ being irreducible. Then, there exist two substitutions θ and γ and a term-graph G such that : [G True] θ[g φ] γ satifies φ σ = θγ G γ = H

60 Modal Logic and Graph Transformation motivations Specify graph shapes (data-structures) Circular list Balanced tree Graph properties can be specified within several logics, such as : Separation Logic, Monadic second order logic, Modal logics (e.g., LTL, CTL, µ-calculus, etc). Verification of graph transformation: Invariant Reachability Need to define new logics able to specify rule application and graph transformation.

61 Dynamic Logic Agents Knowledge Actions Evaluate a formula in a model Transform the considered model

62 A Modal Logic for Graph Rewriting G = φ G! R = φ where G! R is the normal form of G G! R = φ iff G = [R ]φ

63 A Modal Logic for Graph Rewriting : L gr Language Formulas : φ ::= p φ φ φ [α]φ Actions : α ::= a α α; α α α modifiers. [α]φ : After performing actions α, formula Φ holds.

64 Modifiers Add a new node Add a new label (to current node) remove a label from the current node Add the label a to the edges going from a Φ-node to a Ψ-node.... Graph modifiers : U n n φ? (ω := g φ) (ω := l φ) (a + (φ, ψ)) (a (φ, ψ))

65 Modifiers Example [p := g ][p := l ](p [a]( p q)) p a p, q a q p a q a q a q a q

66 Modal Logic: L gr Semantics (informally) G r = p iff p holds at node r G r = [a]ϕ iff G n = ϕ for all nodes n such that the edge r a n G. G r = [ω := g ][ω := l ] ϕ iff H r = ϕ, H is obtained from G by tagging the node r by ω (ω does not hold outside node r).

67 Modal Logic: L gr Semantics G r = [a (φ, ψ)]ϕ iff H r = ϕ, H is obtained from G by erasing the edges n a m, such that G n = φ and G m = ψ. G r = [a + (φ, ψ)]ϕ iff H r = ϕ, H is obtained from G by adding the edges n a m, such that G n = φ and G m = ψ. G r = [f?]ϕ iff G r = ϕ and f holds at node r. G r = [ nw]ϕ iff H nw = ϕ. H is obtained from G by adding a new node nw.

68 Examples of L gr Specified Properties Class of all a-cycle-free rooted termgraphs. [ω := g ][U][ω := l ][a + ]ω Class of all a-circular rooted termgraphs [ω := g ][U][ω := l ] a + ω. Class of all (a, b)-binary rooted termgraphs [ω := g ][U][ω := l ][a][π := g ][(a b) ][π := l ][U](ω [b][(a b) ]π). a Let R G (a) = {(n 1, n 2 ): the edge n 1 n2 G}. G = [ω := g ][U][ω := l ][a] ω iff R G (a) is irreflexive. Classes of circular lists, balanced trees,...

69 Hamiltonian Graphs The following formula expresses the existence of a Hamiltonian cycle. α stands for a 1... a n, where the a i s are the possible features used in the graph (F = {a 1,..., a n }): ω := g ; π := g ; ω := l ; π := l ; (α; ω?; ω := l ) (π [U] ω).

70 Decidability With * and without nw : the problem of model checking (G = Φ) is decidable.

71 Modal Logic L gr and Graph Rewriting Expressing pattern-matching in L gr Proposition: Let G r be a term-graph with root r (a distinguished node). There exists a -free action α G and a -free formula φ G such that for all finite rooted termgraphs G r, G r = α G φ G iff there exists a graph homomorphism from G to G r. We define the action α G and the formula φ G as follows: β G = (π 0 := g );... ; (π N 1 := g ), (N being the number of nodes in G) for all non-negative integers i, if i < N then γ i G = ( π 0... π i 1 )?; (π i := l ); U, α G = β G ; γ 0 G ;... ; γn 1 G,

72 Modal Logic L gr and Graph Rewriting We define the formula φ G as follows: for all non-negative integers i, if i < N then ψ i G = if Ln (i) is defined then U (π i L n (i)) else, for all non-negative integers i, j, if i, j < N then χ i,j G = if there exists an edge e E such that S(e) = i and T (e) = j then U (π i L e (e) π j ) else, φ G = ψg 0... ψn 1 G χ 0,0 G... χn 1,N 1 G.

73 Modal Logic L gr and Graph Rewriting Actions representing the right-hand sides can be expressed by the following elementary formulas : Action n : f (a 1 n 1,..., a k n k ) U; π n?; (f := l ); (a 1 + (π n, π n1 ));... ; (a k + (π n, π nk )). Action n a m (a (π n, )); (a + (π n, π m )). Action n m) (for a-edges) (λ a := g ); (λ a := g a π n ); (a (, π n )); (a + (λ a, π m )).

74 Modal Logic L gr and Graph Rewriting Firing a rule ρ = L R β ρ = α L ; α R Normal form of graph G r satisfies ϕ: Let R = ( β ρi ) G r = [R ]([R] ϕ) Rule ρ preserves the property ϕ: = (ϕ [β ρ ]ϕ)

75 Conclussion and perspectives Admissible termgraphs seem to be a good trade-off to ensure confluence and efficient strategies Cloning and algebraic approaches (sesqui-pushout) Narrowing Visual Programming and Termgraph Rewriting Proof Techniques Applications