Abstract
I articulate a novel modal argument for P=NP.
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Notes
See http://www.claymath.org/millennium. There are six other “millennium” problems; each of these is also associated with a $1M prize.
As many readers know, the history of the problem is littered with failed attempts to provide non-constructive substantiation of the received view that \({\mathbf{P}}\!\ne\! {\mathbf{NP}}\).
His position is communicated in a stunningly prescient letter he wrote to von Neumann in 1950; this letter is reproduced, in English, in Sipser (1992). Gödel, writing of course before the modern P=?NP framework, inquires as to von Neumann’s thoughts about what is today known as the k-symbol provability problem. Let \(\phi \) be a formula of \({\mathcal {L}}\) \(_I\) (a formula of first-order logic, or just FOL). I write \(\vdash _k \phi \) provided there is a first-order proof of \(\phi \) of \(\le k\) symbols. Gödel apparently believed that it might well be possible to answer questions of the form “\(\vdash _k \phi \)?” in linear or quadratic time. When the set here is made explicit and configured so as to allow for encoding on a Turing-machine tape, it’s patent that it’s NP-complete. Gödel was quite at home with the idea that as logic and mathematics progress, machines would increasingly take over the “Yes-No” part of the enterprise. Any notion that Gödel would have embraced an argument by analogy from the undecidability of FOL to the perpetual intractability of the k-symbol provability problem is utterly misguided: He writes: “[I]t would obviously mean that in spite of the undecidability of the Entscheidungsproblem, the mental work of a mathematician concerning Yes-or-No questions could be completely replaced by a machine.”
An exactly parallel point obviously holds of all those incorporeal models known to be equivalent to TMs: register machines, the \(\lambda \)-calculus, abaci, etc. As more philosophically inclined readers will know, there isn’t consensus as to what a physical instantiation of a mathematical machine is. However, given the orientation of the present paper, we can rise above such turbulence. The fact is, a cornerstone of computer science is that idealized machines can be physically realized: we teach computability theory because we presuppose this. For a sustained discussion of the relationship between abstract machines, and theorems regarding them, and their embodied counterparts (see Bringsjord and Zenzen 1997).
STP is NP-hard when the metric is non-discretized, and NP-complete when the metric is discrete.
Analog computers are nothing new, though they don’t get much air time these days. An elegant example is Vannevar Bush’s famous differential analyzer, which solves ordinary differential equations. A nice discussion of the analyzer (in connection not with Bush, but rather with Claude Shannon) can be found in Earman (1986). A colorful discussion of the speedup that can be achieved through analog computation is provided by Dewdney (1984), who describes a series of analog computers, including something close to the soapfilm computer central to the present paper, and who also in some sense pointed in at least the general direction of my argument, by writing:
Throughout the foregoing discussion [of analog computers for small inputs] I have dodged the important issue of the feasibility of constructing [such] gadgets. Although each of the gadgets can be built and persuaded to work, after a fashion, on small problems, it would be silly to suggest that one construct them with serious computations in mind. Yet, considered in the context of an ideal world in which ideal materials are available, each gadget works, by definition, exactly as described. (Dewdney 1984, p. 25; emphasis mine)
One of the oldest discussions of analog computers is presented by Courant and Robbins (1941), who tinkered with various wireframe-in-soapfilm analog computers, in connection with STP, and other problems.
Just as we have the metaphor-clothed ‘Traveling Salesmen Problem’ (an NP-complete problem analyzed e.g. in Lewis and Papadimitriou 1981), STP can be conceived as the problem of building a road system (possibly with intersections outside the towns themselves) that connects n towns, where that system is of minimum length.
E.g., 1 is short for \(\exists x (A(x) \wedge M \text{ solves } \ldots )\).
I assume for certification a natural-deduction calculus, with rules for introducing and eliminating truth-functional connectives and quantifiers; a nice system of this sort for FOL is \({\mathcal {F}}\), from Barwise and Etchemendy (1999). For natural deduction in modal logic, consult (Konyndyk 1986).
The double modal operator is key. While \(\models _w \Diamond _p \phi \) means in the standard formal semantics that \(\phi \) is true at some possible world \(w' \in {\mathcal {W}}_p \,\subsetneq \,{\mathcal {W}}\) (where \({\mathcal {W}}\) is the set of all possible worlds logically/mathematically accessible from w, and \({\mathcal {W}}_p\) the proper subset of \({\mathcal {W}}\) preserving the physical laws of w), and \(\models _w \Diamond \phi \) holds iff \(\phi \) is true at some world in \({\mathcal {W}}\), \(\models _w \Diamond \Diamond _p \phi \) means that \(\Diamond _p \phi \) holds at some world \(w' \in {\mathcal {W}}\).
Michael Zenzen has pointed out to me that the so-called Principle of Least Action is a prime candidate for something that is at the core of what is carried over from world to world in order to reach to \(w^\star \) (for discussion of these principles, see the remarkable Castigliano 1966). This principle would be what Jim Fahey has referred to as a ‘proto’-law of nature. It’s a profound waste of time to build gadgets that, in \(w^\alpha \), compute ever larger initial intervals, if the aim of such activity is to seek evidence that counts either for or against my argument: The conditions in such seat-of-the-pants experiments are not controlled (even Courant (1941), when running his soapfilm experiments, realized the need to idealize), and all my proof needs is truth at quite-removed \(w^\star \), not our actual world.
I assume a normal S5 version of the \(\Diamond \) operator.
The transformation preserves polynomial-time processing, as cognoscenti know. For others, a sketch: Use an n-dimension TM in which n is high enough to sufficiently represent the CA which is the universe. Let each cell of the TM’s tapes represent a cell of the CA. The alphabet of the TM will contain some representation of all states for the CA’s cells. The computation performed by the CA is finite, so the TM’s states are as well. It is known that the transformation from multidimensional Turing machines to standard Turing machines is a polynomial transformation. If the computation performed in each cell of the CA is in class P, the equivalent TM will be in class P.
Physical phenomena that can be rigorously modeled only via information processing above the Turing Limit (Siegelmann 1999; Bringsjord and Zenzen 2003) would be phenomena calling for hypercomputational machines. Just as the class of mathematical devices equivalent to TMs is infinite, so also there are an infinite number of hypercomputational machines. Examples include analog chaotic neural nets (Siegelmann and Sontag 1994) and infinite time Turing machines (Hamkins and Lewis 2000). Other examples include analog “knob” TMs (Bringsjord 2001) and accelerated TMs (Copeland 1998).
The cardinality of sets produced by successive application of the power set operator \({\mathcal {P}}\) grows geometrically, but we can imagine \(2^n\) processors that share memory in which the input \(I = \{i_1, i_2, \ldots , i_n\}\) is placed, each of which is connected to a printer. Starting at \(i_1\) and working up in parallel, each processor does or doesn’t issue a print command, depending upon whether the \(i_k\) matches its unique identifier. In linear time \({\mathcal {P}}(I)\) is printed.
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I’m greatly indebted to Michael Zenzen for many valuable discussions about the P=?NP problem and physics (simpliciter and digital), and to Jim Fahey for discussions about such physics and mixed-mode dual-diamond operators in modal logic. The presentation of the core arguments herein to editions of Bringsjord’s graduate seminar, Logic & Artificial Intelligence, and his guest lectures on P=?NP in Formal Foundations of Cognitive Science graduate seminars, sparked a number of helpful objections and suggestions, for which I’m grateful. I’m indebted as well to two anonymous referees for trenchant comments. Though the two arguments herein (the second of which seems to establish P=NP) are for weal or woe Bringsjord’s, Joshua Taylor’s astute objections catalyzed much thought and crucial refinements.
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Bringsjord, S. An Argument for P=NP . Minds & Machines 27, 663–672 (2017). https://doi.org/10.1007/s11023-017-9454-1
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DOI: https://doi.org/10.1007/s11023-017-9454-1