Introduction to Term Rewriting

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Introduction to Term Rewriting

• Stefan Kahrs

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Ancestry
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Definition

• A Term Rewriting System is a pair (?,R)
• signature ?(S,), S set of symbols,
• S?N (ranked alphabet)
• set T? of first-order terms (with variables)
• rules R? T? T?, t?s ? R such that
• t?Var
• vars(t)?vars(s)

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Example

• ?(add,succ,0,)
• (add)2, (succ)1, (0)0
• R add(succ(x),y)?succ(add(x,y)),
• add(0,x)?x

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Semantics

• of a TRS is the abstract reduction system (T?,?R)
  (ARS same as transition system)
• where ?R is the smallest rewrite relation
  containing R (the rewrite closure of R)
• a relation S on terms is a rewrite relation iff
• t S u implies ts S us for any substitution s
• t S u implies Ct S Cu for any context C_

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Basics

• substitution f(t1,...,tn)s f(t1s,...,tns)
• homomorphism on the term algebra
• context C_ is a term with a hole ?
• CtC_s where ?st, xsx

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Notations useful of ARS

• R binary relation on some set...
• R equivalence closure of R
• R-1, inverse of R
• ?R, complement of R
• R reflexive transitive closure
• ?RR(R-1) joinability
• ?R(R-1)R common source

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Properties of ARS

• Church-Rosser R ? ?R
• equivalent terms have a common reduct
• Confluence ?R ? ?R
• diverting computations can be re-joined
• Strong Normalisation of R ?S.S? RS ? S?
• all R-chains must end
• Weak Normalisation of R (RR ?) ?
• all terms have normal forms

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Properties of ARS, continued...

• Weak Confluence, WCR R-1R ? ?R
• one-step diverting computations can be re-joined
• Uniqueness of Normal Forms (UN)
• normal forms in the R relation are identical
• ... w.r.t. reduction (UN?)
• normal forms in the ?R relation are identical

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TRSs and the word problem of Universal Algebra

• CR SN if we have both then the TRS gives us a
  decision procedure for the equational theory
  (assuming finiteness)
• Newmans Lemma CRSN ?WCRSN
• Critical Pair lemma ?R is WCR iff for all
  critical pairs (x,y) of R x ?Ry

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Critical Pair

• suppose a?b, c?d are rules of the TRS
• if there exist C_, t with Ctc, t?Var
• and if there exist s,q such that tsaq then the
  terms ds, Csbq form a critical pair
• explanation
• ds?csCtsCstsCsaq ?Csbq
• s,q mgu, C? ? if rules the same

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Whats the big deal?

• Knuth Bendix turned the critical pair lemma
  around, to construct a decision procedure for an
  equational theory
• central ideas
• turn equations into rewrite rules
• maintain termination
• maintain WCR by turning critical pairs into
  equations

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KB algorithm, sketch

• E equations ? R rules
• remove ab from E and compute their normal forms
  a, b w.r.t. R (stop on ?)
• if ab continue at 1, otherwise...
• add a?b or b?a to R
• add all critical pairs of the new rule with R
  (including itself) to E back to 1

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Catches

• quite a few things can go wrong
• orienting into a rule (step 3) may not be
  possible without losing termination
• termination check of R is tricky
• the algorithm itself (in the described form) is
  quite likely to fail to terminate we need a few
  tweaks to give it a fighting chance

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Example

• The Theory of Free Groups
• (xy)z x(yz)
• -xx 0
• 0x x
• Orient into rules (from left to right). All three
  rules have a critical pair with the first rule,
  but only one is not joinable (-xx)zz

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Eventually

• 0x?x
• -xx ?0
• (xy)z ?x(yz)
• -x(xz) ?z
• x0 ?x

--x ? x -0 ? 0 x -x ? 0 x(-xy) ? y -(xy) ?
-y -x
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Aside

• I implemented a bit of KB in Sicstus prolog, put
  in the group example, and then proved
  inconsistency, i.e. that every group is trivial.
• Problem the left-hand sides of the rules
• -xx?0,
• -x(xz)?z
• are unifiable when we consider them to be regular
  trees, assigning x to xzw, with the critical
  pair (z,0).

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Inductionless Induction

• show that equation X is an inductive consequence
  of E
• Run KB on E giving R
• Run KB on EX (same ordering) giving R
• Check that all left-hand sides of R are
  ground-reducible by R.

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Example

• App(Nil,x) x
• App(Cons(x,y),z) Cons(x,App(y,z))
• Rev(Nil) Nil
• Rev(Cons(x,y)) App(Rev(y),Cons(x,Nil))
• trying to prove Rev(Rev(x))x

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Result

• KB of the first 4 equations orients them from
  left to right.
• Similar for KB of all 5, but we get in addition
• Rev(App(x,Cons(y,Nil)))?Cons(y,Rev(x))
• Rev(App(Rev(x),Cons(y,Nil)))?Cons(y,x)
• The lhs of all additional three rules are
  ground-reducible by the first 4 rules, hence we
  found an inductive theorem.

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Termination

• Generally hard, traditional techniques are based
  on simplification ordering
• A simplification ordering is a rewrite relation ?
  such that Ctgtt (where gt is ?-?)
• By a theorem of Kruskal it is known that
  simplification orderings over finite signatures
  are well-founded.

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Orthogonal TRS

• no critical pairs (hence WCR)
• left-linear
• A ?C(A) C(x)?D(x,C(x)) D(x,x)?E not CR
• weaker notions weakly/almost orthogonal
• normalising strategy
• OTRS parallel outermost,
• left-normal OTRS leftmost outermost

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Modularity (since late 1980s)

• disjoint union A?B of two TRSs join the
  signatures and rules, provided the symbol sets
  are disjoint (if not rename)
• a property j of TRSs is modular iff
• j holds for A and j holds for B iff
• j holds for A?B

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Properties

• CR is modular (Toyamas theorem, 1987 the second
  most exciting TRS paper)
• SN is not
• CR SN is not
• CR SN LL is
• simplifying is

Collapsed Tree Rewriting the first three
modularity results are exactly the opposite!
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Extensions, modifications

• CTRS, various flavours
• Term graph rewriting
• rewriting modulo E
• transfinite rewriting, infinite terms
• higher-order rewriting (rewrite modulo a
  substitution calculus)

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More Information

• handbooks
• of TCS (Dershowitz, Jouannaud), Vol B
• of Logic in Computer Science (Klop), Vol II
• textbooks
• Baader Nipkow, Ohlebusch, Bachmair
• if you read German Bündgen, Drosten, Avenhaus
• conference RTA