TY - JOUR

T1 - The influence of domain interpretations on computational models

AU - Boker, Udi

AU - Dershowitz, Nachum

PY - 2009/10/15

Y1 - 2009/10/15

N2 - Computational models are usually defined over specific domains. For example, Turing machines are defined over strings, and the recursive functions over the natural numbers. Nevertheless, one often uses one computational model to compute functions over another domain, in which case, one is obliged to employ a representation, mapping elements of one domain into the other. For instance, Turing machines (or modern computers) are understood as computing numerical functions, by interpreting strings as numbers, via a binary or decimal representation, say. We ask: Is the choice of the domain interpretation important? Clearly, complexity is influenced, but does the representation also affect computability? Can it be that the same model computes strictly more functions via one representation than another? We show that the answer is "yes", and further analyze the influence of domain interpretation on the extensionality of computational models (that is, on the set of functions computed by the model). We introduce the notion of interpretation-completeness for computational models that are basically unaffected by the choice of domain interpretation, and prove that Turing machines and the recursive functions are interpretation-complete, while two-counter machines are incomplete. We continue by examining issues based on model extensionality that are influenced by the domain interpretation. We suggest a notion for comparing computational power of models operating over arbitrary domains, as well as an interpretation of the Church-Turing Thesis over arbitrary domains.

AB - Computational models are usually defined over specific domains. For example, Turing machines are defined over strings, and the recursive functions over the natural numbers. Nevertheless, one often uses one computational model to compute functions over another domain, in which case, one is obliged to employ a representation, mapping elements of one domain into the other. For instance, Turing machines (or modern computers) are understood as computing numerical functions, by interpreting strings as numbers, via a binary or decimal representation, say. We ask: Is the choice of the domain interpretation important? Clearly, complexity is influenced, but does the representation also affect computability? Can it be that the same model computes strictly more functions via one representation than another? We show that the answer is "yes", and further analyze the influence of domain interpretation on the extensionality of computational models (that is, on the set of functions computed by the model). We introduce the notion of interpretation-completeness for computational models that are basically unaffected by the choice of domain interpretation, and prove that Turing machines and the recursive functions are interpretation-complete, while two-counter machines are incomplete. We continue by examining issues based on model extensionality that are influenced by the domain interpretation. We suggest a notion for comparing computational power of models operating over arbitrary domains, as well as an interpretation of the Church-Turing Thesis over arbitrary domains.

KW - Computability

KW - Computational comparison

KW - Computational models

KW - Computational power

KW - Domain interpretation

KW - Domain representation

KW - Hypercomputation

KW - Turing machine

UR - http://www.scopus.com/inward/record.url?scp=77955309739&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2009.04.063

DO - 10.1016/j.amc.2009.04.063

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:77955309739

SN - 0096-3003

VL - 215

SP - 1323

EP - 1339

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

IS - 4

ER -