Operational Semantics This course Why semantics? y := 1; • precise specification of software and while ¬ ( x = 1) do ( y := x ∗ y ; x := x − 1) hardware First we assign 1 to y , then we test whether • facilitate reasoning about systems: x is 1 or not. If it is then we stop and testing may reveal errors but not their otherwise we update y to be the product of absense x and the previous value of y and then we • form the basis for prototype imple- decrement x by one. Now we test whether mentations, e.g. interpreters and com- the new value of x is 1 or not · · · pilers Two kinds of operational semantics: Why functional programming? • Natural Semantics • based on mathematical notation, not on the von Neuman architecture • Structural Operational Semantics • excelent for prototype definitions I.1 I.2
Denotational Semantics Axiomatic Semantics y := 1; while ¬ ( x = 1) do ( y := x ∗ y ; x := x − 1) y := 1; The program computes a partial function while ¬ ( x = 1) do ( y := x ∗ y ; x := x − 1) from states to states: the final state will If x = n holds before the program is exe- be equal to the initial state except that the cuted then y = n ! will hold when the exe- value of x will be 1 and the value of y will cution terminates (if it terminates) be equal to the factorial of the value of x in the initial state. Two kinds of axiomatic semantics: • Partial Correctness Two kinds of denotational semantics: • Total Correctness • Direct Style Semantics • Continuation Style Semantics I.3 I.4
Which approach? Selection criteria • constructs of the language Programming Language – imperative ↓ – functional – concurrent/parallel Semantics – object oriented • natural semantics – non-deterministic • structural operational semantics – · · · • what is the semantics used for • direct style denotational semantics – understanding the language • continuation style denotational se- mantics – verification of programs – prototyping • partial correctness axiomatic seman- tics – compiler construction – program analysis • total correctness axiomatic semantics – · · · I.5 I.6
Approach While language Theoretical Development S ::= x := a | skip | S 1 ; S 2 concrete LL(1)/LALR(1) grammar | if b then S 1 else S 2 syntax concrete syntax trees | while b do S | repeat S until b ↓ struct. op. natural abstract syntactic categories semantics semantics syntax abstract syntax trees ↓ direct style cont. style den. sem. den. sem. semantics semantic categories semantic definitions partial total correctness correctness I.7 I.8
Miranda Syntactic Categories for While language • numerals n ∈ Num Programming languages • variables • imperative (procedural) x ∈ Var • applicative (functional) • arithmetic expressions a ∈ Aexp – lazy languages a ::= n | x | a 1 + a 2 Miranda | a 1 ∗ a 2 | a 1 − a 2 – eager languages Standard ML • booleans expressions Lisp?, Scheme? b ∈ Bexp b ::= true | false | a 1 = a 2 • declarative (logical) | a 1 ≤ a 2 | ¬ b | b 1 ∧ b 2 • object-oriented • statements • concurrent / parallel S ∈ Stm S ::= x := a | skip | S 1 ; S 2 • · · · | if b then S 1 else S 2 | while b do S I.9 II.1
Semantic Categories Meanings of the syntactic categories Natural numbers N = { 0 , 1 , 2 , · · ·} Numerals N : Num → N Truth values T = { tt, ff } Variables s ∈ State = Var → N States Arithmetic expressions • State = Var → N • State ′ = (Var × N) ∗ A : Aexp → (State → N) • State ′′ = Var ∗ × N ∗ Boolean expressions Lookup in a state: s x B : Bexp → (State → T) Update a state: s ′ = s [ y �→ v ] Statements S : Stm → (State ֒ → State) s x if x � = y s ′ x = if x = y v II.2 II.3
A : Aexp → State → N B : Bexp → State → T B [ true ] s = tt B [ false ] s = ff tt if A [ a 1 ] s = A [ a 2 ] s B [ a 1 = a 2 ] s = ff if A [ a 1 ] s � = A [ a 2 ] s A [ n ] s N [ n ] = tt if A [ a 1 ] s ≤ A [ a 2 ] s A [ x ] s = s x B [ a 1 ≤ a 2 ] s = ff if A [ a 1 ] s �≤ A [ a 2 ] s A [ a 1 + a 2 ] s A [ a 1 ] s + A [ a 2 ] s = tt if B [ b ] s = ff A [ a 1 ∗ a 2 ] s A [ a 1 ] s ∗ A [ a 2 ] s = B [ ¬ b ] s = if B [ b ] s = tt ff A [ a 1 − a 2 ] s A [ a 1 ] s − A [ a 2 ] s = tt if B [ b 1 ] s = tt and B [ b 2 ] s = tt B [ b 1 ∧ b 2 ] s = if B [ b 1 ] s = ff ff or B [ b 2 ] s = ff II.4 II.5
Compositional Definitions Statements Syntactic Category S ::= x := a | skip | S 1 ; S 2 • The syntactic categories are specified | if b then S 1 else S 2 by an abstract syntax giving a unique | while b do S decomposition of each element into its constituents. Meaning of the syntactic category: • The semantics is defined by composi- S : Stm → (State ֒ → State) tional definitions of functions. Two operational semantics Since the decomposition of the ele- • Natural Semantics ments is unique this means that the semantics is well-defined. • Structural Operational Semantics specified by transition systems II.6 II.7
Transition System Natural semantics Idea: describe how the overall result of the computation is obtained Transition system: (Γ , T, → ) • Γ = { ( S, s ) | S ∈ While, s ∈ State } (Γ , T, > ) ∪ State • Γ : a set of configurations • T = State • T : a set of terminal configurations • → ⊆ { ( S, s ) | S ∈ While, s ∈ State } T ⊆ Γ × State • > : a transition relation Typical transition: > ⊆ Γ × Γ ( S, s ) → s ′ where S is the program s is the initial state s ′ is the final state II.8 II.9
Natural Semantics ( x := a, s ) → s [ x �→ A [ a ] s ] ( skip , s ) → s ( S 1 , s ) → s ′ , ( S 2 , s ′ ) → s ′′ ( S 1 ; S 2 , s ) → s ′′ ( S 1 , s ) → s ′ ( if b then S 1 else S 2 , s ) → s ′ if B [ b ] s = tt ( S 2 , s ) → s ′ ( if b then S 1 else S 2 , s ) → s ′ if B [ b ] s = ff ( S, s ) → s ′ , ( while b do S, s ′ ) → s ′′ ( while b do S, s ) → s ′′ if B [ b ] s = tt ( while b do S, s ) → s if B [ b ] s = ff II.10
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