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**Hoare’s Correctness Triplets Dijkstra’s Predicate Transformers**

Axiomatic Semantics Hoare’s Correctness Triplets Dijkstra’s Predicate Transformers Calculating with programs : Reasoning reduced to symbol manipulation. Helps determine precise Boundary conditions. Formalizing intuitions. Other approaches: Denotational Semantics: Real meaning in terms of functions on N. Equivalence: f(x) = f(x) f(x) = if f(x) ==1 then 0 else 1 unsatisfiable (“non-sense”) f(x) = f(x) f(x) = if f(x) ==1 then 1 else f(x) multiple solutions (“no information”) McCarthy’s 91-function Operational Semantics: Abstract interpreter based

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**gcd-lcm algorithm w/ invariant**

{PRE: (x = n) and (y = m)} u := x; v := y; while {INV: 2*m*n = x*v + y*u} (x <> y) do if x > y then x := x - y; u := u + v else y := y - x; v := v + u fi od {POST:(x = gcd(m, n)) and (lcm(m, n) = (u+v) div 2)} cs784(tk/pm)

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**Goal of a program = IO Relation**

Problem Specification Properties satisfied by the input and expected of the output (usually described using “assertions”). E.g., Sorting problem Input : Sequence of numbers Output: Permutation of input that is ordered. View Point All other properties are ignored. Timing behavior Resource consumption … cs784(tk/pm)

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**ax·i·om n.1. A self-evident or universally recognized truth; a maxim**

2. An established rule, principle, or law. 3. A self-evident principle or one that is accepted as true without proof as the basis for argument; a postulate. From a dictionary cs784(tk/pm)

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Axiomatic Semantics Capture the semantics of the elements of the PL as axioms Capture the semantics of composition as a rule of inference. Apply the standard rules/logic of inference. Consider termination separately. cs784(tk/pm)

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**States and Assertions States: Variables mapped to Values**

Includes all variables Files etc. are considered “global” variables. No notion of value-undefined variables At a given moment in execution An assertion is a logic formula involving program variables, arithmetic/boolean operations, etc. All assertions are attached to a control point. Assertions: States mapped to Boolean Boolean connectives: and, or, not, implies, … For-all, There-exists Special predicates defined just for use in assertions (not for use in the program). cs784(tk/pm)

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**Hoare’s Logic Hoare Triplets: {P} S {Q}**

P, pre-condition assertion; S, statements of a PL; Q, post-condition assertion If S begins executing in a state satisfying P, upon completion of S, the resulting state satisfies Q. {P} S {Q} has no relevance if S is begun otherwise. A Hoare triplet is either true or false. Never undefined. The entire {P}S{Q} is considered true if the resulting state satisfies Q if and when S terminates. If not, the entire {P}S{Q} is false. cs784(tk/pm)

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**Hoare Triplet Examples**

true triplets {x = 11 } x := 0 { x = 0 } we can give a weaker precondition {x = 0 } x := x + 1 { x = 1 } {y = 0} if x <> y then x:= y fi { x = 0 } {false } x := 0 { x = 111 } correct because “we cannot begin” no state satisfies false post condition can be any thing you dream {true} while true do od {x = 0} true is the weakest of all predicates correct because control never reaches post {false} while true do od {x = 0} false is the strongest of all predicates false triplet {true} if x < 0 then x:= -x fi { x > 0 } 1. False = empty set of states. Precondition unsatisfiable, so Hoare triple trivially valid. 2. Strong precondition 4. Nontermination. 5. Multipath program : Else null statement; 6. Partially correct because the IF-part of definition not met, so there are no guarantees from THEN. 7. Modify the precondition to get a valid triple. Material Implication : IF sun rises in the west THEN there will be snow in July in Mexico City. cs784(tk/pm)

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Weaker/Stronger An assertion R is said to be weaker than assertion P if the truth of P implies the truth of R written: P→R equivalently not P or R. For arbitrary A, B we have: A and B → B This general idea is from Propositional Calculus n > 0 is of course weaker than n = 1, but this follows from Number Theory. cs784(tk/pm)

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**Weaker/Stronger Q’ stronger P’ weaker Q’ Q P P’ States States P’ Q**

The program transforms a state into another state. (point to point map) Assertions characterize a collection of states. cs784(tk/pm)

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**Partial vs Total Correctness**

Are P and S such that termination is guaranteed? S is partially correct for P and Q iff whenever the execution terminates, the resulting state satisfies Q. S is totally correct for P and Q iff the execution is guaranteed to terminate, and the resulting state satisfies Q. Logical Implication : IF false THEN (1 = 2) is valid. cs784(tk/pm)

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**Hoare Triplet Examples**

Totally correct (hence, partially correct) {x = 11} x := 0 {x = 0} {x = 0} x := x + 1 {x = 1} {y = 0}if x <> y then x:= y fi {x = 0} {false} while true do S od {x = 0} {false} x := 0 {x = 111} Not totally correct, but partially correct {true} while true do S od {x = 0} Not partially correct {true} if x < 0 then x:= -x fi {x > 0} False = empty set of states. Precondition unsatisfiable, so Hoare triple trivially valid. 1a. Unecessarily Strong precondition 1d and 2. Nontermination. 3. Multipath program : Else null statement; 6. Partially correct because the IF-part of definition not met, so there are no guarantees from THEN. 7. Modify the precondition to get a valid triple. Material Implication : IF sun rises in the west THEN there will be snow in July in Mexico City. cs784(tk/pm)

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**Assignment axiom {Q(e)} x := e {Q(x)} Q(x) has free occurrences of x.**

Q(e): every free x in Q replaced with e Assumption: e has no side effects. Caveats If x is not a “whole” variable (e.g., a[2]), we need to work harder. PL is assumed to not facilitate aliasing. cs784(tk/pm)

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**Inference Rules Rules are written as Can also be stated as:**

Hypotheses: H1, H2, H Conclusion: C1 Can also be stated as: H1 and H2 and H3 implies C1 Given H1, H2, and H3, we can conclude C1. cs784(tk/pm)

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**Soundness and Completeness**

Soundness is about “validity” Completeness is about “deducibililty” Ideally in a formal system, we should have both. Godel’s Incompleteness Theorem: Cannot have both Inference Rules ought to be sound What we proved/ inferred/ deduced is valid Examples of Unsound Rules A and B and C not B x > y implies x > y+1 (in the context of numbers) All the rules we present from now on are sound cs784(tk/pm)

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**Rule of Consequence Suppose: {P’} S {Q’}, P=>P’, Q’=>Q**

Conclude: {P} S {Q} Replace precondition by a stronger one postcondition by a weaker one cs784(tk/pm)

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**Statement Composition Rule**

{P} S1 {R}, {R} S2 {Q} {P} S1;S2 {Q} Using Rule of Consequence {P} S1 {R1}, R1 R2, {R2} S2 {Q} {P} S1;S2 {Q} Reasoning turned into symbol manipulation : Substitution. Confusing with constructs such as: {??} x == x++ * 5 { x = y } cs784(tk/pm)

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**if-then-else-fi Hoare’s Triplets**

{P and B} S1 {Q} {P and not B} S2 {Q} {P} if B then S1 else S2 fi {Q} We assumed that B is side-effect free Execution of B does not alter state cs784(tk/pm)

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Invariants An invariant is an assertion whose truth-value does not change Recall: All assertions are attached to a control point. An Example: x > y The values of x and y may or may not change each time control reaches that point. But suppose the > relationship remains true. Then x > y is an invariant Focus on what is essential for the problem at hand rather than the weakest conditiion. Invariant = truth preserved. cs784(tk/pm)

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**Loop Invariants Semantics of while-loop {I and B} S {I}**

{I} while B do S od {I and not B} Termination of while-loop is not included in the above. We assumed that B is side-effect free. cs784(tk/pm)

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**Data Invariants Well-defined OOP classes**

Public methods ought to have a pre- and post-conditions defined There is a common portion across all public methods That common portion is known as the data invariant of the class. cs784(tk/pm)

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**while-loop: Hoare’s Approach**

Wish to prove: {P} while B do S od {Q} Given: P, B, S and Q Not given: a loop invariant I The I is expected to be true just before testing B To prove {P} while B do S od {Q}, discover a strong enough loop invariant I so that P => I {I and B} S {I} I and not B => Q We used the Rule of Consequence twice Focus on what is essential for the problem at hand rather than the weakest conditiion. Invariant = truth preserved. cs784(tk/pm)

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**A while-loop example { n > 0 and x = 1 and y = 1}**

while (n > y) do y := y + 1; x := x*y od {x = n!} Choosing invariant requires insight and is goal driven. Invariant must hold in the loop. So cannot have (n = y) or (x = n!) etc n > 0 implies n >= 1 because n is of type natural-number cs784(tk/pm)

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**while-loop: Choose the Invariant**

Invariant I should be such that I and not B Q I and not (n > y) (x = n!) Choose (n ≥ y and x = y!) as our I Precondition Invariant n > 0 and x=1 and y=1 n ≥ 1 and 1=1! Choosing invariant requires insight and is goal driven. Invariant must hold in the loop. So cannot have (n = y) or (x = n!) etc n > 0 implies n >= 1 because n is of type natural-number cs784(tk/pm)

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**while-loop: Verify Invariant**

I === n ≥ y and x = y! Verify: {I and n > y} y:= y + 1; x:=x*y {I} {I and n > y} y:= y + 1 {n ≥ y and x*y = y!} {I and n > y} y:= y + 1 {n ≥ y and x= (y-1)!} (I and n > y) (n ≥ y+1 and x= (y+1-1)!) (I and n > y) (n > y and x= y!) (n ≥ y and x = y! and n > y) (n > y and x= y!) QED Hoare’s triplets, but also using wp(). cs784(tk/pm)

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**while-loop: I and not B Q**

I === n ≥ y and x = y! n ≥ y and x = y! and not (n > y) x = n! n = y and x = y! x = n! QED Hoare’s triplets, but also using wp(). cs784(tk/pm)

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**while-loop: Termination**

Termination is not part of Hoare’s Triplets General technique: Find a quantity that decreases in every iteration. And, has a lower bound The quantity may or may not be computed by the algorithm For our example: Consider n – y values of y: 1, 2, …, n-1, n values of n - y: n-1, n-2, …, 1, 0 Hoare’s triplets, but also using wp(). cs784(tk/pm)

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**Weakest Preconditions**

We want to determine minimally what must be true immediately before executing S so that assertion Q is true after S terminates. S is guaranteed to terminate The Weakest-Precondition of S is a mathematical function mapping any post condition Q to the "weakest" precondition Pw. Pw is a condition on the initial state ensuring that execution of S terminates in a final state satisfying R. Among all such conditions Pw is the weakest wp(S, Q) = Pw cs784(tk/pm)

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**Dijkstra’s wp(S, Q) Let Pw = wp(S, Q)**

Def of wp(S, Q): Weakest precondition such that if S is started in a state satisfying Pw, S is guaranteed to terminate and Q holds in the resulting state. Consider all predicates Pi so that {Pi}S{Q}. Discard any Pi that does not guarantee termination of S. Among the Pi remaining, choose the weakest. This is Pw. {P} S {Q} versus P => wp(S, Q) {Pw} S {Q} is true. But, the semantics of {Pw} S {Q} does not include termination. If P => wp(S, Q) then {P}S{Q} also, and furthermore S terminates. cs784(tk/pm)

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**Properties of wp Law of the Excluded Miracle wp(S, false) = false**

Distributivity of Conjunction wp(S, P and Q) = wp(S,P) and wp(S,Q) Law of Monotonicity (Q→R) → (wp(S,Q)→wp(S,R)) Distributivity of Disjunction wp(S,P) or wp(S, Q) → wp(S,P or Q) cs784(tk/pm)

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**Predicate Transformers**

Assignment wp(x := e, Q(x)) = Q(e) Composition wp(S1;S2, Q) = wp(S1, wp(S2,Q)) For programs without loops (and recursion), Hoare’s and Dijkstra’s approach converge. Hoare’s approach generates triples while Dijkstra’s approach tries to capture the semantics Unaddressed Questions: Why weakest precondition rather than strongest post-condition? Expressiveness in FOL for while? Application: Distributed computing program proofs. cs784(tk/pm)

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**A Correctness Proof {x=0 and y=0} x:=x+1;y:=y+1 {x = y}**

wp(x:=x+1;y:=y+1, x = y) wp(x:=x+1, wp(y:=y+1, x = y)) === wp(x:=x+1, x = y+1) === x+1 = y+1 === x = y x = 0 and y = 0 x = y use a better example === at the meta level, = part of assertion language syntax cs784(tk/pm)

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**if-then-else-fi in Dijkstra’s wp**

wp(if B then S1 else S2 fi, Q) === (B wp(S1,Q)) and (not B wp(S2,Q)) === (B and wp(S1,Q)) or (not B and wp(S2,Q)) cs784(tk/pm)

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**wp-semantics of while-loops**

DO == while B do S od IF == if B then S fi Let k stand for the number of iterations of S Clearly, k >= 0 If k > 0, while B do S od is the same as: if B then S fi; while B do S od cs784(tk/pm)

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**while-loop: wp Approach**

wp(DO, Q) = P0 or there-exists k > 0: Pk States satisfying Pi cause i-iterations of while-loop before halting in a state in Q. Pi defined inductively P0 = not B and Q … cs784(tk/pm)

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**wp(DO, Q) There exists a k, k ≥ 0, such that H(k, Q)**

H is defined as follows H(0, Q) = not B and Q H(k, Q) = H(0, Q) or wp(IF, H(k-1, Q)) cs784(tk/pm)

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**Example (same as before)**

{ n>0 and x=1 and y=1} while (n > y) do y := y + 1; x := x*y od {x = n!} cs784(tk/pm)

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**Example: while-loop correctness**

Pre === n>0 and x=1 and y=1 P0 === y >= n and x = n! Pk === B and wp(S, Pk-1) P1 === y > n and y+1>=n and x*(y+1) = n! Pk === y=n-k and x=(n-k)! Weakest Precondition: W === there exists k >= 0 such that P0 or Pk Verification : For k = n-1: Pre => W cs784(tk/pm)

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**Induction Proof Hypothesis : Pk = (y=n-k and x=(n-k)!)**

Pk+1 = B and wp(S,Pk) = y<n and (y+1 = n-k) and (x*(y+1)=(n-k)!) = y<n and (y = n-k-1) and (x = (n-k-1)!) = y<n and (y = n- k+1) and (x = (n- k+1)!) = (y = n - k+1) and (x = (n - k+1)!) Examples of Valid preconditions: { n = 4 and y = 2 and x = 2 } (k = 2) { n = 5 and x = 5! and y = 6} (no iteration) cs784(tk/pm)

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**Detailed Work wp(y:=y+1;x:=x*y, x=y!and n>=y)**

= wp(y:=y+1, x*y=y! and n>=y) = wp(y:=y+1, x=(y-1)! and n>=y) = x=(y+1-1)! and n>=y+1) = x=y! and n>y cs784(tk/pm)

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**gcd-lcm algorithm w/ invariant**

{PRE: (x = n) and (y = m)} u := x; v := y; while {INV: 2*m*n = x*v + y*u} (x <> y) do if x > y then x := x - y; u := u + v else y := y - x; v := v + u fi od {POST:(x = gcd(m, n)) and (lcm(m, n) = (u+v) div 2)} cs784(tk/pm)

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**gcd-lcm algorithm proof**

PRE implies Loop Invariant (x = n) and (y = m) implies 2*m*n = x*v + y*u {Invariant and B} Loop-Body {Invariant} {2*n*m = x*v + y*u and x <> y} loop-body {2*n*m = x*v + y*u} Invariant and not B implies POST 2*n*m = x*v + y*u and x == y implies (x = gcd(n,m)) and (lcm(n,m) = (u+v) div 2) cs784(tk/pm)

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**gcd-lcm algorithm proof**

Invariant and not B implies POST 2*m*n = x*v + y*u and x == y implies (x = gcd(m, n)) and (lcm(m, n) = (u+v) div 2) Simplifying 2*m*n = x*(u + v) and x == y implies (x = gcd(m, n)) and (lcm(m, n) = (u+v) div 2) cs784(tk/pm)

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**gcd-lcm algorithm proof**

Simplifying 2*m*n = x*(u + v) and x == y implies (x = gcd(m, n)) and (x*lcm(m, n) = m*n) cs784(tk/pm)

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**Some Properties of lcm-gcd**

gcd() and lcm() are symmetric gcd(m, n) = gcd(n, m) lcm(m, n) = lcm(n, m) gcd(m, n) = gcd(m + k*n, n) where k is a natural number. gcd(m, n) * lcm(m, n) = m * n cs784(tk/pm)

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