Abstract
After an introduction to the syntax of Gödel systemT, we present its naive denotational semantics in the domain of lazy natural numbers and show an adequacy property relating syntax and semantics. We recall the natural restrictions of systemT delineating primitive recursion as a subsystem. We discuss the weakness of primitive recursion by exhibiting a simple unary algorithm whose denotation is not the semantics of a primitive recursive algorithm. This algorithm can nevertheless be programmed in systemT by using the power of higher-order (functional) definitions. Generalizing this example, we obtain a representation theorem, asserting that every “reasonable” algorithm of typeN →N can be programmed in systemT. We conclude by discussing what is known in the case of higher arities.
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Colson, L. A unary representation result for systemT . Ann Math Artif Intell 16, 385–403 (1996). https://doi.org/10.1007/BF02127804
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DOI: https://doi.org/10.1007/BF02127804