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I am reading a book about history of mathematics, and it inspired to think about that formal languages do not need to specify time to transfer a message. I am thinking about DNA as a formal language, about linguistic (although I specify a comma as time interval, that could be interpreted subjectively be the "hardware", as in music), about software.

Why there is no need to specify time to execute (or express) an instruction? Which is the role of time for an hardware in correctly interpreting a software, conceptually?

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    DNA isn't a formal language because it doesn't have a semantics. You can't specify an "interpretation" of the "sentences" of DNA that make some of those sentences "true". More generally, you don't have to have "time" in a formal language, because you can use formal languages to state eternal truths, such as, e.g. "2 + 2 = 4" – shane Mar 27 '16 at 0:57
  • @shane A formal language need not have semantics, that's why it's hanndled as an uninterpreted calculus. When we are doing propositional calculus, we abandon meaning and blindly apply derivation rules. Moreover, the DNA codon table is a perfect candidate of a finite-state transducer. Many formal systems don't have the concept of truth. They have just derivability. – prash Apr 3 '16 at 11:16
  • Though I am familiar with all the concepts you're talking about, I'm not sure what you're asking. The definition of the density of an object (mass/volume) has no need for time either. Some things make implicit assumptions about time, others don't. – prash Apr 3 '16 at 18:54
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    A string of uninterpreted symbols isn't a language because you can't "say" anything with it. And note too that all the interesting metalogical properties about formal languages like completeness, soundness and so forth are semantic properties. Definitions of concept are typically timeless because the content of a concept doesn't change. A circle just is a circle--there's no reference to time at all. Most logical and mathematical statement are similarly timeless. – shane Apr 4 '16 at 1:52
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    plato.stanford.edu/entries/logic-classical the first paragraph there makes basically the same claim I am making about semantics. However Wikipedia thinks you don't need semantics for a logic. Perhaps this tracks a difference between philosophers and computer scientists. But I've always seen the semantic component included in the definition of a logic. I don't know what the use of having a connective like "and" in your language would be if you didn' specify that "A and B" is true iff A is true and B is true. – shane Apr 4 '16 at 12:14
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DNA is analogous to computer languages, which are formal but not declarative: their texts consist of instructions that have no truth values, nor does the whole text have a truth value (the assessment of computer program as "correct" is external, computer needs none of that to execute it). DNA is similarly a sequence of instructions for producing a specimen, and it does have an alphabet and composition rules. Time intervals are not specified because there is no point to do so, execution time depends on "hardware" and environment, will differ depending on it, often can not be controlled, and therefore has to be irrelevant. In other words, instructions must work when executed in sequence regardless of the execution time.

Of course, some instructions may be time sensitive, and some computer languages that work with real time applications do have means to constrain execution time. For example, if your browser can not establish connection with a server for prescribed period of time it will abort, and display the message like "connection timed out". Epigenetic mechanisms may perform some such functions for DNA.

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