[Haskell-cafe] Semantic Domain, Function, and denotational model.....

Daryoush, Hopefully, the other replies about proving the monad laws already answered your previous question: yes! As for notions of semantic domain and denotational model, these ideas go back quite a ways; but, were given solid footing by Dana Scotthttp://en.wikipedia.org/wiki/Dana_Scott. In a nutshell, we have essentially three views of a computation - Operational http://en.wikipedia.org/wiki/Operational_semantics -- computation is captured in a program and rules for executing it - Logical http://en.wikipedia.org/wiki/Proof-theoretic_semantics -- computation is captured in a proof and rules for normalizing it - Denotational http://en.wikipedia.org/wiki/Denotational_semantics -- computation is captured as a (completely unfolded) mathematical structure In the latter view we think of computations/programs as denoting some (usually infinitary) mathematical object. Our aimhttp://en.wikipedia.org/wiki/Domain_theoryis to completely define the meaning of programs in terms of maps between their syntactic representation and the mathematical objects their syntax is intended to denote. (Notationally, one often writes such maps as [| - |] : Program -> Denotation.) For example, we model terms in the lambda calculus as elements in a D-infinity domain or Bohm trees or ... . Or, in more modern parlance, we model terms in the lambda calculus as morphisms in some Cartesian closed category, and the types of those terms as objects in said category. The gold standard of quality of such a model is full abstractionhttp://en.wikipedia.org/wiki/Denotational_semantics#Full_abstraction-- when the denotational notion of equivalence exactly coincides with an intended operational notion of equivalence. In symbols, - P ~ Q <=> [| P |] = [| Q |], where ~ and = are the operational and denotational notions of equivalence, respectively The techniques of denotational semantics have been very fruitful, but greatly improved by having to rub elbows with operational characterizations. The original proposals for denotational models of the lambda calculus were much too arms length from the intensional structure implicit in the notion of computation and thus failed to achieve full abstraction even for toy models of lambda enriched with naturals and booleans (cf the so-called full abstraction for PCF problemhttp://en.wikipedia.org/wiki/Programming_language_for_Computable_Functions). On the flip side, providing a better syntactic exposure of the use of resources -- ala linear logic -- aided the denotational programme. More generally, operational models fit neatly into two ready-made notions of equivalence - contextual equivalencehttp://encyclopedia2.thefreedictionary.com/Contextual+equivalence-- behaves the same in all contexts - bisimulation http://en.wikipedia.org/wiki/Bisimulation -- behaves the same under all observations Moreover, these can be seen to coincide with ready-made notions of equivalence under the logical view of programs. Except for syntactically derived initial/final denotational models -- the current theory usually finds a more "home-grown" denotational notion of equivalence. In fact, i submit that it is this very distance from the syntactic presentation that has weakened the more popular understanding of "denotational" models to just this -- whenever you have some compositionally defined map from one representation to another that respects some form of equivalence, the target is viewed as the denotation. Best wishes, --greg Date: Mon, 15 Sep 2008 10:13:53 -0700 From: "Daryoush Mehrtash" Subject: Re: [Haskell-cafe] Semantic Domain, Function, and denotational model..... To: "Ryan Ingram" Cc: Haskell Cafe Message-ID: Content-Type: text/plain; charset=ISO-8859-1 Interestingly, I was trying to read his paper when I realized that I needed to figure out the meaning of denotational model, semantic domain, semantic functions. Other Haskell books didn't talk about design in those terms, but obviously for him this is how he is driving his design. I am looking for a simpler tutorial, text book like reference on the topic. Daryoush On Mon, Sep 15, 2008 at 1:33 AM, Ryan Ingram wrote:

I recommend reading Conal Elliott's "Efficient Functional Reactivity" paper for an in-depth real-world example.

http://www.conal.net/papers/simply-reactive

-- ryan

On Sun, Sep 14, 2008 at 11:31 AM, Daryoush Mehrtash wrote:

...

I have been told that for a Haskell/Functional programmer the process of design starts with defining Semantic Domain, Function, and denotational model of the problem. I have done some googling on the topic but haven't found a good reference on it. I would appreciate any good references on the topic.

thanks,

daryoush

ps. I have found referneces like http://en.wikibooks.org/wiki/Haskell/Denotational_semantics which talks about semantic domain for "the Haskell programs 10, 9+1, 2*5" which doesn't do any good for me. I need something with a more real examples. _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe

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