Program Transformation and Constraint-based Verification Valerio Senni Department of Computer Science, Systems and Production University of Rome ‘Tor Vergata’, Italy joint work with E. De Angelis, F . Fioravanti, A. Pettorossi, M. Proietti CPmeetsCAV 2012, Turunç 1 / 99
Introduction Rule-Based Program Transformation - Origins rule-based �− → . . . �− → initial P 0 P n final model-preserving M ( P 0 ) = . . . = M ( P n ) (local) rewriting An approach to developing correct & efficient programs [Burstall-Darlington 77] → ∗ P 0 �− P n ’easy-to-prove-correct’ correct & efficient M ( P 0 ) = M ( P n ) → ∗ �− ≡ optimization Separate correctness concerns from efficiency concerns → ∗ constructed according to a strategy �− 2 / 99
Introduction Constraint Logic Programming Programs as sets of rules (clauses) of the form: H ← c ∧ B (meaning, H holds if c is satisfiable in T and B holds) ∧ Example: ordered([]) ordered([ x ]) ordered([ x 1 , x 2 | L ]) ← x 1 ≤ x 2 ordered([ x 2 | L ]) ∧ ∧ � �� � � �� � solver for T resolution Query evaluation: ⇒ k d ∧ ∧ G = c 1 ∧ ∧ . . . ∧ ∧ c n ∧ c n T -satisfiable) (with c 1 ∧ ∧ . . . ∧ ρ (and ρ = ϑ 1 · . . . · ϑ k ) 1. d ∧ ∧ G = d ∧ ∧ A ∧ ∧ R current goal ⇒ 1 = is 2. ( A = H ) ϑ find unifying head ϑ 3. ( d ∧ ∧ c ∧ ∧ B ∧ ∧ R ) ϑ rewrite 3 / 99
Introduction Transformation of Constraint Logic Programs A program as a first order theory: theory transformation (changing the axioms of a theory, while preserving the model) Syntax Semantics logic programs least Herbrand model + negation perfect model, stable models + constraints least/perfect D -model 4 / 99
Introduction Rules Definition Introduction �− → newp ( x ) ← c ( x ) ∧ p 1 ( x ) ∧ . . . ∧ p n ( x ) Unfolding p ( x ) ← d 1 ∧ B 1 H ← c ∧ d 1 ∧ B 1 ∧ R . . . . H ← c ∧ p ( x ) ∧ R �− → . . p ( x ) ← d n ∧ B n H ← c ∧ d n ∧ B n ∧ R Folding c ⊑ d p ( x ) ← d ∧ B H ← c ∧ B ∧ R �− → H ← c ∧ p ( x ) ∧ R Clause Removal 1. H ← c ∧ B �− → H ← d if c ⊑ d ( c entails d ) H ← d 2. H ← c ∧ B �− → ∅ if c is unsatisfiable Rearrangement, Addition/Deletion, Constraint Rewriting 5 / 99
Introduction An Introductory Example 6 / 99
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Introduction An Introductory Example 13 / 99
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Introduction An Introductory Example 18 / 99
Introduction Transformation Strategies Directed by syntactic features of programs: Specializing programs to the context of use (pre-computing) snm(S) ← near_match( [2,0] , 2 ,S) Avoiding the computation of unnecessary values ,K,S) ← a( L ,T,S) ∧ near_match(P ∧ a(Q,R,T) ∧ ∧ near(P ,K,Q) Avoding multiple visits of data structures and repeated computations near_match(P ,K,S) ← a(L, T ,S) ∧ ∧ a( Q ,R, T ) ∧ ∧ near(P ,K, Q ) Reducing nondeterminism (avoid multiple matchings per call-pattern) snm([X|S]) ← 0 ≤ X ≤ 4 ∧ ∧ a(Q,R,S) ∧ ∧ n([0],2,Q) snm([X|S]) ← 0 ≤ X ≤ 4 ∧ ∧ a(L,T,S) ∧ ∧ a(Q,R,T) ∧ ∧ near([2,0],2,Q) ⇓ snm([X|S]) ← 0 ≤ X ≤ 4 ∧ ∧ new1(S) 19 / 99
Introduction Transformation Strategies Directed by syntactic features of programs: Specializing programs to the context of use (pre-computing) snm(S) ← near_match( [2,0] , 2 ,S) Avoiding the computation of unnecessary values ,K,S) ← a( L ,T,S) ∧ near_match(P ∧ a(Q,R,T) ∧ ∧ near(P ,K,Q) Avoding multiple visits of data structures and repeated computations near_match(P ,K,S) ← a(L, T ,S) ∧ ∧ a( Q ,R, T ) ∧ ∧ near(P ,K, Q ) Reducing nondeterminism (avoid multiple matchings per call-pattern) snm([X|S]) ← 0 ≤ X ≤ 4 ∧ ∧ a(Q,R,S) ∧ ∧ n([0],2,Q) snm([X|S]) ← 0 ≤ X ≤ 4 ∧ ∧ a(L,T,S) ∧ ∧ a(Q,R,T) ∧ ∧ near([2,0],2,Q) ⇓ snm([X|S]) ← 0 ≤ X ≤ 4 ∧ ∧ new1(S) 20 / 99
Introduction Transformation Strategies Directed by syntactic features of programs: Specializing programs to the context of use (pre-computing) snm(S) ← near_match( [2,0] , 2 ,S) Avoiding the computation of unnecessary values ,K,S) ← a( L ,T,S) ∧ near_match(P ∧ a(Q,R,T) ∧ ∧ near(P ,K,Q) Avoding multiple visits of data structures and repeated computations near_match(P ,K,S) ← a(L, T ,S) ∧ ∧ a( Q ,R, T ) ∧ ∧ near(P ,K, Q ) Reducing nondeterminism (avoid multiple matchings per call-pattern) snm([X|S]) ← 0 ≤ X ≤ 4 ∧ ∧ a(Q,R,S) ∧ ∧ n([0],2,Q) snm([X|S]) ← 0 ≤ X ≤ 4 ∧ ∧ a(L,T,S) ∧ ∧ a(Q,R,T) ∧ ∧ near([2,0],2,Q) ⇓ snm([X|S]) ← 0 ≤ X ≤ 4 ∧ ∧ new1(S) 21 / 99
Introduction Transformation Strategies Directed by syntactic features of programs: Specializing programs to the context of use (pre-computing) snm(S) ← near_match( [2,0] , 2 ,S) Avoiding the computation of unnecessary values ,K,S) ← a( L ,T,S) ∧ near_match(P ∧ a(Q,R,T) ∧ ∧ near(P ,K,Q) Avoding multiple visits of data structures and repeated computations near_match(P ,K,S) ← a(L, T ,S) ∧ ∧ a( Q ,R, T ) ∧ ∧ near(P ,K, Q ) Reducing nondeterminism (avoid multiple matchings per call-pattern) snm([X|S]) ← 0 ≤ X ≤ 4 ∧ ∧ a(Q,R,S) ∧ ∧ n([0],2,Q) snm([X|S]) ← 0 ≤ X ≤ 4 ∧ ∧ a(L,T,S) ∧ ∧ a(Q,R,T) ∧ ∧ near([2,0],2,Q) ⇓ snm([X|S]) ← 0 ≤ X ≤ 4 ∧ ∧ new1(S) 22 / 99
Introduction Rule-Based Program Transformation - More P 0 �− → . . . �− → P n What do we preserve? M ( P 0 ) = . . . = M ( P n ) a model A ∈ M ( P 0 ) A ∈ M ( P n ) iff . . . iff selected predicates M ( P 0 ) � ϕ iff . . . iff M ( P n ) � ϕ a class of formulas → ∗ ≡ deduction �− Depending on the choice of the set of rules and the transformation strategy 23 / 99
Introduction Applications Theorem Proving [Kott,P ,P ,Roychoudhury,Seki] → ∗ S ∪ { p ←} T ∪ { p ← ϕ } �− Program Verification [Albert,Gallagher,Puebla] → ∗ Q ∪ { p ←} ] ∪ { p ← ϕ } �− [ [ P ] where [ [ P ] ] is an encoding of the program semantics (e.g., an interpreter) Program Synthesis [Darlington,Deville,Flener,Hogger,Lau,Manna,Waldinger] → ∗ P T ∪ { p ( x ) ← ϕ ( x ) } �− s.t. T | = ϕ ( a ) iff p ( a ) ∈ M ( P ) 24 / 99
Improving Infinite-state and Software Model Checking Improving Infinite-state Systems Model Checking 25 / 99
Improving Infinite-state and Software Model Checking Specialization-based Model Checking Program Specialization P : I 1 × I 2 − → O By partial evaluation P 1 : I 2 − → O A faster residual program Take advantage of static knowledge Specialization-based Symbolic Model Checking CTL : TS × ϕ × Parameters − → yes/no By partial evaluation MC ϕ − → yes/no TS : Parameters An ad-hoc model checker 26 / 99
Improving Infinite-state and Software Model Checking Specialization-based Model Checking Program Specialization P : I 1 × I 2 − → O By partial evaluation P 1 : I 2 − → O A faster residual program Take advantage of static knowledge Specialization-based Symbolic Model Checking CTL : TS × ϕ × Parameters − → yes/no By partial evaluation MC ϕ − → yes/no TS : Parameters An ad-hoc model checker 27 / 99
Improving Infinite-state and Software Model Checking Specialization Strategy Input : P and a clause δ 0 : p sp ( x ) ← c ( x ) ∧ p ( x ) p sp ( z ) ∈ M ( P ∪ { δ 0 } ) iff p sp ( z ) ∈ M ( SpecP ) Output : SpecP s.t. SpecP := ∅ ; Defs := { δ 0 } ; while ∃ δ ∈ Defs do Γ := Unfold δ ∆ := Simplify Γ (Φ , NewDefs ) := Generalize & Fold ∆ := ( Defs − { δ } ) ∪ NewDefs Defs := SpecP ∪ Φ SpecP od Test 28 / 99
Improving Infinite-state and Software Model Checking Specialization Strategy Input : P and a clause δ 0 : p sp ( x ) ← c ( x ) ∧ p ( x ) p sp ( z ) ∈ M ( P ∪ { δ 0 } ) iff p sp ( z ) ∈ M ( SpecP ) Output : SpecP s.t. SpecP := ∅ ; Defs := { δ 0 } ; while ∃ δ ∈ Defs do Γ := Unfold δ ∆ := Simplify Γ (Φ , NewDefs ) := Generalize & Fold ∆ := ( Defs − { δ } ) ∪ NewDefs Defs := SpecP ∪ Φ SpecP od Test 29 / 99
Improving Infinite-state and Software Model Checking Specialization Strategy Input : P and a clause δ 0 : p sp ( x ) ← c ( x ) ∧ p ( x ) p sp ( z ) ∈ M ( P ∪ { δ 0 } ) iff p sp ( z ) ∈ M ( SpecP ) Output : SpecP s.t. SpecP := ∅ ; Defs := { δ 0 } ; while ∃ δ ∈ Defs do Γ := Unfold δ (Propagate Context) ∆ := Simplify Γ (Φ , NewDefs ) := Generalize & Fold ∆ (Apply Induction) := ( Defs − { δ } ) ∪ NewDefs Defs := SpecP ∪ Φ SpecP od Test 30 / 99
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