The Calculator CS571. Abstract syntax of correct button push sequences. The Button Layout. Notes 16 Denotational Semantics of a Simple Calculator

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1 CS571 Notes 16 Denotational Semantics of a Simple Calculator The Calculator Two functions: + and * Unbounded natural numbers (no negatives Conditional: if-then-else Parentheses One memory register 1of 19 2of 19 The Button Layout Abstract syntax of correct button push sequences Display (unlimited Retrieves value in memory Separates expressions in conditional ON OFF , 5 6 ( LASTANSWR < TOTAL + * IF > Scroll buttons Arithmetic and conditional valuates expression P Program S xpression-sequence xpression N Numeral D Digit P :: ON S S :: TOTAL S TOTAL OFF :: * 2 N ( IF 1, 2, 3 LASTANSWR N :: N D D D :: of 19 4of 19

2 xample program ON TOTAL 3 * LASTANSWR TOTAL OFF Result is a sequence of values: (3, 9 The denotation of a program will be this sequence, represented as a list in the semantic algebras Semantic algebras Natural numbers, with constants, addition, multiplication and equality: zero, one etc. plus, times, equals Booleans with constants, and the conditional form: true, false _ 5of 19 6of 19 Algebra for a List Domain: List A * (A is any domain Operations: nil: List hd: List A tl: List List cons: A List List Valuation functions - Functionality P: Program Nat * S: xpression-sequence Nat Nat * : xpression Nat Nat N: Numeral Nat D: Digit Nat 7of 19 8of 19

3 Valuation functions - numbers D D 0 zero 9 nine N D ( ten times N plus D D DD N N D N Valuation functions simple expressions N ( n NN n ( ( ( n LASTANSWR ( n n n is the value of the memory ( n ( n plus ( n ( n ( n times ( n 9of of 19 conditional form IF 1, 2, 3( n ( ( n equals zero ( n ( n ach expression produces it value when the TOTAL button is pushed as in the expression sequence expression sequence TOTAL OFF ( n ( n S TOTAL S ( n let n ( n in n cons S S ( n S n is the value put into memory for the evaluation of the rest of the sequence 11 of of 19

4 program P S ON S S ( zero zero is the initial value of the memory passed to the first expression A Sample derivation Program is: ON TOTAL 3 * LASTANSWR TOTAL OFF S P ON TOTAL 3 * LASTANSWR TOTAL OFF TOTAL 3 * LASTANSWR TOTAL OFF ( zero let n ( zero in ncons S 3 * LASTANSWR TOTAL OFF ( n 13 of of 19 Work on the first expression: N D ( zero 1( zero plus 2( zero 1 plus N 1 1 plus D 1 one plus two Reduce two plus one to three. The derivation then continues: ( 3 LASTANSWR ( three cons S 3 LASTANSWR TOTAL OFF ( three three cons three 15 of of 19

5 Again work on the expression separately: 3 LASTANSWR ( three three times 3times three D3times three N 3( LASTANSWR( three three times three nine The Derivation Finished The final result is (the program s denotation is: three cons nine Without reducing the algebraic expressions it is the compiled version: ( one plus two cons ( three times ( one plus two 17 of of 19 Lambda forms (alternative form for valuation functions Can push extra arguments to right-hand side as lambda parameters: λ TOTAL OFF λn. ( n S TOTAL S n.let n ( n in n conss S ( n S Can sometimes omit same argument on both sides: ( 19 of 19