Arrays

1. Arrays Daniel Kroening and OferStrichman Decision Procedure

2. Array logic • Arrays are a basic data structure that is used in most software programs • modeling memories and caches in hardware design. • Array logic permits expressions over arrays. • Arrays can be formalized as maps from an index type to an element type. • Array logic has two basic operations • Reading • Writing

3. Contents • Introduction • Arrays as uninterpreted function • A reduction algorithm for array logic • Conclusion

4. Array logic • Array is the set of functions that map an index value to an element. • Index type TI • Element type TE • Array type • Let denote an array. • Two basic operations on arrays: • Reading : reads an element with index from a. The value of the element of index i is denoted by a[i]. • Writing : writes an element with . Let denote the value to be written. The array a where element i has been replaced by e is denoted by a{i e}.

5. Index logic • Index logic • The index logic should permit existential and universal quantification. • Existential quantification • “There exists an array element that is zero” • Universal quantification • “All elements of the array are greater than zero” • Array logic is defined by index logic and element logic • Validity for array logic is undecidable, since index logic permits existential and universal quantification. • First-order-logic is undecidable.

6. The syntax of array logic • atomI denotes an atom in the index logic • atomE denotes an atom in the element logic

7. Read-over-write axiom • Read-over-write axiom • McCarthy proposed. • After the value e has been written into array a at index i, the value of this array at index i is e. The value at any index j ≠ imatches that in the array before the write operation at index j:

8. Example • The main step of correctness argument is to show that the invariant in the line 7 is maintained by the assignment in line 6. • A common way to do so is to generate verification conditions.

9. Example (cond.) • We can obtain the following verification condition for the claim: • Proving validity of this formula shows that the loop invariant is maintained. • Prove by means of read-over-write axiom

10. Arrays as uninterpreted functions • Here we consider a method to check satisfiability of array logic formula whose quantifications are removed by a reduction mechanism. • We will see the reduction scheme. • array logic formula with quantification  array logic formula without quantification • Here • array logic formula without quantification  equality formula with uninterpreted functions • How to reduce? • array type  uninterpreted function • index type  function argument

11. Example • Array logic without quantification • Equality formula with an uninterpreted function • Apply Bryant’s reduction

12. A reduction algorithm for array logic • We define here a restricted class of array logic formulas in order to obtain decidability. • Recall arbitrary formula of original array logic is undecidable. • From hereon we will learn how to reduce given an array logic formula to the decidable array logic formula. • We consider formulas that are Boolean combinations of array property. • The reduced decidable array logic formula has array property.

13. Array property • Def. An array logic formula is called an array property if and only if it is of the form • and satisfies the following conditions: • The predicate , called the index guard, must follow the grammar. • The predicate is called the value constraint. The “integer-identifier” used in “term” must not be .

14. Extensionality rule • Def. Extensionality rule is the elementwise equality of two arrays. • Extensionality is an array property: • index guard is simply TRUE in this case. • is the value constraint of Extensionality rule.

15. Example • Array logic formula • Array property formula • Only second conjunct is not array property formula • Write-rule : Array updates can be handled by replacing each expression of the form by a fresh variable a’ of array type, and by adding two constraints that correspond directly to the two cases of the read-over-write axiom: 1. for the value that is written, 2. for the values that are unchanged.

16. A reduction algorithm • Array-Reduction • Input : An array property formula • Output : A formula with uninterpreted functions • Apply the write rule to remove all array updates from . • Replace all existential quantifications of the form by P(j), where j is a fresh variable. • Replace all universal quantifications of the form by • Replace the array read operations by uninterpreted functions and obtain . • return .

17. A reduction algorithm • In step 3 • The universal quantifiers replaced by the conjunction , where the set denotes the index variables that i might possibly be equal to in the formula . • set contains the following element : • All variables used as an array index in that are not quantified variables. • All variables used inside index guards in that are not quantified variables. • If contains none of the above, is {0} in order to obtain a nonempty set of index variables.

18. Example • We want to check satisfiability of 1. Apply the write rule to remove all array updates from . 2. Replace all existential quantifications of the form by P(z), where zis a fresh variable. = = =

19. Example (cond.) 3. Replace all universal quantifications of the form by , where = { i, z } 4. Remove trivially satisfied conjuncts. 5. Replace the two arrays a and a’ by uninterpreted functions Fa and Fa’.

20. Conclusion • Validity of an array logic formula without any reduction is undecidable. • Reduction algorithm needed. • Reduction algorithm outputs a formula with uninterpreted functions