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On the structural simplicity of machines and languages

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Abstract

We employ an algebraic method for comparing the structural simplicity of families of abstract machines. We associate with each family of machines a first-order structure that includes machines as objects, composition operations, which construct larger machines from smaller ones, as functions, and a set of semantic relations. We then compare families of machines by studying the existence of homomorphisms between the associated structures. Given families of machinesL, L′ with associated structuresS, S′, we say thatL issimpler thanL′ if there is a homomorphism ofS intoS′, but notvice versa. We show that across several abstract machine models — finite automata, Turing machines, and logic programs — deterministic machines are simpler than nondeterministic machines and nondeterministic machines are simpler than alternating machines. Our results cross computational complexity boundaries. We show that for Turing machines, finite automata and logic programs every non-deterministic variant is not simpler than any deterministic variant, and that every alternating variant is not simpler than any nondeterministic (non-alternating) variant. In addition, we show that nondeterministic and alternating Turing machines are equivalent in their structural simplicity to iterative and ordinary logic programs, respectively, and that deterministic Turing machines are as simple as deterministic logic programs. We thus establish, in a sense somewhat independent of computational complexity classes and machine models, that determinism is simpler than nondeterminism and that nondeterminism is simpler than alternation. The method of comparison we employ is quite general. It has been applied successfully to the comparison of concurrent programming languages and we expect it to be applicable to other languages and machine models as well.

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Moscowitz, Y., Shapiro, E. On the structural simplicity of machines and languages. Ann Math Artif Intell 15, 379–405 (1995). https://doi.org/10.1007/BF01536402

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