Predicting the BNF of a Normal Form

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Predicting the BNF of a Normal Form

• Alan Bundy
• (Joint Work with Predrag Janicic and Alan Smaill)
• University of Edinburgh

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The Problem
Rewrite Rules
Output BNF?
Input BNF?
Normalise
Input Formula
Output Formulae
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Decision Procedure Synthesis
Input Formula
Remove
BNF
Stratify
BNF
Thin
BNF
Assoc
BNF
Trivial Formula
BNF
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Example Remove

• Input BNF
• Term Atom f(Term) g(Term)
• Rewrite Rule
• g(x) ?f(f(x))
• Output BNF
• Norm Atom f(Norm)

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Example Thin

• Input BNF
• Term Atom f(Term)
• Rewrite Rule
• f(f(x)) ?x
• Output BNF
• Norm Atom f(Atom)

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Example Assoc

• Input BNF
• Term Atom h(Term,Term)
• Rewrite Rule
• h(h(x,y),z) ?h(x,h(y,z))
• Output BNF
• Norm Atom h(Atom,Norm)

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Example Stratify

• Input BNF
• Term Atom f(Term)h(Term,Term)
• Rewrite Rule
• f(h(x,y)) ?h(f(x),f(y))
• Output BNF
• Norm Atomf(Sub)h(Norm,Norm)
• Sub Atomf(Sub)

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Predrags Counterexample

• Input BNF Term ? a.Term.b
• Rewrite Rules b ?d.c, c.d ?d.c
• Output set an.dn.cn
• Cannot be expressed as context-free BNF.
• So desired BNF may not always exist.

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Unfolding and Separation

• Thin Rewrite Rule f(f(x)) ?x
• Input BNF Term Atom f(Term)
• Unfold
• Term Atom f(Atom) f(f(Term))
• Replace matched terms
• Norm Atom f(Atom) Norm
• Simplify Norm Atom f(Atom)

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Separation match and unify

• Replace terms that match f(f(Term))
• or omit them?
• Retain terms that dont unify f(Atom)
• Unfold terms that unify, but dont match f(Term)

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Infinite Regress Avoided

• Input BNF Term Atom h(Term,Term)
• Rewrite Rule h(h(x,y),z) ?h(x,h(y,z))
• Unfold and Replace
• Term Atom h(Atom,Term) h(Term,h(Term,Term))
• Term Atom h(Atom,Term) h(Atom,h(Term,Term))
  h(Term,h(Term,h(Term,Term)))
• Unfold and Omit
• Term Atom h(Atom,Term)

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Smaill Algorithm Background

• Assume output BNF ? input BNF.
• Only singleton, terminating rule set l ?r.
• Input BNF Term D1 Dn.
• Standardise apart Terms.
• Construct Norm recursively for each Di by cases.
• Di is Di with each Termj replaced by Normj.
• Di is Di with each Termj replaced by Subj.

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Smaill Algorithm Cases

• Di and l match omit ith disjunct.
• Di and l dont unify include Di.
• Di and l unify but dont match
• include Di.
• unfold to Di Di1 Dik,
• add as new context sensitive equation,
• apply algorithm recursively.

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Example Stratify

• Input BNF Term Atomf(Term)h(Term,Term)
• Rewrite Rule f(h(x,y)) ?h(f(x),f(y))
• Apply algorithm
• Norm Atomf(Term1)h(Norm2,Norm3)
• f(Term1) f(Atom)f(f(Term2))f(h(Term3,Term4))
• Recurse on 2nd equation
• f(Sub1) f(Atom)f(f(Sub2))
• Simplify 2nd equation
• Sub1 Atomf(Sub2)

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Theoretical Questions

• Is algorithm sound?
• Is algorithm complete?
• Does algorithm terminate
• when rewrite rules do?
• Can we lift restrictions
• to singleton rule sets?
• to output BNF ? input BNF?

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Conclusion

• Output BNF prediction
• Important for decision procedure synthesis.
• Separation and unfolding as key.
• Algorithm based on this.
• Theoretical questions.