This Lecture • Introduction to Denotational Semantics G54FOP: Lecture 15 • The relation between Operational and Denotational Semantics and Domain Theory I Denotational Semantics Henrik Nilsson University of Nottingham, UK G54FOP: Lecture 15 – p.1/16 G54FOP: Lecture 15 – p.2/16 Denotational Semantics (1) Denotational Semantics (2) Operational Semantics (review): Denotational Semantics: • The meaning of a term is the term it • Idea: Semantic function maps (abstract) (ultimately) reduces to, if any: syntax directly to meaning in a semantic - Stuck terms domain. - Infinite reduction sequences • Domains consist of appropriate semantic • No inherent meaning (structure) beyond objects (Booleans, numbers, functions, ...) syntax of terms. and have structure; in particular, an information ordering. • Focus on behaviour; important aspects of semantics (such as non-termination) • The semantic functions are total; in particular, emerges from the behaviour. even a diverging computation is mapped to an element in the semantic domain. G54FOP: Lecture 15 – p.3/16 G54FOP: Lecture 15 – p.4/16 Denotational Semantics (3) Compositionality (1) Example: • It is usually required that a denotational [[(1+2) * 3]] = 9 semantics is compositional: that the meaning of a program fragment is given in abstract syntax terms of the meaning of its parts. • Compositionality ensures that meaning/denotation - the semantics is well-defined (here, semantic domain is Z) - important meta-theoretical properties hold [[·]], or variations like E[[·]], C[[·]]: semantic functions [[ and ]]: Scott brackets or semantic brackets. G54FOP: Lecture 15 – p.5/16 G54FOP: Lecture 15 – p.6/16 Compositionality (2) Definition Example: Formally, a denotational semantics for a language L is given by a pair [[(1+2) * 3]] = [[1+2]] × [[3]] hD, [[·]]i abstract syntax where • D is the semantic domain subterms (abstract syntax) • [[·]] : L → D is the valuation function or semantic multiplication semantic function. Z Z Z (× : × → ) In simple cases D may be a set, but usually The meaning of the whole is given by composing more structure is required leading to domains as the meaning of the parts. defined in domain theory. G54FOP: Lecture 15 – p.7/16 G54FOP: Lecture 15 – p.8/16 Example: Simple Expr. Language (1) Example: Simple Expr. Language (2) Develop a denotational semantics hD, [[·]]i for E, e → expressions: picking N as the semantic domain for simplicity: | true constant true N | false constant false D = N | if e then e else e conditional [[·]] : e → | 0 constant zero However, as there are both Booleans and natural | succ e successor numbers in the object language, a more refined | pred e predecessor choice for the semantics at the meta level would have been N ⊎ B, the disjoint union of natural | iszero e zero test numbers and Booleans. (On whiteboard) G54FOP: Lecture 15 – p.9/16 G54FOP: Lecture 15 – p.10/16 Exercises (1) Exercises (2) 1. Find the denotation of 2. Consider the following language extension: if (iszero (succ 0)) then true else false e → expressions: . | not e logical negation | e && e logical conjunction | e + e addition | e - e subtraction | e * e multiplication Extend the denotational semantics accordingly. G54FOP: Lecture 15 – p.11/16 G54FOP: Lecture 15 – p.12/16 Operational and Denotational Sem. (1) Operational and Denotational Sem. (2) Given a language L, suppose we have: Closed terms t1,t2 ∈ L are semantically or • a big-step operational semantics denotationally equivalent iff ⇓⊆ L × V [[t1]] = [[t2]] where V ⊆ L is the set of values Assume D is ground (no functions; i.e., our closed terms are programs that output • a denotational semantics something “printable”). We adopt a function hD, [[·]]i · : D → V where [[·]] : L → D that maps a semantic value d ∈ D to its term How should these be related? representation v ∈ V . G54FOP: Lecture 15 – p.13/16 G54FOP: Lecture 15 – p.14/16 Operational and Denotational Sem. (3) Operational and Denotational Sem. (4) Assuming termination: Assuming termination: • Correctness of operational semantics w.r.t. • Computational adequacy of operational denotational semantics: semantics w.r.t. denotational semantics (or vice versa, depending on point of view): t ⇓ v ⇒ [[t]] = [[v]] t ⇓ v ⇔ [[t]] = [[v]] • Completeness of operational semantics w.r.t. denotational semantics: [[t]] = d ⇒ t ⇓ d G54FOP: Lecture 15 – p.15/16 G54FOP: Lecture 15 – p.16/16.