The Influence of Domain Interpretations on Computational Models

1. The Influence of Domain Interpretationson Computational Models Udi Boker and Nachum Dershowitz Tel-Aviv, Israel Physics and Computation WorkshopUC 2008

2. The Need for Interpretations • Computational models are usually defined over specific domains • E.g. Turing machines over strings and the recursive functions over the natural numbers • Yet, we often use them over different domains • E.g. Turing machines (or modern computers) for functions over the natural numbers • We do it by representing one domain by another • E.g. Representing natural numbers as strings via binary or decimal notation

3. Representation and Interpretation • ‘Representation’ and ‘Interpretation’ are dual: (5)=“101” 6 5 N 5 is represented by “101” via   “101” is interpreted as 5 via  {0,1}* “110” “001” has no interpretation via  “101” “001” • Representations are usually injections

4. Function and Model Interpretation • Interpretations extend to functions and computational models: 6 5 N  The behavior of gout of Image is irrelevant {0,1}* “110” “101” “001” • The interpretation of a model A via the representation is

5. Does the Choice of Interpretation Matter? • Intuitively it matters for complexity • What about computability? • Can it be that one interpretation, , is better than another, , by strictly containing it? • Can the original extensionality be enlarged?

6. Interpretations Do Matter! • Yes! Interpretations do matter to the extensionality. A model might be enlarged! • Example: Two counter machines (2CM) • 2CM are known to be as powerful as the recursive functions (REC) [Minsky 60’s] • 2CM cannot compute n.n2 [R. Schroeppel 72, F. Yao, J. Barzdins] • The key is the interpretation: • Interpreting two counter machines via the representation  strictly enlarges its extensionality

7. So, What is the Problem? • How “serious” is the interpretation issue, in the sense that a model can be enlarged?[BD 2005] • How should we compare the power of computational models?[Rogers 66, Sommerhalder & Westrehenen 88, …, BD 2005, 2006] • Can Turing machines be interpreted to compute more than the recursive functions? [BD 2005] • What are the “proper” representations?[Rogers 66, Shapiro 82, Weihrauch 2000, …, BD 2008] • What is effectiveness over an arbitrary domain?[Montague 60, Rabin 60, Shapiro 82, Weihrauch 2000, Rescorla 2007, …, BD 2008]

8. There are basically two possible directions: Handling the problem Problem:Model extensionality is sensitive to the domain interpretation Trying to eliminate the problem Learning to live with it

9. Yet again, two basic directions: Trying to Eliminate the Problem Sensitivity to the domain interpretation Trying to eliminate the problem Learning to live with it Restricting theacceptable models Restricting theinterpretations • Starting point: • Interpretations: Via any injective representation • Computational model: Any object computing a set of partial functions

10. Restricting Interpretations • To understand the possible influence, it is enough to check the model’s interpretations over its original domain • Only very limited representations are “harmless”, eliminating the problem for all computational models: • Thm. [BD 05] A representation is harmless if and only if it is a “narrow permutation” (bounded cycles)

11. Restricting Models • The model of two counter machines is a “respectful” member, yet has the problem • Thm.[BD 05]The problem exists even when using only bijective representations and allowing only models closed under functional composition, including the identity function, all constant functions, and, for models over the naturals, the successor function

12. The Problem is Stubborn Sensitivity to the domain interpretation Trying to eliminate the problem Learning to live with it Restricting theacceptable models Restricting theinterpretations This direction is shown to end up withtoo restricted interpretations or models

13. Learning to Live with It Sensitivity to the domain interpretation Learning to live with it Trying to eliminate the problem Restricting theacceptable models Restricting theinterpretations • Investigating the variety of interpretations • Identifying “immune” models (“complete” / “stable”) • Seeking proper interpretations This direction is shown to end up withtoo restricted interpretations or models Effectivenessover arbitrary domains Comparing computational power

14. The Natural Questions • How varied is the set of interpretations? • Is there always a maximal interpretation? • How to choose a proper interpretation? • Are there models already maximally interpreted? • Can Turing machines be interpreted as computing more than the recursive functions? Restricting to bijective representations

15. The Variety of Interpretations • Thm.[BD 05,08]Interpretations might be very varied: • Better & worse than the original extensionality • Infinitely many interpretations containing one another • Thm.[BD 05,08] There are models with maximal interpretations and others without • Some models are alreadymaximally interpreted Restricting to bijective representations

16. Interpretation-Completeness and Stability • We focus on the property denoting that the model is already in its maximal interpretation • Interpretation-complete – with respect to any representation: • Interpretation-stable – w.r.t. bijective representations: Stable Complete

17. Interpretation-Stability • Thm.[BD 05] Sticking to bijective representations, there are exactly two options for the interpretation influence: • Stable model – immune to the influence of bijective representations; no better nor worse interpretations. • Unstable model – there is no maximum, nor minimum, interpretation via bijective representations. Restricting to bijective representations Stable Unstable

18. Interpretation-Completeness • Thm.[BD 05,06,08] Non-bijective interpretations provide a more varied behavior • There are stable models that are incomplete • There are complete models with worse interpretations • Isomorphism preserves interpretation-completeness Restricting to bijective representations Maximal interpretations Minimal interpretations Complete

19. Standard Models Thm. [BD 05,06,08] • Turing machines are complete (both halting and non-halting) • The recursive functions are complete (both total and partial) • Two counter machines are incomplete • The recursive functions are a maximal interpretation of two counter machines • One, three -or more-, counter machines are complete • Two stack machines are complete • -calculus, over all -terms, is incomplete

20. Comparing Computational Power Sensitivity to the domain interpretation Learning to live with it Trying to eliminate the problem Restricting theacceptable models Restricting theinterpretations • Investigating the variety of interpretations • Identifying “immune” models (“complete” / “stable”) • Seeking proper interpretations This direction is shown to end up withtoo restricted interpretations or models Effectivenessover arbitrary domains Comparing computational power

21. The Comparison Approach

22. Comparing Over Different Domains • Basically, B A if B can compute whatever A can. • But, hey… what about different domains? In general, B should have an interpretation that contains A [Rogers 66, Sommerhalder & Westrehenen 88, …]

23. Stronger • B is stronger than A if it computes more • But! Because of the sensitivity to the domain interpretation:Strict containment is not good enough! • Define (“go by the book”):

24. Comparing with Complete Models • When the weaker model is interpretation-complete, comparison is simpler. • strict containment is good enough

25. Hypercomputation • A modelA is hypercomputational if • Since Turing machines are interpretation-complete, we have that a model A is hypercomputational if there is a representation , such that • (Note that computability could not be defined in such a way with respect to an interpretation-incomplete model as 2CM)

26. Hypercomputation & Completeness • Is every hypercomputational model complete? Thm. [BD 08] • No • But, Yes if preserving the closure properties to composition, primitive recursion, minimalization • TMs with oracles are interpretation-complete

27. Effectiveness Sensitivity to the domain interpretation Learning to live with it Trying to eliminate the problem Restricting theacceptable models Restricting theinterpretations • Investigating the variety of interpretations • Identifying “immune” models (“complete” / “stable”) • Seeking proper interpretations This direction is shown to end up withtoo restricted interpretations or models Effectivenessover arbitrary domains Comparing computational power

28. Effectiveness over Arbitrary Domains • Let f be a function over some domain D What does “f is not computable” mean? • It usually means that f is not Turing-computable via some representation of D by strings • But what if f is Turing computable via another representation? • Every decision function has some representation via which it is “computable” [Shapiro 82] • Apparently, effectiveness, and the Church-Turing Thesis, are not well defined over arbitrary domains [Montague 60, Rescorla 2007]

29. The Role of Representations • Richard Montague [Montague 60]: “Turing's notion of computability applies directly only to functions on and to the set of natural numbers. … [for] another denumerable set S… [it] depends on what correspondence between S and the set of natural numbers is chosen”

30. “Effective” Representations • Restricting to “effective” representations is circular. • Richard Montague [Montague 60] continues: “The natural procedure is to restrict consideration to those correspondences which are in some sense ‘effective’, … But the notion of effectiveness remains to be analyzed, and would indeed seem to coincide with computability.” • Michael Rescorla [Rescorla 2007] adds:“I will suggest that purported conceptual analysis involving Church's thesis generate a subtle yet ineliminable circularity”

31. The Solution • Incomputability over an arbitrary domain is a property of a set of functions, not of a single function [cf. Myhill 52] • The Church-Turing Thesis over arbitrary domains is interpreted accordingly as CTT: There is no “effective” computational model stronger than Turing machines • For a constructible domain, we can define computability of a single function together with its domain constructors

32. Effectiveness Over Constructible Domains • Here, a “proper representation” is definable: • Define. A representation :N* is proper iff the representation of the successor, (S ), is Turing computable [cf. Shapiro 82, Weihrauch 2000] • Thm.[BD 08] For every injection :N*, • For other constructible domains, the constructing functions take the place of the successor in the above definition

33. What to take Home • Model extensionality is a very interesting set (of functions) • varies from “fluid” sets for which containment does not mean more power, as Cantor found for ordinary sets • to explicit, complete, sets for which nothing can be added without additional power • Get to know your model – is it stable / complete? • Compare computational power properly • Turing machines and the recursive functions are shown again to be robust, this time w.r.t. representations • Effectiveness is of a set of functions, or of a single function together with its domain constructors Thanks

34. References • U. Boker & N. Dershowitz. How to compare the power of computational models. CiE2005, LNCS vol. 3526, 2005. • U. Boker & N. Dershowitz. Comparing computational power. Logic Journal of the IGPL, 14(5), 2006. • U. Boker & N. Dershowitz. A hypercomputational alien. Applied Mathematics and Computation, 178(1), 2006. • U. Boker & N. Dershowitz. The Church-Turing thesis over arbitrary domains. In Pillars of Computer Science, LNCS vol. 4800, 2008. • U. Boker & N. Dershowitz. The influence of domain interpretations on computational models. Unconventional computation, 2008. • N. D. Jones. Computability and Complexity from a Programming Perspective, 1997. • M. L. Minsky. Computation: Finite and Infinite Machines, 1967. • R. Montague. Towards a general theory of computability. Synthese, 12(4), 1960. • J. Myhill. Some philosophical implications of mathematical logic. three classes of ideas. The Review of Metaphysics, 6(2), 1952. • M. O. Rabin. Computable algebra, general theory and theory of computable fields. Transactions of the American Mathematical Society, 95(2), 1960. • M. Rescorla. Church’s thesis and the conceptual analysis of computability. Notre Dame Journal of Formal Logic, 48(2), 2007. • H. Rogers, Jr. Theory of Recursive Functions and Effective Computability. 1966. • R. Schroeppel. A two counter machine cannot calculate 2n. MIT technical report, 1972. • S. Shapiro. Acceptable notation. Notre Dame Journal of Formal Logic, 23(1), 1982. • R. Sommerhalder and S. C. van Westrhenen. The Theory of Computability: Programs, Machines, Effectiveness and Feasibility. 1988. • K. Weihrauch. Computable Analysis — An introduction. 2000.