paper pdf - Lab for Automated Reasoning and Analysis - LARA

Proof Sketch. It remains to eliminate undetermined shapevariables from β. This process is similar to term algebraquantifier elimination [25, Section 3.4]; the key ingredientis Lemma 10, which relies on the fact that undeterminedparameter variables may take on infinitely many values. Wetherefore ensure that the conjuncts outside shapeBase donot constrain the undetermined parameter shape variablesto denote the values from a finite set.Consider an undetermined parameter shape variable u s .u s does not occur in termHom, because all term variablesare determined and a conjunct u s =sh(u) would implythat u s is determined as well. u s can thus occur onlyin cardin within some cardinality constraint |φ| u s=k or|φ| u s≥k. Moreover, formula φ in each such cardinality constraintis closed: otherwise φ would contain some free termvariable u and since all term variables are determined, u swould be determined as well.Let u s denote some shape s. Because φ is a closed formula,|φ| is equal to 0 if φ C =false and to the shape sizem = |leaves(s)| if φ C =true. (The fact that closed formulasreduce to the constraints on the domain size appears in[30, Theorem 3.36, Page 13]. In term powers, these constraintsbecome constraints on the size of the shape.) Wetransform β into the disjunction β 1 ∨ β 2 of base formulaswhere β 1 ≡ β ∧ prim(φ) and β 2 ≡ β ∧ prim(¬φ). Constraintsof the form prim(¬φ) ∧ |φ| u s=k reduce to 0=k,we replace them by true if k≡0 and false if k≢0. Onthe other hand, prim(φ) ∧ |φ| u s=k denotes the constraintm = k and prim(φ) ∧ |φ| u s≥k denotes m≥k. Hence,by repeating this process for every formula φ which appearsin some cardinality constraint |φ| u s=k or |φ| u s≥k,we obtain a conjunction of linear constraints of the formm = k and m ≥ k. These constraints specify a finiteor infinite set S ⊆ {0, 1, . . .} of possible sizes m. LetA = {s | |leaves(s)| ∈ S}. By nature of our constraints,if the set S is infinite then it contains an infinite intervalof form {m 0 , m 0 + 1, . . .}, so the set A is infinite. If Σcontains a unary constructor and S is nonempty, then A isalso infinite. If Σ contains no unary constructors and S isfinite then A is finite and we can effectively compute A.The cardinality constraints containing u s are thus equivalentto ∨ pi=1 us = t s i where A = {ts 1 , . . . , ts p }. Transform∨the structural base formula β into a disjunction of formulaspi=1 β i where β i results from β by replacing the cardinalityconstraints containing u s with u s = t s i . Convert each β ito a structural base formula by labelling the subterms of t s iwith internal shape variables using UNF rules, and by doingcase analysis on the equality between the new internalshape variables, using ShDis rule. By repeating this processfor all shape variables u s where the set S is finite, we obtainbase formulas where the set A is infinite for every undeterminedparameter shape variable u s . We may then eliminateall undetermined parameter and non-parameter shape variablesalong with the conjuncts that contain them. The resultis an equivalent formula because Lemma 10 implies that itis always possible to find the values of eliminated parametervariables, so their existence is a redundant condition. Wetherefore eliminate all undetermined shape variables and theresulting structural base formulas contain only determinedvariables.Proposition 25 (Struct. Base to Quantifier-Free) Everystructural base formula β can be effectively transformed toan equivalent well-defined quantifier-free formula φ.Proof. Apply Corollary 19, then Lemma 23, and thenLemma 24. All variables in the resulting disjunction ofstructural base formulas are determined, so each of themis equivalent to some quantifier free formula φ i by Corollary17. The disjunction ∨ i φ i is the desired quantifier-freeformula φ.Summary of Our Quantifier Elimination Algorithm.Consider a closed L P -formula φ. Convert φ to an extendedterm-powerformula φ 1 using (2). Convert φ 1 to prenexform φ 2 . Eliminate all quantifiers from φ 2 starting from theinnermost one, as follows. If φ 2 ≡ 〈Q i u ∗ i 〉 i∃v ∗ . ψ where ψis quantifier-free then apply Proposition 13, Proposition 12and then Proposition 25. If φ 2 ≡ 〈Q i u ∗ i 〉∀v∗ . ψ then consider〈Q i u ∗ i 〉.¬∃v∗ . ¬ψ and proceed as in the previous case.By applying Proposition 13 and Proposition 25 to the resultingvariable-free formula we obtain a propositional combinationof prim(φ) formulas. Theorem 3 then follows by (3).Acknowledgements We thank Albert Meyer, Jens Palsberg,Tim Priesnitz, Stefan Ratschan, Jakob Rehof, ZhendongSu, and anonymous reviewers for useful comments.References[1] A. Aiken, D. Kozen, and E. Wimmers. Decidability ofsystems of set constraints with negative constraints.Information and Computation, 122, 1995.[2] A. Aiken, E. L. Wimmers, and T. K. Lakshman. Soft typingwith conditional types. In Proc. 21st ACM POPL, pages163–173, New York, NY, 1994.[3] H. Ait-Kaci, A. Podelski, and G. Smolka. A featureconstraint system for logic programming with entailment.In Theoretical Computer Science, volume 122, pages263–283, January 1994.[4] R. M. Amadio and L. Cardelli. Subtyping recursive types.Transactions on Programming Languages and Systems,15(4):575–631, 1993.[5] L. O. Andersen. Program Analysis and Specialization ofthe C Programming Language. PhD thesis, DIKU,University of Copenhagen, 1994.[6] R. Backofen. A complete axiomatization of a theory withfeature and arity constraints. Journal of LogicProgramming, 24:37–72, 1995.10