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Program Analysis and Verification Noam Rinetzky Lecture 2: Program Semantics Slides credit: Roman Manevich, Mooly Sagiv, Eran Yahav

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Good manners Mobiles 2

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Admin Grades – 4 Assignments (30%) 1 involves programming – 1 Lesson summary (10%) – Toar I: Final exam (60%) Must pass – Toar II: Project (60%) Scribes (this week) ?Scribes (nextweek) 3

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Today What does semantics mean? – Why do we need it? – How is it related to analysis/verification? Operational semantics – Natural operational semantics – Structural operational semantics 4

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Motivation: Verifying absence of bugs static OSStatus SSLVerifySignedServerKeyExchange(SSLContext *ctx, bool isRsa, SSLBuffer signedParams, uint8_t *signature, UInt16 signatureLen) { OSStatus err;... if ((err = SSLHashSHA1.update(&hashCtx, &serverRandom)) != 0) goto fail; if ((err = SSLHashSHA1.update(&hashCtx, &signedParams)) != 0) goto fail; if ((err = SSLHashSHA1.final(&hashCtx, &hashOut)) != 0) goto fail;... fail: SSLFreeBuffer(&signedHashes); SSLFreeBuffer(&hashCtx); return err; } 5

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What can we do about it? Monitoring Testing Static analysis Formal verification Specification Run time Design Time 6

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What can we do about it? Monitoring Testing Static analysis Formal verification Specification Run time Design Time 7

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Program analysis & verification y = ? x = ? if (x > 0) { y = 42; } else { y = 73; foo(); } assert (y == 42); Is assertion true? No/?/Yes 8

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Program analysis & verification Can we prove this? Automatically? Bad news: problem is generally undecidable Yes/?/No y = ? x = ? x = y * 2 if (x % 2 == 0) { y = 42; } else { y = 73; foo(); } assert (y == 42); 9

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universe Main idea: use over-approximation Exact set of configurations/ Behaviors/stat es Over Approximation 10

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Main idea: find (properties of) all reachable states* initial states bad states reachable states *or of something else … 11

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Technique: find (properties of) more than all reachable states* initial states bad states reachable states *or of something else … 12

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Program analysis & verification y = ?; x = ?; x = y * 2 if (x % 2 == 0) { y = 42; } else { y = 73; foo(); } assert (y == 42); ? 13

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What does P do? y = ?; x = ?; x = y * 2 if (x % 2 == 0) { y = 42; } else { y = 73; foo(); } assert (y == 42); ? 14

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What does P mean? y = ?; x = ?; x = y * 2 if (x % 2 == 0) { y = 42; } else { y = 73; foo(); } assert (y == 42); … syntax semantics 15

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“Standard” semantics y = ?; x = y * 2 if (x % 2 == 0) { y = 42; } else { y = 73; foo(); } assert (y == 42); …-1,0,1, … yx 16

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“Standard” semantics (“state transformer”) y = ?; x = y * 2 if (x % 2 == 0) { y = 42; } else { y = 73; foo(); } assert (y == 42); …-1,0,1, … yx 17

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“Standard” semantics (“state transformer”) y = ?; y=3, x=9 x = y * 2 if (x % 2 == 0) { y = 42; } else { y = 73; foo(); } assert (y == 42); …-1,0,1, … yx 18

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“Standard” semantics (“state transformer”) y = ?; y=3, x=9 x = y * 2y=3, x=6 if (x % 2 == 0) {y=3, x=6 y = 42;y=42, x=6 } else { y = 73;… foo();… } assert (y == 42); y=42, x=6 …-1,0,1, … yx 19

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“State transformer” semantics initial states bad states reachable states y=3,x=9 y=3,x=6 20

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“State transformer” semantics initial states bad states reachable states y=4,x=1 y=4,x=8 21

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“State transformer” semantics initial states bad states reachable states y=4…,x=… 22

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initial states bad states reachable states y=3,x=9 y=3,x=6 y=4,x=1 y=4,x=8 y=3,x=6 y=4,x=8 “State transformer” semantics Main idea: find (properties of) all reachable states* y=4…,x=… 23

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“Standard” (collecting) semantics (“sets-of states-transformer”) y = ?; x = ?;{(y,x) | y,x ∈ Nat} x = y * 2 if (x % 2 == 0) { y = 42; } else { y = 73; foo(); } assert (y == 42); 24

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“Standard” (collecting) semantics (“sets-of states-transformer”) y = ?; {(y=3, x=9),(y=4,x=1),(y=…, x=…)} x = y * 2 {(y=3, x=6),(y=4,x=8),(y=…, x=…)} if (x % 2 == 0) { {(y=3, x=6),(y=4,x=8),(y=…, x=…)} y = 42; {(y=42, x=6),(y=42,x=8),(y=42, x=…)} } else { y = 73; { } foo(); { } } assert (y == 42); {(y=42, x=6),(y=42,x=8),(y=42, x=…)} Yes 25

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“Set-of-states transformer” semantics initial states bad states reachable states y=3,x=9 y=3,x=6 y=4,x=1 y=3,x=6 y=4,x=1 26

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“Abstract-state transformer” semantics y = ?; y=T, x=T x = y * 2 if (x % 2 == 0) { y = 42; } else { y = 73; foo(); } assert (y == 42); T OE T yx T OE T (y=E,x=E)={(0,0), (0,2), (-4,10),…} 27

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“Abstract-state transformer” semantics y = ?; y=T, x=T x = y * 2y=T, x=E if (x % 2 == 0) {y=T, x=E y = 42;y=T, x=E } else { y = 73;… foo();… } assert (y == 42); y=E, x=E Yes/?/No T OE T yx T OE T (y=E,x=E)={(0,0), (0,2), (-4,10),…} 28

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“Abstract-state transformer” semantics y = ?; y=T, x=T x = y * 2y=T, x=E if (x % 2 == 0) {y=T, x=E y = 42;y=T, x=E } else { y = 73;… foo();… } assert (y == 42); y=E, x=E Yes/?/No T OE T yx T OE T (y=E,x=E)={(0,0), (0,2), (-4,10),…} 29

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“Abstract-state transformer” semantics y = ?; y=T, x=T x = y * 2y=T, x=E if (x % 2 == 0) {y=T, x=E y = 42;y=E, x=E } else { y = 73;… foo();… } assert (y%2 == 0) y=E, x=E ? T OE T yx T OE T (y=E,x=E)={(0,0), (0,2), (-4,10),…} 30

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“Abstract-state transformer” semantics initial states bad states 31 reachable states

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“Abstract-state transformer” semantics initial states bad states 32 reachable states

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“Abstract-state transformer” semantics initial states bad states 33 reachable states

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“Abstract-state transformer” semantics initial states bad states 34 reachable states

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Technique: explore abstract states initial states bad states 35 reachable states

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Sound: cover all reachable states 36 initial states bad states reachable states

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37 Imprecise abstraction initial states bad states 37 reachable states False alarms

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What does P mean? y = ?; x = ?; x = y * 2 if (x % 2 == 0) { y = 42; } else { y = 73; foo(); } assert (y == 42); … syntax semantics Abs 38

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Programming Languages Syntax “how do I write a program?” – BNF – “Parsing” Semantics “What does my program mean?” – … 39

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Program semantics State-transformer – Set-of-states transformer – Trace transformer Predicate-transformer Functions 40

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Program semantics State-transformer – Set-of-states transformer – Trace transformer Predicate-transformer Functions Cat-transformer 41

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What semantics do we want? Captures the aspects of computations we care about – “adequate” Hides irrelevant details – “fully abstract” Compositional 42

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Operational Semantics

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Recap

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Program semantics State-transformer – Set-of-states transformer – Trace transformer Predicate-transformer Functions Cat-transformer

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What semantics do we want? Captures the aspects of computations we care about – “adequate” Hides irrelevant details – “fully abstract” Compositional

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Formal semantics 47 “Formal semantics is concerned with rigorously specifying the meaning, or behavior, of programs, pieces of hardware, etc.” / page 1

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Formal semantics 48 “This theory allows a program to be manipulated like a formula – that is to say, its properties can be calculated.” Gérard Huet & Philippe Flajolet homage to Gilles Kahn

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Why formal semantics? Implementation-independent definition of a programming language Automatically generating interpreters (and some day maybe full fledged compilers) Verification and debugging – if you don’t know what it does, how do you know its incorrect? 49

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Levels of abstractions and applications 50 Program Semantics Assembly-level Semantics (Small-step) Static Analysis (abstract semantics)

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Semantic description methods Operational semantics – Natural semantics (big step) [G. Kahn] – Structural semantics (small step) [G. Plotkin] Trace semantics Collecting semantics [Instrumented semantics] Denotational semantics [D. Scott, C. Strachy] Axiomatic semantics [C. A. R. Hoare, R. Floyd] 51

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Operational Semantics

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A simple imperative language: While Abstract syntax: a ::= n | x | a 1 + a 2 | a 1 a 2 | a 1 – a 2 b ::= true | false | a 1 = a 2 | a 1 a 2 | b | b 1 b 2 S ::= x := a | skip | S 1 ; S 2 | if b then S 1 else S 2 | while b do S 54

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Concrete Syntax vs. Abstract Syntax 55 a y := x S a z y Sa x z S ; S ; S a x z S a y x S ; S ; S a z y S z:=x; x:=y; y:=z z:=x; (x:=y; y:=z)(z:=x; x:=y); y:=z

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Exercise: draw an AST 56 SS ; S y:=1; while (x=1) do (y:=y*x; x:=x-1)

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Syntactic categories n Numnumerals x Varprogram variables a Aexparithmetic expressions b Bexpboolean expressions S Stmstatements 57

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Semantic categories Z Integers {0, 1, -1, 2, -2, …} T Truth values { ff, tt } State Var Z Example state:s=[ x 5, y 7, z 0] Lookup: s x = 5 Update: s[ x 6] = [ x 6, y 7, z 0] 58

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Example state manipulations [ x 1, y 7, z 16] y = [ x 1, y 7, z 16] t = [ x 1, y 7, z 16][ x 5] = [ x 1, y 7, z 16][ x 5] x = [ x 1, y 7, z 16][ x 5] y = 59

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Semantics of arithmetic expressions Arithmetic expressions are side-effect free Semantic function A Aexp : State Z Defined by induction on the syntax tree A n s = n A x s = s x A a 1 + a 2 s = A a 1 s + A a 2 s A a 1 - a 2 s = A a 1 s - A a 2 s A a 1 * a 2 s = A a 1 s A a 2 s A (a 1 ) s = A a 1 s --- not needed A - a s = 0 - A a 1 s Compositional Properties can be proved by structural induction 60

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Arithmetic expression exercise Suppose s x = 3 Evaluate A x+1 s 61

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Semantics of boolean expressions Boolean expressions are side-effect free Semantic function B Bexp : State T Defined by induction on the syntax tree B true s = tt B false s = ff B a 1 = a 2 s = B a 1 a 2 s = B b 1 b 2 s = B b s = 62

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Operational semantics Concerned with how to execute programs – How statements modify state – Define transition relation between configurations Two flavors – Natural semantics: describes how the overall results of executions are obtained So-called “big-step” semantics – Structural operational semantics: describes how the individual steps of a computations take place So-called “small-step” semantics 63

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Natural operating semantics (NS) 64

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Natural operating semantics (NS) aka “Large-step semantics” 65 S, s s’ all steps

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Natural operating semantics Developed by Gilles Kahn [STACS 1987]STACS 1987 Configurations S, s Statement S is about to execute on state s sTerminal (final) state Transitions S, s s’Execution of S from s will terminate with the result state s’ – Ignores non-terminating computations 66

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Natural operating semantics defined by rules of the form The meaning of compound statements is defined using the meaning immediate constituent statements 67 S 1, s 1 s 1 ’, …, S n, s n s n ’ S, s s’ if… premise conclusion side condition

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Natural semantics for While 68 x := a, s s[x A a s] [ass ns ] skip, s s [skip ns ] S 1, s s’, S 2, s’ s’’ S 1 ; S 2, s s’’ [comp ns ] S 1, s s’ if b then S 1 else S 2, s s’ if B b s = tt [if tt ns ] S 2, s s’ if b then S 1 else S 2, s s’ if B b s = ff [if ff ns ] axioms

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Natural semantics for While 69 S, s s’, while b do S, s’ s’’ while b do S, s s’’ if B b s = tt [while tt ns ] while b do S, s s if B b s = ff [while ff ns ] Non-compositional

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Example Let s 0 be the state which assigns zero to all program variables 70 x:=x+1, s 0 skip, s 0 s 0 s 0 [x 1] skip, s 0 s 0, x:=x+1, s 0 s 0 [x 1] skip ; x:=x+1, s 0 s 0 [x 1] x:=x+1, s 0 s 0 [x 1] if x=0 then x:=x+1 else skip, s 0 s 0 [x 1]

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Derivation trees Using axioms and rules to derive a transition S, s s’ gives a derivation tree – Root: S, s s’ – Leaves: axioms – Internal nodes: conclusions of rules Immediate children: matching rule premises 71

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Derivation tree example 1 Assumes 0 =[x 5, y 7, z 0] s 1 =[x 5, y 7, z 5] s 2 =[x 7, y 7, z 5] s 3 =[x 7, y 5, z 5] 72 ( z:=x; x:=y); y:=z, s 0 s 3 ( z:=x; x:=y), s 0 s 2 y:=z, s 2 s 3 z:=x, s 0 s 1 x:=y, s 1 s 2 [ass ns ] [comp ns ]

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Derivation tree example 1 Assumes 0 =[x 5, y 7, z 0] s 1 =[x 5, y 7, z 5] s 2 =[x 7, y 7, z 5] s 3 =[x 7, y 5, z 5] 73 ( z:=x; x:=y); y:=z, s 0 s 3 ( z:=x; x:=y), s 0 s 2 y:=z, s 2 s 3 z:=x, s 0 s 1 x:=y, s 1 s 2 [ass ns ] [comp ns ]

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Top-down evaluation via derivation trees Given a statement S and an input state s find an output state s’ such that S, s s’ Start with the root and repeatedly apply rules until the axioms are reached – Inspect different alternatives in order In While s’ and the derivation tree is unique 74

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Top-down evaluation example Factorial program with s x = 2 Shorthand: W= while (x=1) do (y:=y*x; x:=x-1) 75 y:=1; while (x=1) do (y:=y*x; x:=x-1), s s[y 2][x 1] y:=1, s s[y 1] W, s[y 1] s[y 2, x 1] y:=y*x; x:=x-1, s[y 1] s[y 2][x 1] W, s[y 2][x 1] s[y 2, x 1] y:=y*x, s[y 1] s[y 2] x:=x-1, s[y 2] s[y 2][x 1] [ass ns ] [comp ns ] [ass ns ] [comp ns ] [while ff ns ] [while tt ns ] [ass ns ]

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Program termination Given a statement S and input s – S terminates on s if there exists a state s’ such that S, s s’ – S loops on s if there is no state s’ such that S, s s’ Given a statement S – S always terminates if for every input state s, S terminates on s – S always loops if for every input state s, S loops on s 76

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Semantic equivalence S 1 and S 2 are semantically equivalent if for all s and s’ S 1, s s’ if and only if S 2, s s’ Simple example while b do S is semantically equivalent to: if b then (S; while b do S) else skip – Read proof in pages

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Properties of natural semantics Equivalence of program constructs – skip; skip is semantically equivalent to skip – ((S 1 ; S 2 ); S 3 ) is semantically equivalent to (S 1 ; (S 2 ; S 3 )) – (x:=5; y:=x*8) is semantically equivalent to (x:=5; y:=40) 78

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Equivalence of (S 1 ; S 2 ); S 3 and S 1 ; (S 2 ; S 3 ) 79

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Equivalence of (S 1 ; S 2 ); S 3 and S 1 ; (S 2 ; S 3 ) 80 (S 1 ; S 2 ), s s 12, S 3, s 12 s’ (S 1 ; S 2 ); S 3, s s’ S 1, s s 1, S 2, s 1 s 12 S 1, s s 1, (S 2 ; S 3 ), s 12 s’ (S 1 ; S 2 ); S 3, s s’ S 2, s 1 s 12, S 3, s 12 s’ Assume (S 1 ; S 2 ); S 3, s s’ then the following unique derivation tree exists: Using the rule applications above, we can construct the following derivation tree: And vice versa.

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Deterministic semantics for While Theorem: for all statements S and states s 1, s 2 if S, s s 1 and S, s s 2 then s 1 =s 2 The proof uses induction on the shape of derivation trees (pages 29-30) – Prove that the property holds for all simple derivation trees by showing it holds for axioms – Prove that the property holds for all composite trees: For each rule assume that the property holds for its premises (induction hypothesis) and prove it holds for the conclusion of the rule 81 single node #nodes>1

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The semantic function S ns The meaning of a statement S is defined as a partial function from State to State S ns : Stm (State State) Examples: S ns skip s = s S ns x:=1 s = s [x 1] S ns while true do skip s = undefined 82 S ns S s = s’ if S, s s’ undefined else

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Structural operating semantics (SOS) aka “Small-step semantics” 83 S, s S’, s’ first step

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Structural operational semantics Developed by Gordon Plotkin Configurations: has one of two forms: S, s Statement S is about to execute on state s sTerminal (final) state Transitions S, s = S’, s’ Execution of S from s is not completed and remaining computation proceeds from intermediate configuration = s’Execution of S from s has terminated and the final state is s’ S, s is stuck if there is no such that S, s 84 first step

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Structural semantics for While 85 x:=a, s s[x A a s] [ass sos ] skip, s s [skip sos ] S 1, s S 1 ’, s’ S 1 ; S 2, s S 1 ’; S 2, s’ [comp 1 sos ] if b then S 1 else S 2, s S 1, s if B b s = tt [if tt sos ] if b then S 1 else S 2, s S 2, s if B b s = ff [if ff sos ] S 1, s s’ S 1 ; S 2, s S 2, s’ [comp 2 sos ] When does this happen?

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Structural semantics for While 86 while b do S, s if b then S; while b do S ) else skip, s [while sos ]

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Derivation sequences A derivation sequence of a statement S starting in state s is either A finite sequence 0, 1, 2 …, k such that 1. 0 = S, s 2. i i+1 3. k is either stuck configuration or a final state An infinite sequence 0, 1, 2, … such that 1. 0 = S, s 2. i i+1 Notations: – 0 k k 0 derives k in k steps – 0 * 0 derives in a finite number of steps For each step there is a corresponding derivation tree 87

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Derivation sequence example Assume s 0 =[x 5, y 7, z 0] 88 (z:=x; x:=y); y:=z, s 0 x:=y; y:=z, s 0 [z 5] y:=z, (s 0 [z 5])[x 7] ((s 0 [z 5])[x 7])[y 5] (z:=x; x:=y); y:=z, s 0 x:=y; y:=z, s 0 [z 5] z:=x; x:=y, s 0 x:=y, s 0 [z 5] z:=x, s 0 s 0 [z 5] Derivation tree for first step:

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Evaluation via derivation sequences For any While statement S and state s it is always possible to find at least one derivation sequence from S, s – Apply axioms and rules forever or until a terminal or stuck configuration is reached Proposition: there are no stuck configurations in While 89

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Factorial (n!) example Input state s such that s x = 3 y := 1; while (x=1) do (y := y * x; x := x – 1) 90 y :=1 ; W, s W, s[y 1] if (x =1) then ((y := y * x; x := x – 1); W else skip), s[y 1] ((y := y * x; x := x – 1); W), s[y 1] (x := x – 1; W), s[y 3] W, s[y 3][x 2] if (x =1) then ((y := y * x; x := x – 1); W else skip), s[y 3][x 2] ((y := y * x; x := x – 1); W), s[y 3] [x 2] (x := x – 1; W), s[y 6] [x 2] W, s[y 6][x 1] if (x =1) then ((y := y * x; x := x – 1); W else skip, s[y 6][x 1] skip, s[y 6][x 1] s[y 6][x 1]

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Program termination Given a statement S and input s – S terminates on s if there exists a finite derivation sequence starting at S, s – S terminates successfully on s if there exists a finite derivation sequence starting at S, s leading to a final state – S loops on s if there exists an infinite derivation sequence starting at S, s 91

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Properties of structural operational semantics S 1 and S 2 are semantically equivalent if: – for all s and which is either final or stuck, S 1, s * if and only if S 2, s * – for all s, there is an infinite derivation sequence starting at S 1, s if and only if there is an infinite derivation sequence starting at S 2, s Theorem: While is deterministic: – If S, s * s 1 and S, s * s 2 then s 1 =s 2 92

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Sequential composition Lemma: If S 1 ; S 2, s k s’’ then there exists s’ and k=m+n such that S 1, s m s’ and S 2, s’ n s’’ The proof (pages 37-38) uses induction on the length of derivation sequences – Prove that the property holds for all derivation sequences of length 0 – Prove that the property holds for all other derivation sequences: Show that the property holds for sequences of length k+1 using the fact it holds on all sequences of length k (induction hypothesis) 93

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The semantic function S sos The meaning of a statement S is defined as a partial function from State to State S sos : Stm (State State) Examples: S sos skip s = s S sos x:=1 s = s [x 1] S sos while true do skip s = undefined 94 S sos S s = s’ if S, s * s’ undefined else

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An equivalence result For every statement in While S ns S = S sos S Proof in pages

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The End

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