calculus, theorems etc'

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?-calculus, theorems etc.

• Stefan Kahrs

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?-Theories

• a ?-theory E allows us to prove equations
• that is E tu
• equivalence, i.e. E tu ? E ut etc.
• congruence, i.e. E tu ? E ?x.t ?x.u
• contains ?, i.e. E (?x.t) u txu
• the theory ? has nothing else in it
• ?? ?x.t xt (if x?FV(t))

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Consistency

• Logic cannot prove every sentence
• ?-theory cannot prove every closed equation
• is ? itself consistent?
• not, if we used naive substitution
• t (?x.t) u (?y.(?x.y)) t u (?x.(?y.(?x.y))
  x) t u (?x.(?x.x)) t u (?x.x) u u

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Proving Consistency

• Church-Rosser property implies consistency
• t ?? u ? ?s.t ?? s ?u ?? s (?? reflexive and
  transitive closure of ??, ?? equivalence closure)
• note that ? tu ? t ?? u

t?t1?t2?t3?t4?... ?tn?u
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CR versus Confluence

• Confluence t ??u ? t ??v ? ?s. u ??s ? v
  ??s
• obviously, CR ? Confluence
• but also, Confluence ? CR

t?t1?t2?t3?t4?... ?tn?u
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How to prove confluence

• standard strategy
• prove local confluence (weaker premise t ?? u ? t
  ?? v) ?
• prove termination ? ?. (?? ?) ? ? ? Ø ?
• alternative, diamond property (weaker concl.)
• t ?? u ? t ?? v ? ?s. u ??s ? v ??s ?
• 2nd alternative
• prove diamond property of any relation ? with
  ?? ? ? ? ??.

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Tait Martin-Löf

• t ? t
• t ? u ? ?x.t ? ?x.u
• t ? u ? r ? s ? (t r) ? (u s)
• t ? u ? r ? s ? (?x.t) r ? uxs
• proving diamond property by induction on the
  definition of ?

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Lemmas

• prove t ? u ? r ? s ? txr ? uxs
• inversion lemmas (easy!)
• ?x.t ? u implies u?x.t and t ? t
• (a b) ? u implies either
• u(a b) with a ? a and b ? b , or
• a?x.t, utxb with t ? t and b ? b

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Example (lemma)

• Case t ? u is an instance of the last case of ?.
  
• Then t(?y.a) b, uayb with a ? a , b ?
  b
• Thus txr (?y.axr) bxr
• applying the induction hypothesis gives us
  axr ? axs and bxr ? bxs
• using the definition txr ?
  axsybxs
• substitution lemma axsybxs
  aybxs
• obvious uxs aybxs

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Strategies

• various definitions possible, e.g.
• F ??? such that t ??F(t)
• R ? (??)
• the above with provisos about non-NF
• F ????, where t F t 0 F t i ?? F t (i1)
  (or F t iF t (i1))
• one-step strategies
• recursive or effective strategies

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Properties of Strategies

• F is normalizing
• if t ??u, u? NF? then there exists n such that
  Fn(t)u in other words t F u, viewing F as a
  relation
• F is hyper-normalizing
• if t ??u, u? NF? then t (??F ??) u
• optimal strategies (constraints one-step, rec.)
• time
• space

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Examples I

• leftmost outermost, Flo
• Flo(t) ?1(A(t))
• A(?y.a) (?y.a,c) where A(a)(a,c)
• A(x) (x,0)
• A(t u) (axu,1), if t ?x.a
• A(t u) (t u,1), if t??x.a, A(t)(t,1)
• A(t u) (t u,c), if t??x.a, ?2(A(t))0,
  A(u)(u,c)

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Examples II

• leftmost Innermost, Fli
• Fli(t) ?1(B(t))
• B(?y.a) (?y.a,c) where B(a)(a,c)
• B(t u) (axu,1), if t?x.a, ?2(B(t))?2(B
  (u))0
• B(t u) (t u,1), if B(t)(t,1)
• B(t u) (t u,1), if ?2(B(t))0, B(u)(u,1)
• B(t) (t,0), otherwise

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Examples III

• Gross-Knuth strategy, FGK
• FGK(?y.a) ?y. FGK (a)
• FGK(x) x
• FGK((?x.a) u) FGK(a)x FGK(u)
• FGK(t u) FGK(t) FGK(u), if t??x.a
• note t ? FGK(t)

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Some results

• Fli and Flo are one-step strategies, FGK is not
• all are recursive
• Flo and FGK are normalizing (and
  hyper-normalizing), Fli is not
• FGK is cofinal ?? ? FGK(??)-1 (if you divert
  from Gross-Knuth, you can always rejoin)

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Standardisation Theorem

• standard reduction the empty reduction sequence
  is standard, otherwise s ?? t ??u is standard if
• t ??u is standard
• redexes occurring left to the one contracted in s
  ?? t are not contracted in t ??u
• theorem if t ??u then there is a standard
  reduction from t to u

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Consequences

• Flo always produces a standard reduction, since
  there is no redex left to the ones we contract
• Flo is normalising - it gives us the only
  standard reduction from a term to its normal form

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Optimal Strategies

• elusive
• Fli is not (simply because it is not normalising)
• Flo is not (often outperformed by Fli)
• change of representation (Lévy 1978)
• graph reduction, Wadsworth 1971 lazy evaluation
• bookkeeping work, Lamping POPL 1990
• Asperti et.al. 1995, interaction
  systems Bologna Optimal Higher-Order Machine

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Name conflict resolution

• name-capture unsound
• solutions
• dont reduce under ?-binders
• built-in ??-conversion
• combinators
• different kind of binders

implementations textbook Miranda, Schönfinkel de
Bruijn
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Combinators

• define (untyped) functions
• I x x
• K x y x
• S x y z (x z) (y z)
• view ?-terms as abbreviations
• ?x.xI
• ?x.yK y, if x?y
• ?x.(a b)S(?x.a)(?x.b)

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De Bruijn indices

• replace variable names by numbers (their
  relative addresses on a stack)
• Example ?h.(?x.h(x x))(?x.h(x x)) becomes
  ?(?2(1 1))(?2(1 1))
• Consequence ?-convertible terms become identical
• Name conflict resolution all the time

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Explicit Substitutions

• flourishing since 1990 (Abadi et. al.
  ??-calculus now there are many variations, even
  regular workshops)
• idea remove substitutions from the meta-theory
• include them instead in the syntax
• operating on de Bruijn style substitutions

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Syntax

• terms a,b 1 a b ?a as
• substitutions s,t id ?? a ? s s ? t
• Ideas as is substitution application and
• id is the identity substitution
• ? increases the indices of all free variables by
  1
• a ? s replaces the first variable by a and the
  others using the substitution s
• s ? t composition of substitutions

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Dynamics, Terms

• (?a)b ? ab ? id
• 1id ? 1
• 1a ? s ? a
• (a b)s ? (as bs)
• (?a)s ? ?(a1 ? (s ? ?))
• ast ? as ? t

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Dynamics, Substitutions

• id ? s ? s
• ? ? id ? ?
• ? ? (a ? s ) ? s
• (a ? s ) ? t ? at ? (s ? t)
• (s1 ? s2) ? s3 ? s1 ? (s2 ? s3)

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Remarks

• this rewrite system gives a complete
  implementation of the ?-calculus
• it is still confluent
• the substitution part (without the ? rule) is
  confluent and terminating
• it has proved difficult to come up with typed
  calculi of explicit substitutions which are nice