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Techniques de rcriture et transformations Horatiu CIRSTEA and Claude KIRCHNER 27 janvier 2007 Horatiu CIRSTEA and Claude KIRCHNER Techniques de rcriture et transformations 27 janvier 2007 1 / 166 Mathematics is frequently described as

  1. Defining term rewriting Rewriting Definition of rewriting It relies on 5 notions : ➲ The objects : terms and rewrite rules ➲ The actions matching substitutions replacement and, given a rule and a term, it consists in : ➥ finding a subterm of the term ➥ that matches the left hand side of the rule ➥ and replacing that subterm by the right hand side of the rule instanciated by the match Formally Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 21 / 166

  2. Defining term rewriting Extended notions of rewriting Conditional rules l → r if c l , r ∈ T ( F , X ) , c a boolean term V ar ( r ) ∪ V ar ( c ) ⊆ V ar ( l ) The rule applies on a term t provided the matching substitution σ allows c σ to reduce to true . Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 22 / 166

  3. Defining term rewriting Extended notions of rewriting Applying a conditional rewrite rule even ( 0 ) → true → even ( s ( x )) odd ( x ) odd ( x ) → true if not ( even ( x )) → odd ( x ) false if even ( x ) even ( s ( 0 )) − → odd ( 0 ) − → false Generalized rules Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 23 / 166

  4. Defining term rewriting On the use of rewriting Expressiveness of rewriting [Max Dauchet 1989] A Turing machine can be simulated by a single rewrite rule. This unique rewrite rule can further be left linear and regular ! Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 24 / 166

  5. Defining term rewriting On the use of rewriting On the use of term rewriting for programming (ASF , ELAN, MAUDE, ML, OBJ, Stratego, TOM, . . . ) for proving (Completion procedures, proof systems, . . . ) for solving (Constraint manipulations, . . . ) for verifying (exhaustive (intelligent) search) Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 25 / 166

  6. Defining term rewriting Rewriting modulo Matching modulo Finding a substitution σ such that σ ( l ) = t is called the matching problem from l to t (denoted l ≪ ? t ). Finding a substitution σ such that σ ( l ) = E t is called the matching problem from l to t (denoted l ≪ ? E t ). Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 26 / 166

  7. Defining term rewriting Rewriting modulo Examples (commutative symbol(s)) F = { a ( 0 ) , b ( 0 ) , c ( 0 ) , f ( 2 ) , g ( 2 ) , h ( 1 ) } f is assumed to be commutative (the other symbols have no property). C ( f ) : ∀ x , y ∈ T ( F , X ) f ( x , y ) = f ( y , x ) f ( a , b ) = f ( b , a ) — yes g ( f ( a , b ) , a ) = g ( f ( b , a ) , a ) — yes g ( f ( a , b ) , a ) = g ( a , f ( b , a )) — no f ( a , f ( a , b )) = f ( f ( b , a ) , a ) — yes f ( a , f ( b , c )) = f ( f ( c , b ) , a ) — yes f ( f ( a , b ) , c ) = f ( a , f ( b , c )) — no Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 27 / 166

  8. Defining term rewriting Rewriting modulo Matching modulo C : examples Solve the following problems : f ( x , y ) ≪ ? C f ( a , b ) σ = { x �→ a , y �→ b } σ = { x �→ b , y �→ a } f ( y , f ( x , x )) ≪ ? C f ( f ( f ( a , b ) , f ( b , a )) , f ( b , a )) σ = { x �→ a , y �→ f ( f ( a , b ) , f ( b , a )) } σ = { x �→ b , y �→ f ( f ( a , b ) , f ( b , a )) } σ = { x �→ f ( a , b ) , y �→ f ( a , b ) } Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 28 / 166

  9. Defining term rewriting Rewriting modulo Matching modulo C : A rule based description t ≪ ? t ∧ P Delete �→ �→ P f ( t 1 , . . . , t n ) ≪ ? f ( t ′ 1 , . . . , t ′ Decomposition n ) ∧ P � i = 1 ,..., n t i ≪ ? t ′ �→ �→ i ∧ P f ( t 1 , . . . , t n ) ≪ ? g ( t ′ 1 , . . . , t ′ m ) ∧ P SymbolClash �→ �→ Fail if f � = g f ( t 1 , . . . , t n ) ≪ ? x ∧ P SymbolVariableClash �→ �→ Fail if x ∈ X x ≪ ? t ∧ x ≪ ? t ′ ∧ P MergingClash if t � = t ′ �→ �→ Fail Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 29 / 166

  10. Defining term rewriting Rewriting modulo Assume + commutative t 1 + t ′ C t ′ 1 + t ′ 2 ≪ ? C − Decomposition 2 ∧ P ( t 1 ≪ ? C t ′ 1 ∧ t 2 ≪ ? C t ′ 2 ) ∨ ( t 1 ≪ ? C t ′ 2 ∧ t 2 ≪ ? C t ′ �→ �→ 1 ) Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 30 / 166

  11. Defining term rewriting Rewriting modulo Find a match x ∗ ( 3 + y ) ≪ ? C 1 ∗ ( 4 + 3 ) → Decomposition x ≪ ? C 1 ∧ 3 + y ≪ ? C 4 + 3 → C (+) − Decomposition x ≪ ? C 1 ∧ (( 3 ≪ ? C 4 ∧ y ≪ ? C 3 ) ∨ ( 3 ≪ ? C 3 ∧ y ≪ ? C 4 )) → MergingClash x ≪ ? C 1 ∧ ( Fail ∨ ( 3 ≪ ? C 3 ∧ y ≪ ? C 4 )) → Delete x ≪ ? C 1 ∧ ( Fail ∨ ( y ≪ ? C 4 )) → Bool x ≪ ? C 1 ∧ y ≪ ? C 4 Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 31 / 166

  12. Defining term rewriting Rewriting modulo Matching rules Does it terminate ? Do we always get the same result ? Theorem The normal form by the rules in Commutative − Match , of any matching problem t ≪ ? t ′ such that V ar ( t ) ∩ V ar ( t ′ ) = ∅ , exists and is unique. If it is Fail , then there is no match from t to t ′ . 1 If it is of the form � � i ∈ I x k i ≪ ? C t k with I , K � = ∅ , the 2 k ∈ K i substitutions σ k = { x k i �→ t k i } i ∈ I are all the matches from t to t ′ . If it is empty then t and t ′ are identical : t = t ′ . 3 Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 32 / 166

  13. Defining term rewriting Rewriting modulo Matching modulo associativity-commutativity (1) ∪ is assumed to be an associative commutative (AC) symbol : ∀ x , y , z , ∪ ( x , ∪ ( y , z )) = ∪ ( ∪ ( x , y ) , z ) and ∀ x , y , ∪ ( x , y ) = ∪ ( y , x ) . { i } ∪ s ≪ ? AC { 1 } ∪ { 2 } ∪ { 3 } ∪ { 4 } ∪ { 5 } { 1 } ∪ { 2 } ∪ { 3 } ∪ { 4 } ∪ { 5 } = AC { 2 } ∪ { 3 } ∪ { 4 } ∪ { 5 } ∪ { 1 } = AC . . . { 5 } ∪ { 1 } ∪ { 2 } ∪ { 3 } ∪ { 4 } 5 different and non AC -equivalent matches. Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 33 / 166

  14. Defining term rewriting Rewriting modulo Rewriting modulo : definition A class rewrite system R / A is composed of a set of rewrite rules R and a set of equalities A , such that A and R are disjoint sets. → x + 0 x x + ( 0 + y ) → x + y x + ( − x ) → 0 x + (( − x ) + y ) → y − − x → x − 0 → 0 − ( x + y ) → ( − x ) + ( − y ) x + y = y + x ( x + y ) + z = x + ( y + z ) Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 34 / 166

  15. Defining term rewriting Rewriting modulo → R / A t ( R / A ) -rewrites to t ′ if t = A t 1 → R t 2 = A t ′ To be more effective, consider any relation → RA such that : → R ⊆ → RA ⊆ → R / A Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 35 / 166

  16. Defining term rewriting Rewriting modulo → R , A A term rewrite system R (a set of rewrite rules) determines a relation on terms denoted − → R , A [Peterson & Stickel,81] u → R , A v iff there exist l → r ∈ R , an occurrence ω in t , such that u | ω = A σ ( l ) and v = u [ σ ( r )] ω USUALLY, when defining the rewriting relation, one requires the all rewrite rules satisfy Var ( r ) ⊆ Var ( l ) . Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 36 / 166

  17. Defining term rewriting Rewriting modulo For example Let ∪ be an AC symbol, such that { i } ∪ x → i { 1 } ∪ { 2 } ∪ { 3 } ∪ { 4 } ∪ { 5 } = AC { 2 } ∪ { 3 } ∪ { 4 } ∪ { 5 } ∪ { 1 } = AC . . . { 5 } ∪ { 1 } ∪ { 2 } ∪ { 3 } ∪ { 4 } Since this term matches the lefthand side of the rewriting rule in 5 different and non AC -equivalent ways, the rewrite rule applies in 5 different ways. Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 37 / 166

  18. Defining term rewriting Rewriting modulo Examples Assume + to be AC (associative and commutative) R = { a + a → a } R / E -rewrite the term ( a + c ) + a a+c R , E -rewrite the term ( a + c ) + a R = { a + a → a ( a + a ) + x → a + x } R / E -rewrite the term ( a + c ) + a a+c R , E -rewrite the term ( a + c ) + a a+c Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 38 / 166

  19. Rewriting for verifying A smooth introduction 1 2 Defining term rewriting Terms Matching Rewriting Extended notions of rewriting On the use of rewriting Rewriting modulo 3 Rewriting for verifying 4 Properties of term rewrite systems Abstract rewrite systems Termination of TRS Confluence of TRS 5 Rewriting calculus Introduction Syntax and semantics Expressiveness Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 39 / 166

  20. Rewriting for verifying Communication protocol : is this a good one ? Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 40 / 166

  21. Rewriting for verifying may be not . . . Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 41 / 166

  22. Rewriting for verifying Needham-Schroeder public-key protocol The Needham-Schroeder public-key protocol aims to establish a mutual authentication between an initiator and a responder that communicate via an insecure network. Initiator Responder Net A → B : { N A , A } K ( B ) 1. A SLEEP B SLEEP ∅ A WAIT B SLEEP { N A , A } K ( B ) B → A : { N A , N B } K ( A ) B { N A , A } K ( B ) 2. A WAIT ∅ A WAIT B WAIT { N A , N B } K ( A ) 3. A → B : { N B } K ( B ) A { N A , N B } K ( A ) B WAIT ∅ A COMMIT B WAIT { N B } K ( B ) 4. N B is the session key A COMMIT B COMMIT ∅ Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 42 / 166

  23. Rewriting for verifying Rewrite rules for the agents ➲ initiator starts the communication with a responder [initiator-1] x+SLEEP+resp || E <> y+std+init || D <> I <> ls => x+WAIT+N(x,y) || E <> y+std+init || D <> I <> x-->y:K(y)[N(x,y),DN,A(x)] & ls end Data structures Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 43 / 166

  24. Rewriting for verifying Rewrite rules for the agents ➲ responder reads the message and sends the acknowledgement [responder-1] E <> y+SLEEP+init || D <> I <> w-->y:K(y)[N(n1,n3),N(n2,n4),A(z)] & ls=> E <> y+WAIT+N(y,z) || D <> I <> y-->z:K(z)[N(n1,n3),N(y,z),A(y)] & ls end Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 44 / 166

  25. Rewriting for verifying Rewrite rules for the agents ➲ initiator receives the acknowledgement and checks its validity [initiator-2] x+WAIT+N(x,v) || E <> D <> I <> w-->x:K(x)[N(x,v),N(n2,n4),A(z)] & ls => x+COMMIT+N(x,v) || E <> D <> I <> x-->v:K(v)[N(n2,n4),DN,DA] &ls end [initiator-2] x+WAIT+N(x,v) || E <> D <> I <> w-->x:K(x)[N(n1,n3),N(n2,n4),A(z)] &ls => ERROR if x!=n1 or v!=n3 end Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 45 / 166

  26. Rewriting for verifying Rewrite rules for the intruder ➲ the intruder intercepts all the messages in the network but the messages generated by itself and stores or decrypts them. [intruder-1] E <> D <> w#l#ll <> z-->x:K(w)[N(n1,n3),N(n2,n4),A(v)] & ls => E <> D <> w#N(n1,n3) | N(n2,n4) | l#ll <> ls if w!=z end [intruder-1] E <> D <> w#l#ll <> !w-->x:K(w)[N(n1,n3),N(n2,n4),A(v)] & ls => E <> D <> w#N(n1,n3) | N(n2,n4) | l#ll <> ls end Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 46 / 166

  27. Rewriting for verifying Rewrite rules for the intruder ➲ the nonces obtained previously by the intruder are used in order to generate fake messages that are sent to all the agents. [intruder-4] E <> D <> w # resp | l # ll <> ls => E <> D <> w # l # ll <> w-->y:K(y)[resp,DN,A(xadd)] & ls where (Agent)y+std+dn :=(extAgent) elemIA(D || E) where (Agent)xadd+std1+dn1 :=(extAgent) elemIA(D || E) end Generalized rules Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 47 / 166

  28. Rewriting for verifying The invariants ➲ authenticity of the responder : if an initiator x committed with a responder y , then y has really been involved in the protocol. [attack-1] x+COMMIT+N(x,y) || E <> D <> i#l#ll <> ls => ATTACK if y!=i if not(existAgent(y+WAIT+N(y,x),D)) and not(existAgent(y+COMMIT+N(y,x),D)) end ➲ authenticity of the initiator : if a responder y committed with an initiator x then the initiator have committed as well with y . [attack-2] E <> y+COMMIT+N(y,x) || D <> i#l#ll <> ls => ATTACK if x!=i if not(existAgent(x+COMMIT+N(x,y),E)) end Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 48 / 166

  29. Rewriting for verifying The strategy We apply repeatedly all the rewrite rules in any order and in all the possible ways until one of the attack rules can be applied. []attStrat => repeat*( dk( attack-1, attack-2, intruder-1, intruder-2, intruder-3, intruder-4, initiator-1, initiator-2, responder-1, responder-2 ) ); attackFound end where [attackFound] ATTACK => ATTACK end Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 49 / 166

  30. Rewriting for verifying The attack I.1. A → I : { N A , A } K ( I ) II.1. I ( A ) → B : { N A , A } K ( B ) B → I ( A ) : { N A , N B } K ( A ) II.2. I.2. I → A : { N A , N B } K ( A ) I.3. A → I : { N B } K ( I ) I ( A ) → B : { N B } K ( B ) II.3. Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 50 / 166

  31. Rewriting for verifying The corrected protocol 1. A → B : { N A , A } K ( B ) B → A : { N A , N B , B } K ( A ) 2. A → B : { N B } K ( B ) 3. Modified rule : initiator-2 [initiator-2] x+WAIT+N(x,v) || E <> D <> I <> w-->x:K(x)[N(x,v),N(n2,n4),A(v)] & ls => x+COMMIT+N(x,v) || E <> D <> I <> x-->v:K(v)[N(n2,n4),DN,DA] & ls end Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 51 / 166

  32. Properties of term rewrite systems A smooth introduction 1 2 Defining term rewriting Terms Matching Rewriting Extended notions of rewriting On the use of rewriting Rewriting modulo 3 Rewriting for verifying 4 Properties of term rewrite systems Abstract rewrite systems Termination of TRS Confluence of TRS 5 Rewriting calculus Introduction Syntax and semantics Expressiveness Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 52 / 166

  33. Properties of term rewrite systems Abstract rewrite systems Think abstractly The properties of this relation could be studied in an abstract way : ⇒ Abstract rewrite systems Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 53 / 166

  34. Properties of term rewrite systems Abstract rewrite systems Showing normalization A (partial) order on T is a reflexive, antisymetric and transitive relation. An ordering is total on T when two terms are always comparable > is well-founded or Noetherian on T if there is no infinite decreasing sequence on T : t 1 > t 2 > t 3 > . . . Theorem Consider an ARS ( A , → ). → is terminating iff there exists a well-founded (partial) order > on T and a mapping φ s.t. for all rewrite rule a → a ′ implies φ ( a ) > φ ( a ′ ) . Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 54 / 166

  35. Properties of term rewrite systems Abstract rewrite systems Example Use the order ( > , N ) which is well-founded. Several choices for strings A = ( • | ◦ ) ∗ φ ( w ) = number of • works for all • -decreasing reductions φ ( w ) = number of ◦ works for all ◦ -decreasing reductions φ ( w ) = number of • and ◦ works for all length-decreasing reductions Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 55 / 166

  36. � � � � � � � � � � � � Properties of term rewrite systems Abstract rewrite systems Relathionships ) Definitions ( Diamond property (DP) Localy confluent (LC) t 0 t 0 � � � � ������� � ������� � � � � � � � � � � � t 1 t 2 t 1 t 2 ∗ ∗ t 3 t 3 Confluent (C) Church Rosser (CR) t 0 + � s t � � � ∗ ∗ � � � � � � � � � � � � ∗ ∗ t 1 t 2 u ∗ ∗ t 3 Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 56 / 166 Relathionships

  37. � � � Properties of term rewrite systems Abstract rewrite systems Local versus global confluence C ⇒ LC 1 LC ⇒ C ? 2 ➥ Consider four distinct �� �� � elements a , b , c , d of T �� �� a b and the relation : a → b b → a c d a → c b → d Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 57 / 166

  38. Properties of term rewrite systems Abstract rewrite systems Newman’s lemma [Newman 1942] Provided the relation → is terminating then → is confluent iff it is locally confluent Proof : locally confluent if confluent ➥ obvious confluent if locally confluent ➥ ? Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 58 / 166

  39. Properties of term rewrite systems Termination of TRS Termination R (or → R ) terminates iff all derivation issued from any term terminate. Termination implies the existence of normal form(s) for any term. Termination is in general undecidable but interesting sufficient condition can be found. Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 59 / 166

  40. Properties of term rewrite systems Termination of TRS Proving termination could be tricky . . . f ( a , b , x ) → f ( x , x , x ) is terminating → g ( x , y ) x g ( x , y ) → y , is terminating Is the union terminating ? Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 60 / 166

  41. � Properties of term rewrite systems Termination of TRS f ( a , b , x ) → f ( x , x , x ) g ( x , y ) → x g ( x , y ) → y , We have the derivation : �� �� � f ( a , g ( a , b ) , g ( a , b )) � f ( a , b , g ( a , b )) f ( g ( a , b ) , g ( a , b ) , g ( a , b )) [Toyama 1986] Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 61 / 166

  42. Properties of term rewrite systems Termination of TRS Orderings on terms A Reduction ordering is an ordering on T , stable by context and substitution : ➥ for every context C [ _ ] and for all substitutions σ , if t > s then C [ t ] > C [ s ] and σ ( t ) > σ ( s ) . Theorem R terminates iff there exists a well-founded reduction ordering > s.t. for all rewrite rule ( l → r ) ∈ R , l > r . Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 62 / 166

  43. Properties of term rewrite systems Termination of TRS Example The rules of the game : •• → ◦ ◦◦ → ◦ •◦ → • ◦• → • l > r if | l | > | r | l > r if | l | •◦ > | r | •◦ ( | t | •◦ =number of • and ◦ of the term t built out of • and ◦ ) | f ( f ( x , x ) , y ) | > f ( y , y ) but | f ( f ( x , x ) , f ( x , x )) | � > f ( f ( x , x ) , f ( x , x )) Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 63 / 166

  44. Properties of term rewrite systems Termination of TRS Example The rules of the game : •• → ◦◦ ◦◦ → ◦ •◦ → • ◦• → • l > r if | l | •◦ > | r | •◦ | • •| •◦ = 2 � > 2 = | ◦ ◦| •◦ l > r if | l | • > | r | • | ◦ ◦| • = 0 � > 0 = | ◦ | • Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 64 / 166

  45. Properties of term rewrite systems Termination of TRS Example The rules of the game : •• → ◦◦ ◦◦ → • •◦ → • ◦• → • l > r if | l | •◦ > | r | •◦ | • •| •◦ = 2 � > 2 = | ◦ ◦| •◦ l > r if | l | • > | r | • | ◦ ◦| •◦ + • = 2 � > 2 = | • | •◦ + • Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 65 / 166

  46. Properties of term rewrite systems Termination of TRS Lexicographical extensions Let > be an ordering on T . Its lexicographical extension > lex on T n is defined as : ( s 1 , . . . , s n ) > lex ( t 1 , . . . , t n ) if there exists i , 1 ≤ i ≤ n s.t. s i > i t i , and ∀ j , 1 ≤ j < i , s j = t j . If > is well-founded on T , then > lex is well-founded on T n . FALSE for an infinite product of ordered sets : T = { a , b } with a < b b > lex ab > lex aab > lex aaab > lex . . . Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 66 / 166

  47. Properties of term rewrite systems Termination of TRS Well-founded reduction orderings Syntactic Based on the precedence concept (i.e. a partiel order > F on F ) Recursive or Lexicographic path ordering [Dershowitz, 82] Example : Semantic Terms are interpreted in another structure where a well-founded ordering is known (e.g. the natural numbers) Example : Polynomial interpretations Combinations Ordering combining semantical and syntactical behavior Recursion analysis Induction, dependency pairs Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 67 / 166

  48. Properties of term rewrite systems Confluence of TRS Confluence Allows us to forget about non-determinism : Whatever rewriting is done we will converge later. Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 68 / 166

  49. Properties of term rewrite systems Confluence of TRS Back with the simple game The rules of the game : •• → ◦ ◦◦ → ◦ •◦ → • ◦• → • A starting point : • ◦ • ◦ • ◦ • • • • ◦ ◦ • ◦ ◦ • •◦ From a given start, is the result determinist ? Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 69 / 166

  50. Properties of term rewrite systems Confluence of TRS Analysing the different cases Disjoint redexes : · · · ⊗⊗ · · · ⊗ ⊗ · · · · · · ⊗ · · · ⊗⊗ · · · · · · ⊗ · · · ⊗ · · · is the same as : · · · ⊗ ⊗ · · · ⊗⊗ · · · · · · ⊗⊗ · · · ⊗ · · · · · · ⊗ · · · ⊗ · · · Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 70 / 166

  51. Properties of term rewrite systems Confluence of TRS No disjoint redexes (central black) : · · · ◦• • · · · · · · •• • · · · · · · •• · · · · · · ◦• · · · · · · ◦ · · · · · · • · · · but · · · ◦ •• · · · · · · • •• · · · · · · ◦◦ · · · · · · •◦ · · · · · · ◦ · · · · · · • · · · or · · · ◦• ◦ · · · · · · •• ◦ · · · · · · •◦ · · · · · · ◦◦ · · · · · · • · · · · · · ◦ · · · but · · · ◦ •◦ · · · · · · • •◦ · · · · · · ◦• · · · · · · •• · · · · · · • · · · · · · ◦ · · · Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 71 / 166

  52. Properties of term rewrite systems Confluence of TRS No disjoint redexes (central white) : · · · ◦◦ • · · · · · · •◦ • · · · · · · ◦• · · · · · · •• · · · · · · • · · · · · · ◦ · · · but · · · ◦ ◦• · · · · · · • ◦• · · · · · · ◦• · · · · · · •• · · · · · · • · · · · · · ◦ · · · or · · · ◦◦ ◦ · · · · · · •◦ ◦ · · · · · · ◦◦ · · · · · · •◦ · · · · · · ◦ · · · · · · • · · · but · · · ◦ ◦◦ · · · · · · • ◦◦ · · · · · · ◦◦ · · · · · · •◦ · · · · · · ◦ · · · · · · • · · · Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 72 / 166

  53. � � � � Properties of term rewrite systems Confluence of TRS t 0 t 0 � � � ������� � � ������� � � ∗ � ∗ � � � � � � � � Thus : t 1 t 2 but what about : t 1 t 2 � � � � � � � � � � � � � � ∗ ∗ � � � � t 3 t 3 Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 73 / 166

  54. Properties of term rewrite systems Confluence of TRS Confluence ➥ Undecidable in general, confluence is decidable for finite and terminating rewrite systems. ➥ Assuming termination of the rewrite relation, its confluence is critical pairs . equivalent to the confluence of orthogonal (linear and non-overlapping), then it is ➥ If a rewrite system is confluent. Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 74 / 166

  55. Properties of term rewrite systems Confluence of TRS Other systems What if the system is non-teminating and non-orthogonal ? Theorem Consider a reduction relation → R and let → D s.t. ∗ → R ⊆ → D ⊆ → R → D has the diamond property Then, → R is confluent. Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 75 / 166

  56. Properties of term rewrite systems Confluence of TRS References on rewriting modulo G. Huet. Confluent reductions : Abstract properties and applications to term rewriting systems. Journal of the ACM , 27(4) :797–821, October 1980. G. Peterson and M. E. Stickel. Complete sets of reductions for some equational theories. Journal of the ACM , 28 :233–264, 1981. J.-P . Jouannaud and Hélène Kirchner. Completion of a set of rules modulo a set of equations. SIAM Journal of Computing , 15(4) :1155–1194, 1986. Enno Ohlebusch. Church-Rosser Theorems for Abstract Reduction Modulo an Equivalence Relation RTA, pages 17-31, LNCS 1379, 1998. Claude and Hélène Kirchner. Rewriting Solving Proving www.loria.fr/˜ckirchne/rsp.ps.gz Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 76 / 166

  57. Rewriting calculus A smooth introduction 1 2 Defining term rewriting Terms Matching Rewriting Extended notions of rewriting On the use of rewriting Rewriting modulo 3 Rewriting for verifying 4 Properties of term rewrite systems Abstract rewrite systems Termination of TRS Confluence of TRS 5 Rewriting calculus Introduction Syntax and semantics Expressiveness Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 77 / 166

  58. Rewriting calculus Introduction Why a new calculus ? Rewriting is nice, but the rewrite relation or the rewriting logic are difficult to control non-reducibility is impossible to express Lambda-calculus is great, but lacks of discrimination capabilities Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 78 / 166

  59. Rewriting calculus Introduction A “simple” λ -term. . . � � � � ( λ Y . ( λ y . ( yx ⊥ ( λ X . X ))) Y ) ( λ x 1 .λ z 1 λ z 2 . ( z 2 x 1 ))( λ u 1 λ u 2 . u 1 ) Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 79 / 166

  60. Rewriting calculus Introduction . . . and its meaning � λ Y . ( f ( X ) � X ) Y � � f ( a ) � � �� � � �� � � f ( X ) � X � � f � � a � � � � � � �� � � �� � � �� � ( λ Y . ( λ y . ( yx ⊥ ( λ X . X ))) Y ) ( λ x 1 .λ z 1 λ z 2 . ( z 2 x 1 )) ( λ u 1 λ u 2 . u 1 ) Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 80 / 166

  61. Rewriting calculus Introduction Simple encoding of rewriting in the λ -calculus � λ Y . ( f ( X ) � X ) Y � � f ( a ) � � �� � � �� � � f ( X ) � X � � f � � a � � � � � � �� � � �� � � �� � ( λ Y . ( λ y . ( yx ⊥ ( λ X . X ))) Y ) ( λ x 1 .λ z 1 λ z 2 . ( z 2 x 1 )) ( λ u 1 λ u 2 . u 1 ) �� � � �→ λ Y . ( Yx ⊥ ( λ X . X )) ( λ x 1 .λ z 1 λ z 2 . ( z 2 x 1 ))( λ u 1 λ u 2 . u 1 ) β � �� � �→ λ Y . ( Yx ⊥ ( λ X . X )) λ z 1 λ z 2 . ( z 2 ( λ u 1 λ u 2 . u 1 )) β � � �→ λ z 1 λ z 2 . ( z 2 ( λ u 1 λ u 2 . u 1 )) x ⊥ ( λ X . X ) β � � �→ λ z 2 . ( z 2 ( λ u 1 λ u 2 . u 1 )) ( λ X . X ) β �→ ( λ X . X )( λ u 1 λ u 2 . u 1 ) β �→ ( λ u 1 λ u 2 . u 1 ) β = � a � Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 81 / 166

  62. � ��� Rewriting calculus Introduction Term rewriting f ( x , y ) → x = ⇒ R f ( a , b ) a Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 82 / 166

  63. Rewriting calculus Introduction Rewriting calculus - abstraction f ( X , Y ) � X ➥ Abstraction Operator Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 83 / 166

  64. Rewriting calculus Introduction Rewriting calculus - application � � f ( X , Y ) � X f ( a , b ) ➥ Application Operator Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 84 / 166

  65. � � � � � � � � � � � � � � Rewriting calculus Introduction Rewriting calculus - compute the substitution � � f ( X , Y ) � f ( a , b ) X � � � � � � � � � � � � � � � � σ Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 85 / 166

  66. � Rewriting calculus Introduction Rewriting calculus - replacement � � � f ( X , Y ) f ( a , b ) X σ = { X �→ a , Y �→ b } Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 86 / 166

  67. Rewriting calculus Introduction Rewriting calculus - result � � f ( X , Y ) � f ( a , b ) X �→ σ ( X ) i.e. a Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 87 / 166

  68. � ��� Rewriting calculus Introduction For the rewriting relation f ( x , y ) → x = ⇒ R g ( a , b ) Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 88 / 166

  69. � � � � � � � � � � � � � � Rewriting calculus Introduction For the rewriting calculus � � � f ( X , Y ) g ( a , b ) X � � � � � � � � � � � � � � F Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 89 / 166

  70. Rewriting calculus Introduction A calculus with more explicit features Like for explicit substitution calculi, explicitation of all the ingredients is useful. In “basic” rewriting calculus, rules are first class object application is explicit decision of redex reduction is explicit matching is a main explicit parameter results are first class Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 90 / 166

  71. Rewriting calculus Syntax and semantics The Untyped Syntax P ::= T Patterns T X | K | P � T | T T | T ≀ T ::= Terms T 1 � T 2 is a rule abstraction with pattern T 1 and body T 2 1 ... the free variables of T 1 are bound in T 2 The terms can be also structures built using the symbol “ ≀ ” 2 We work modulo the α -convention and the hygiene-convention 3 Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 91 / 166

  72. Rewriting calculus Syntax and semantics Some ρ -terms ( X � X ) a similar to the λ -term ( λ x . x ) a ( X � X X ) ( X � X X ) the well-known λ -term ( ωω ) ( a � b ) a the application of the rule a � b to the term a ( f ( X , Y ) � g ( X , Y )) f ( a , b ) a classical rewrite rule application Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 92 / 166

  73. Rewriting calculus Syntax and semantics The Simplest Reduction Semantics ( P � A ) B → ρ A θ ( P ≺ if P θ = T B ≺ B ) Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 93 / 166

  74. Rewriting calculus Syntax and semantics Some ρ -reductions ( X � X ) a �→ → ρ a ( X � ( X X )) ( X � ( X X )) �→ → ρ { ω ω } �→ → δ . . . ρ σ ( a � b ) a �→ → ρ b ( f ( X , Y ) � g ( X , Y )) ( f ( a , b )) �→ → ρ g ( a , b ) ( f ( X , Y ) � g ( X , Y )) ( g ( a , b )) Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 94 / 166

  75. Rewriting calculus Syntax and semantics Non unitary matching � � f AC ( X , Y ) � X f AC ( a , b ) �→ ? a �→ ? b Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 95 / 166

  76. Rewriting calculus Syntax and semantics Reduction produces structures ( P � A ) B → ρ A θ 1 ≀ . . . ≀ A θ n , . . . with { θ 1 , . . . , θ n , . . . } = S ol ( P ≺ ≺ T B ) Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 96 / 166

  77. � ��� � ��� Rewriting calculus Syntax and semantics Nondeterminism For the rewriting reduction � f ( x , y ) → x f ( x , b ) → b f ( x , y ) → x = ⇒ R f ( a , b ) a Non Determinism = ⇒ R f ( a , b ) b f ( x , b ) → b Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 97 / 166

  78. Rewriting calculus Syntax and semantics Basic ρ -calculus ( P � A ) B → ρ A θ 1 ≀ . . . ≀ A θ n , . . . with { θ 1 , . . . , θ n , . . . } = S ol ( P ≺ ≺ T B ) ( A ≀ B ) C → δ A C ≀ B C Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 98 / 166

  79. Rewriting calculus Syntax and semantics Detecting matching failures : the symbol stk The relation P �⊑ A detects (some) definitive matching failures : 1 ∀ σ , ∀ B s.t. σ ( A ) �→ B , � ∃ τ s.t. τ ( P ) = B The relation → stk treats matching failures uniformly : 2 ( P � B ) A → stk if P �⊑ A stk stk ≀ A → A stk A ≀ stk → A stk → stk A stk stk Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 99 / 166

  80. Rewriting calculus Syntax and semantics Failures � ( f ( X , Y ) � X ≀ f ( X , c ) � c f ( a , b ) ➥ ( f ( X , Y ) � X ) f ( a , b ) ≀ ( f ( X , c ) � c ) f ( a , b ) ➥ a ≀ stk ➥ a Horatiu CIRSTEA and Claude KIRCHNER Techniques de réécriture et transformations 27 janvier 2007 100 / 166

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