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CHA2545: WEEK 4

CHA2545: WEEK 4. LECTURE: DENOTIONAL SEMANTICS OF A SIMPLE LANGUAGE TUTORIAL: Do exercises given out DURING THE WEEK: Go through ARJ’s supporting WEB notes (from main web page). The Four Parts of a Semantic Definition:. Syntactic domains Abstract Syntax Semantic Domains

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CHA2545: WEEK 4

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  1. CHA2545: WEEK 4 LECTURE: DENOTIONAL SEMANTICS OF A SIMPLE LANGUAGE TUTORIAL: Do exercises given out DURING THE WEEK: Go through ARJ’s supporting WEB notes (from main web page).

  2. The Four Parts of a Semantic Definition: • Syntactic domains • Abstract Syntax • Semantic Domains • Meaning Functions Example syntax domains (see handout) Nml, Ide, Exp, Cmd

  3. DENOTATIONS • The semantics of the various syntactic structures of a language is defined via meaning functions which take the syntactic structure as a parameter and returns its denotation. • A “denotation” is a mathematical representation of the meaning of a construct - its an object belonging to a SEMANTIC DOMAIN.

  4. SEMANTIC DOMAINS - Examples Simple Semantic Domains - T - truth values N - natural numbers Z - integers Complex Domains are made up by combining Simple Domains with operators such as +, x, ->

  5. COMPLEX DOMAINS - Examples • Disjoint Union Domain of all integers and truth values Z + T • Cross Product Domain of all pairs of integers Z x Z • Function Domain consisting of all functions from integers to truth values Z -> Z and pairs of integers to integers (Z x Z) -> Z

  6. COMPLEX DOMAINS - Another Example All “expressible” values in a simple programming language with truth values and natural numbers: E = N + T + {error}

  7. The STORE • The simplest way to represent the STATE of program execution is by a STORE - representing the values stored at any point in the execution of a program. • The store is represented by a function domain S = [ Ide -> E ] + {error} We usually let ‘s’ be a typical store or state.

  8. Commands • Commands change the program state The domain of commands is therefore the functions between states, i.e. S -> S

  9. Semantic Function: Emeans Emeans takes a syntactically well formed expression, and gives it a denotation in terms of a function between stores and expressible values so Emeans : Exp -> S -> E The definition of Emeans depends on the syntactic subclass to which it is applied.

  10. Semantic Function: Cmeans Cmeans takes a syntactically well formed command, and gives it a denotation in terms of a function between stores and stores so Cmeans : Cmd -> S -> S The definition of Cmeans depends on the syntactic subclass to which it is applied

  11. The Problem with Recursive Functions - what do they mean? Cmeans[[while B do C]] = ls.(Emeans[[B]]s -> Cmeans[[while B do C]](Cmeans[[C]]s), s) OR: Cmeans[[while B do C]] = f where f = ls.(Emeans[[B]]s -> f(Cmeans[[C]]s), s)

  12. Recursive Functions - The Fixpoint Equation Given: f = ls.(Emeans[[B]]s -> f(Cmeans[[C]]s), s) Then: H = lf.ls.(Emeans[[B]]s -> f(Cmeans[[C]]s), s) Hf = (lf.ls.(Emeans[[B]]s -> f(Cmeans[[C]]s), s)) f = ls.(Emeans[[B]]s -> f(Cmeans[[C]]s), s) = f SO WE HAVE Hf = f - the fixpoint equation!!

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