Definability of Accelerated Relations in a Theory of Arrays and its - PowerPoint PPT Presentation

Acceleration Preventing divergence τ 1 τ 0 τ 2 ¯ I l 1 P Find control-flow graph: Precise backward reachability With accelerated transitions (desired behavior) ¯ P τ 2 ¯ P τ 0 v 1 ⊥ τ 2 τ 1 τ 0 v 1 ⊥ τ 0 v 2 ⊥ τ + τ 1 1 τ 1 v + v 2 ⊥ 2 τ 0 v 3 ⊥ τ 1 v + v 3 3 F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 6 / 24

Acceleration Preventing divergence τ 1 τ 0 τ 2 ¯ I l 1 P Find control-flow graph: Precise backward reachability With accelerated transitions (desired behavior) ¯ P τ 2 ¯ P τ 0 v 1 ⊥ τ 2 τ 1 τ 0 v 1 ⊥ τ 0 v 2 ⊥ τ + τ 1 1 τ 1 τ 0 v + v 2 ⊥ 2 τ 0 v 3 ⊥ τ 1 v + v 3 3 F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 6 / 24

Acceleration Preventing divergence τ 1 τ 0 τ 2 ¯ I l 1 P Find control-flow graph: Precise backward reachability With accelerated transitions (desired behavior) ¯ P τ 2 ¯ P τ 0 v 1 ⊥ τ 2 τ 1 τ 0 v 1 ⊥ τ 0 v 2 ⊥ τ + τ 1 1 τ 1 τ 0 v + v 2 ⊥ 2 τ 0 v 3 ⊥ τ 1 τ 1 v + v 3 3 F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 6 / 24

Acceleration Preventing divergence τ 1 τ 0 τ 2 ¯ I l 1 P Find control-flow graph: Precise backward reachability With accelerated transitions (desired behavior) ¯ P τ 2 ¯ P τ 0 v 1 ⊥ τ 2 τ 1 τ 0 v 1 ⊥ τ 0 v 2 ⊥ τ + τ 1 1 τ 1 τ 0 v + v 2 ⊥ 2 τ 0 v 3 ⊥ τ 1 τ 1 v + v 3 3 F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 6 / 24

Acceleration Preventing divergence τ 1 τ 0 τ 2 ¯ I l 1 P Find control-flow graph: Precise backward reachability With accelerated transitions (desired behavior) ¯ P τ 2 ¯ P τ 0 v 1 ⊥ τ 2 τ 1 τ 0 v 1 ⊥ τ 0 v 2 ⊥ τ + τ 1 1 τ 1 τ 0 v + v 2 ⊥ 2 τ 0 v 3 ⊥ τ + τ 1 1 τ 1 v + v 3 3 F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 6 / 24

Acceleration Preventing divergence τ 1 τ 0 τ 2 ¯ I l 1 P Find control-flow graph: Precise backward reachability With accelerated transitions (desired behavior) ¯ P τ 2 ¯ P τ 0 v 1 ⊥ τ 2 τ 1 τ 0 v 1 ⊥ τ 0 v 2 ⊥ τ + τ 1 1 τ 1 τ 0 v + v 2 ⊥ 2 τ 0 v 3 ⊥ τ + τ 1 1 τ 1 v + v 3 3 F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 6 / 24

Acceleration State of the art Acceleration: Transitive closure τ + of transitions τ encoding cyclic actions F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 7 / 24

Acceleration State of the art Acceleration: Transitive closure τ + of transitions τ encoding cyclic actions Challenges: � In general transitive closure cannot be expressed in FOL F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 7 / 24

Acceleration State of the art Acceleration: Transitive closure τ + of transitions τ encoding cyclic actions Challenges: � In general transitive closure cannot be expressed in FOL Only some (important) classes of τ ’s allow the definability of τ + Polling-based systems [BBD + 02] Imperative programs over integers [BIK10] F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 7 / 24

Acceleration State of the art Acceleration: Transitive closure τ + of transitions τ encoding cyclic actions Challenges: � In general transitive closure cannot be expressed in FOL Only some (important) classes of τ ’s allow the definability of τ + Polling-based systems [BBD + 02] Imperative programs over integers [BIK10] What about arrays? F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 7 / 24

Acceleration for arrays Contributions In theory: Identification of classes of transitions τ over arrays admitting definable acceleration F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 8 / 24

Acceleration for arrays Contributions In theory: Identification of classes of transitions τ over arrays admitting definable acceleration Determine the price to pay for expressing τ + F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 8 / 24

Acceleration for arrays Contributions In theory: Identification of classes of transitions τ over arrays admitting definable acceleration Determine the price to pay for expressing τ + In practice: Template-based solution ✔ High degree of automation ✔ Computationally cheap F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 8 / 24

Acceleration for arrays Contributions In theory: Identification of classes of transitions τ over arrays admitting definable acceleration Determine the price to pay for expressing τ + In practice: Template-based solution ✔ High degree of automation ✔ Computationally cheap Combination with abstraction-based frameworks F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 8 / 24

Acceleration for arrays Example i ′ = i + 1 τ 1 := pc = l L ∧ i < L ∧ a [ i ] � = e ∧ � �� � � �� � update guard F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 9 / 24

Acceleration for arrays Example i ′ = i + 1 τ 1 := pc = l L ∧ i < L ∧ a [ i ] � = e ∧ � �� � � �� � update guard ⇓ F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 9 / 24

Acceleration for arrays Example i ′ = i + 1 τ 1 := pc = l L ∧ i < L ∧ a [ i ] � = e ∧ � �� � � �� � update guard ⇓   y > 0 ∧ pc = l L ∧ τ +   1 := ∃ y. ∀ j. ( i ≤ j < i + y → j < L ∧ a [ j ] � = e )   i ′ = i + y F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 9 / 24

Acceleration for arrays Example i ′ = i + 1 τ 1 := pc = l L ∧ i < L ∧ a [ i ] � = e ∧ � �� � � �� � update guard ⇓   y > 0 ∧ pc = l L ∧ τ +   1 := ∃ y. ∀ j. ( i ≤ j < i + y → j < L ∧ a [ j ] � = e )   i ′ = i + y F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 9 / 24

The formal framework Iterators Definition (Iterators) A tuple of m -ary terms u ( x ) is said to be an iterator iff there exists an m -tuple of m + 1-ary terms u ∗ ( x, y ) such that for any natural number n it happens that the formula u n ( x ) = u ∗ ( x, ¯ n ) is valid. F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 10 / 24

The formal framework Iterators Definition (Iterators) A tuple of m -ary terms u ( x ) is said to be an iterator iff there exists an m -tuple of m + 1-ary terms u ∗ ( x, y ) such that for any natural number n it happens that the formula u n ( x ) = u ∗ ( x, ¯ n ) is valid. Example u ( x ) := x + 1 F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 10 / 24

The formal framework Iterators Definition (Iterators) A tuple of m -ary terms u ( x ) is said to be an iterator iff there exists an m -tuple of m + 1-ary terms u ∗ ( x, y ) such that for any natural number n it happens that the formula u n ( x ) = u ∗ ( x, ¯ n ) is valid. Example u ( x ) := x + 1 u ∗ ( x, y ) := x + y F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 10 / 24

The formal framework Selectors Definition (Selectors) Given an iterator u ( x ), an m -ary term κ ( x 1 , . . . , x m ) is a selector for u ( x ) iff there is an m + 1-ary term ι ( x 1 , . . . , x m , y ) yielding the validity of the formula z = κ ( u ∗ ( x, y )) → y = ι ( x, z ) F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 11 / 24

The formal framework Selectors Definition (Selectors) Given an iterator u ( x ), an m -ary term κ ( x 1 , . . . , x m ) is a selector for u ( x ) iff there is an m + 1-ary term ι ( x 1 , . . . , x m , y ) yielding the validity of the formula z = κ ( u ∗ ( x, y )) → y = ι ( x, z ) Most likely κ is a projection F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 11 / 24

The formal framework Selectors Definition (Selectors) Given an iterator u ( x ), an m -ary term κ ( x 1 , . . . , x m ) is a selector for u ( x ) iff there is an m + 1-ary term ι ( x 1 , . . . , x m , y ) yielding the validity of the formula z = κ ( u ∗ ( x, y )) → y = ι ( x, z ) Most likely κ is a projection Can a cell z be reached in m iterations? The number ι ( x, z ) gives “the only possible candidate” y number of iterations z = κ ( u ∗ ( x, y )) checks if the candidate y is correct F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 11 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } iterator: u ( i ) := i + 2 F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } iterator: u ( i ) := i + 2 u ∗ ( i, y ) = i + 2 y F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } iterator: u ( i ) := i + 2 u ∗ ( i, y ) = i + 2 y κ ( x ) := x F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } iterator: u ( i ) := i + 2 u ∗ ( i, y ) = i + 2 y κ ( x ) := x � z − i � ι ( i, z ) := 2 F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } iterator: u ( i ) := i + 2 u ∗ ( i, y ) = i + 2 y κ ( x ) := x � z − i � ι ( i, z ) := 2 Example i = 3 F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } iterator: u ( i ) := i + 2 u ∗ ( i, y ) = i + 2 y κ ( x ) := x � z − i � ι ( i, z ) := 2 Example i = 3 a [7] in 3 iterations? F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } iterator: u ( i ) := i + 2 u ∗ ( i, y ) = i + 2 y κ ( x ) := x � z − i � ι ( i, z ) := 2 Example i = 3 a [7] in 3 iterations? � 7 − 3 � ι ( i, z ) = 2 F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } iterator: u ( i ) := i + 2 u ∗ ( i, y ) = i + 2 y κ ( x ) := x � z − i � ι ( i, z ) := 2 Example i = 3 a [7] in 3 iterations? � 7 − 3 � ι ( i, z ) = = 2 2 F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } iterator: u ( i ) := i + 2 u ∗ ( i, y ) = i + 2 y κ ( x ) := x � z − i � ι ( i, z ) := 2 Example i = 3 a [7] in 3 iterations? � 7 − 3 � ι ( i, z ) = = 2 ✔ 2 F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } iterator: u ( i ) := i + 2 u ∗ ( i, y ) = i + 2 y κ ( x ) := x � z − i � ι ( i, z ) := 2 Example i = 3 a [7] in 3 iterations? � 7 − 3 � ι ( i, z ) = = 2 ✔ 2 u ∗ ( i, 2) = 3 + 2 · 2 = 7 F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } iterator: u ( i ) := i + 2 u ∗ ( i, y ) = i + 2 y κ ( x ) := x � z − i � ι ( i, z ) := 2 Example i = 3 a [7] in 3 iterations? � 7 − 3 � ι ( i, z ) = = 2 ✔ 2 u ∗ ( i, 2) = 3 + 2 · 2 = 7 ✔ F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } iterator: u ( i ) := i + 2 u ∗ ( i, y ) = i + 2 y κ ( x ) := x � z − i � ι ( i, z ) := 2 Example i = 3 i = 3 a [7] in 3 iterations? � 7 − 3 � ι ( i, z ) = = 2 ✔ 2 u ∗ ( i, 2) = 3 + 2 · 2 = 7 ✔ F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } iterator: u ( i ) := i + 2 u ∗ ( i, y ) = i + 2 y κ ( x ) := x � z − i � ι ( i, z ) := 2 Example i = 3 i = 3 a [7] in 3 iterations? a [6] in 3 iterations? � 7 − 3 � ι ( i, z ) = = 2 ✔ 2 u ∗ ( i, 2) = 3 + 2 · 2 = 7 ✔ F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } iterator: u ( i ) := i + 2 u ∗ ( i, y ) = i + 2 y κ ( x ) := x � z − i � ι ( i, z ) := 2 Example i = 3 i = 3 a [7] in 3 iterations? a [6] in 3 iterations? � 7 − 3 � 6 − 3 � � ι ( i, z ) = = 2 ✔ ι ( i, z ) = 2 2 u ∗ ( i, 2) = 3 + 2 · 2 = 7 ✔ F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } iterator: u ( i ) := i + 2 u ∗ ( i, y ) = i + 2 y κ ( x ) := x � z − i � ι ( i, z ) := 2 Example i = 3 i = 3 a [7] in 3 iterations? a [6] in 3 iterations? � 7 − 3 � 6 − 3 � � ι ( i, z ) = = 2 ✔ ι ( i, z ) = = 1 2 2 u ∗ ( i, 2) = 3 + 2 · 2 = 7 ✔ F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } iterator: u ( i ) := i + 2 u ∗ ( i, y ) = i + 2 y κ ( x ) := x � z − i � ι ( i, z ) := 2 Example i = 3 i = 3 a [7] in 3 iterations? a [6] in 3 iterations? � 7 − 3 � 6 − 3 � � ι ( i, z ) = = 2 ✔ ι ( i, z ) = = 1 ✔ 2 2 u ∗ ( i, 2) = 3 + 2 · 2 = 7 ✔ F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } iterator: u ( i ) := i + 2 u ∗ ( i, y ) = i + 2 y κ ( x ) := x � z − i � ι ( i, z ) := 2 Example i = 3 i = 3 a [7] in 3 iterations? a [6] in 3 iterations? � 7 − 3 � 6 − 3 � � ι ( i, z ) = = 2 ✔ ι ( i, z ) = = 1 ✔ 2 2 u ∗ ( i, 2) = 3 + 2 · 2 = 7 u ∗ ( i, 1) = 3 + 2 · 1 = 5 ✔ F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Example while ( true ) { a [ i ] = 0; i = i + 2; } iterator: u ( i ) := i + 2 u ∗ ( i, y ) = i + 2 y κ ( x ) := x � z − i � ι ( i, z ) := 2 Example i = 3 i = 3 a [7] in 3 iterations? a [6] in 3 iterations? � 7 − 3 � 6 − 3 � � ι ( i, z ) = = 2 ✔ ι ( i, z ) = = 1 ✔ 2 2 u ∗ ( i, 2) = 3 + 2 · 2 = 7 u ∗ ( i, 1) = 3 + 2 · 1 = 5 ✔ ✘ F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 12 / 24

The formal framework Local ground assignments Definition (Local ground assignment) A local ground assignment is a ground assignment of the form pc = l ∧ φ L ( a , c ) ∧ pc ′ = l ∧ a ′ = wr ( a , κ (˜ c ′ = u (˜ c ) ∧ d ′ = d c ) , t ( a , c )) ∧ ˜ F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 13 / 24

The formal framework Local ground assignments Definition (Local ground assignment) A local ground assignment is a ground assignment of the form pc = l ∧ φ L ( a , c ) ∧ pc ′ = l ∧ a ′ = wr ( a , κ (˜ c ′ = u (˜ c ) ∧ d ′ = d c ) , t ( a , c )) ∧ ˜ where (i) c = ˜ c , d ; F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 13 / 24

The formal framework Local ground assignments Definition (Local ground assignment) A local ground assignment is a ground assignment of the form pc = l ∧ φ L ( a , c ) ∧ pc ′ = l ∧ a ′ = wr ( a , κ (˜ c ′ = u (˜ c ) ∧ d ′ = d c ) , t ( a , c )) ∧ ˜ where (i) c = ˜ c , d ; (ii) u = u 1 , . . . , u | ˜ c | is an iterator; F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 13 / 24

The formal framework Local ground assignments Definition (Local ground assignment) A local ground assignment is a ground assignment of the form pc = l ∧ φ L ( a , c ) ∧ pc ′ = l ∧ a ′ = wr ( a , κ (˜ c ′ = u (˜ c ) ∧ d ′ = d c ) , t ( a , c )) ∧ ˜ where (i) c = ˜ c , d ; (ii) u = u 1 , . . . , u | ˜ c | is an iterator; (iii) the terms κ are a selector assignment for a relative to u ; F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 13 / 24

The formal framework Local ground assignments Definition (Local ground assignment) A local ground assignment is a ground assignment of the form pc = l ∧ φ L ( a , c ) ∧ pc ′ = l ∧ a ′ = wr ( a , κ (˜ c ′ = u (˜ c ) ∧ d ′ = d c ) , t ( a , c )) ∧ ˜ where (i) c = ˜ c , d ; (ii) u = u 1 , . . . , u | ˜ c | is an iterator; (iii) the terms κ are a selector assignment for a relative to u ; (iv) the formula φ L ( a , c ) and the terms t ( a , c ) are purely arithmetical over the set of terms { c , a ( κ (˜ c )) } ∪ { a i ( d j ) } 1 ≤ i ≤ s, 1 ≤ j ≤| d | ; F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 13 / 24

The formal framework Local ground assignments Definition (Local ground assignment) A local ground assignment is a ground assignment of the form pc = l ∧ φ L ( a , c ) ∧ pc ′ = l ∧ a ′ = wr ( a , κ (˜ c ′ = u (˜ c ) ∧ d ′ = d c ) , t ( a , c )) ∧ ˜ where (i) c = ˜ c , d ; (ii) u = u 1 , . . . , u | ˜ c | is an iterator; (iii) the terms κ are a selector assignment for a relative to u ; (iv) the formula φ L ( a , c ) and the terms t ( a , c ) are purely arithmetical over the set of terms { c , a ( κ (˜ c )) } ∪ { a i ( d j ) } 1 ≤ i ≤ s, 1 ≤ j ≤| d | ; (v) the guard φ L contains the conjuncts κ i (˜ c ) � = d j , for 1 ≤ i ≤ s and 1 ≤ j ≤ | d | . F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 13 / 24

The formal framework Contribution Theorem If τ is a local ground assignment, then τ + is a Σ 0 2 -assignment. Francesco Alberti, Silvio Ghilardi, and Natasha Sharygina. Tackling divergence: abstraction and acceleration in array programs. Technical Report 2012/01, University of Lugano, oct 2012. F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 14 / 24

The formal framework Contribution Theorem If τ is a local ground assignment, then τ + is a Σ 0 2 -assignment. Francesco Alberti, Silvio Ghilardi, and Natasha Sharygina. Tackling divergence: abstraction and acceleration in array programs. Technical Report 2012/01, University of Lugano, oct 2012. The proof of the theorem shows the “template” for τ + F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 14 / 24

The formal framework Contribution Theorem If τ is a local ground assignment, then τ + is a Σ 0 2 -assignment. Francesco Alberti, Silvio Ghilardi, and Natasha Sharygina. Tackling divergence: abstraction and acceleration in array programs. Technical Report 2012/01, University of Lugano, oct 2012. The proof of the theorem shows the “template” for τ + The template is parametric with respect to iterators selectors F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 14 / 24

Tool architecture S T = � v , I ( v ) , τ ( v , v ′ ) � � I 1 , I 2 , . . . � , � S 1 , S 2 , . . . � ✔ Loop Model ? Acceleration identification Checker ✘ F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 15 / 24

Acceleration for arrays Practical issue - classification of formulas Different kind of formulas 2 representing the (backward reachable) state-space: ground – formulas of the kind φ ( v ) 2 In all the formulas we admit the term a ( t ) only if t is a variable or a constant. F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 16 / 24

Acceleration for arrays Practical issue - classification of formulas Different kind of formulas 2 representing the (backward reachable) state-space: ground – formulas of the kind φ ( v ) Σ 0 1 – formulas of the kind ∃ i.φ ( i, v ) 2 In all the formulas we admit the term a ( t ) only if t is a variable or a constant. F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 16 / 24

Acceleration for arrays Practical issue - classification of formulas Different kind of formulas 2 representing the (backward reachable) state-space: ground – formulas of the kind φ ( v ) Σ 0 1 – formulas of the kind ∃ i.φ ( i, v ) Σ 0 2 – formulas of the kind ∃ i ∀ j.φ ( i, j, v ) 2 In all the formulas we admit the term a ( t ) only if t is a variable or a constant. F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 16 / 24

Acceleration for arrays Practical issue - classification of formulas Different kind of formulas 2 representing the (backward reachable) state-space: ground – formulas of the kind φ ( v ) Σ 0 1 – formulas of the kind ∃ i.φ ( i, v ) Σ 0 2 – formulas of the kind ∃ i ∀ j.φ ( i, j, v ) � Σ 0 2 -formulas might not fall in any known decidable fragment [BMS06, GdM09] 2 In all the formulas we admit the term a ( t ) only if t is a variable or a constant. F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 16 / 24

Acceleration for arrays Practical issue - classification of transitions Transition formulas can be: ground assignment – transitions of the kind τ ( v , v ′ ) Σ 0 1 -assignment – transitions of the kind ∃ i.τ ( i, v , v ′ ) Σ 0 2 -assignment – transitions of the kind ∃ i ∀ j.τ ( i, j, v , v ′ ) F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 17 / 24

Acceleration for arrays Practical issue - classification of transitions Transition formulas can be: ground assignment – transitions of the kind τ ( v , v ′ ) Σ 0 1 -assignment – transitions of the kind ∃ i.τ ( i, v , v ′ ) Σ 0 2 -assignment – transitions of the kind ∃ i ∀ j.τ ( i, j, v , v ′ ) Preimages with respect to a Σ 0 2 -assignment are Σ 0 2 -formulas F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 17 / 24

Acceleration for arrays Practical issue - classification of transitions Transition formulas can be: ground assignment – transitions of the kind τ ( v , v ′ ) Σ 0 1 -assignment – transitions of the kind ∃ i.τ ( i, v , v ′ ) Σ 0 2 -assignment – transitions of the kind ∃ i ∀ j.τ ( i, j, v , v ′ ) Preimages with respect to a Σ 0 2 -assignment are Σ 0 2 -formulas This prevents the practical application of the theoretical result! F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 17 / 24

Acceleration for arrays Practical issue - classification of transitions Transition formulas can be: ground assignment – transitions of the kind τ ( v , v ′ ) Σ 0 1 -assignment – transitions of the kind ∃ i.τ ( i, v , v ′ ) Σ 0 2 -assignment – transitions of the kind ∃ i ∀ j.τ ( i, j, v , v ′ ) Preimages with respect to a Σ 0 2 -assignment are Σ 0 2 -formulas This prevents the practical application of the theoretical result! Solution: over-approximate problematic Σ 0 2 -formulas with their monotonic abstraction [AGP + 12] F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 17 / 24

Acceleration for arrays Example τ 1 τ 0 τ 2 ¯ I l 1 P ¯ P τ 2 τ 0 v 1 ⊥ τ + τ 1 1 v + v 2 ⊥ 2 v + v 3 3 F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 18 / 24

Acceleration for arrays Example τ 1 τ 0 τ 2 ¯ I l 1 P ¯ P τ 2 τ 0 v 1 ⊥ τ + τ 1 1 v + v 2 ⊥ 2 v + v 3 3 F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 18 / 24

Acceleration for arrays Example v 1 τ + 1 τ 1 Instantiate ∀ j over ∃ i v + v + v 2 2 2 Might produce spurious counterexamples F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 19 / 24

Acceleration for arrays Example v 1 τ + 1 τ 1 This is a Instantiate ∀ j over ∃ i Σ 0 2 -formula v + v + v 2 2 2 Might produce spurious counterexamples F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 19 / 24

Acceleration for arrays Example v 1 τ + 1 τ 1 This is a Instantiate ∀ j over ∃ i Σ 0 2 -formula v + v + v 2 2 2   pc = l L ∧ y > 0 ∧ Might produce spurious counterexamples ( i ≤ j < i + y → j < L ∧ a [ j ] � = e ) ∧ ∃ x, y ∀ j.     0 ≤ x < i ∧ a [ x ] = e ∧ i + y ≥ L F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 19 / 24

Acceleration for arrays Example v 1 τ + Instantiation 1 τ 1 Instantiate j over { x, y, i, i + y, . . . } pushes it back to Σ 0 1 v + v + v 2 2 2   pc = l L ∧ y > 0 ∧ Might produce spurious counterexamples ∃ x, y ∀ j. ( i ≤ j < i + y → j < L ∧ a [ j ] � = e ) ∧     0 ≤ x < i ∧ a [ x ] = e ∧ i + y ≥ L F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 19 / 24

Acceleration for arrays Example v 1 τ + 1 τ 1 Instantiate ∀ j over ∃ i v + v + v + v 2 2 2 2 Might produce spurious counterexamples F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 19 / 24

Acceleration for arrays Example v 1 τ + 1 τ 1 Instantiate ∀ j over ∃ i More instantiations (more precise) v + v + v + v 2 2 2 2 Might produce spurious counterexamples F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 19 / 24

Acceleration for arrays Example v 1 τ + 1 τ 1 Instantiate ∀ j over ∃ i More instantiations (more precise) v + v + v + v 2 2 2 2 Less instantiations (less precise) Might produce spurious counterexamples F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 19 / 24

Acceleration for arrays Example v 1 τ + 1 τ 1 Instantiate ∀ j over ∃ i v + v + v + v 2 2 2 2 Might produce spurious counterexamples F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 19 / 24

Acceleration for arrays Ad-hoc refinement for monotonic abstraction ¬ P τ + J τ i i K + K I F. Alberti Definability of Accelerated Relations in a Theory of Arrays . . . 20 / 24