Degrees of unsolvability: a survey Stephen G. Simpson Pennsylvania State University, Vanderbilt University http://www.math.psu.edu/simpson stephen.g.simpson@vanderbilt.edu sgslogic@gmail.com Proof Theory, Modal Logic, Reflection Principles University of Barcelona November 5–8, 2019 1
Motivation: a non-rigorous “calculus of problems.” We begin with a non-rigorous idea. Given a “problem” P , it is natural to seek a “solution” of P . If P has no “easy solution,” it is natural to ask “how difficult” P is. Let us say that P is “ reducible ” to another “problem” Q if ANY “solution” of Q “leads easily to” SOME “solution” of P . Let us say that P and Q have the same “ degree of difficulty ” if P and Q are “reducible” to each other. The equivalence class of P is called “deg( P ).” Note that “deg( P )” measures the “difficulty” of P as compared with other “problems.” —— This non-rigorous idea can become rigorous, in various ways. We focus on two closely related degree structures: the Turing degrees , D T , and the Muchnik degrees , D w . Turing degrees are used to measure the difficulty of decision problems . Muchnik degrees are used to measure the difficulty of mass problems . The Muchnik degrees are the completion of the Turing degrees. 2
Turing degrees versus Muchnik degrees. A decision problem has only one solution. A mass problem may have many different solutions. A decision problem is a real X ∈ N N . Intuitively, X represents the problem of “finding” or “computing” X . Such a problem has only one solution, namely, X . For X, Y ∈ N N we say that X is Turing reducible to Y , abbreviated X ≤ T Y , if X is computable using Y as a Turing oracle. A Turing degree is an equivalence class of decision problems under mutual Turing reducibility. The Turing degree of X is denoted deg T ( X ). The partial ordering of all Turing degrees is denoted D T . A mass problem is a subset of N N . Intuitively, P ⊆ N N represents the problem of “finding” or “computing” some member of P . Thus any X ∈ P is a solution of this problem. For P, Q ⊆ N N we say that P is Muchnik reducible to Q , abbreviated P ≤ w Q , if ∀ Y ( Y ∈ Q ⇒ ∃ X ( X ∈ P and X ≤ T Y )). In other words, using any solution of Q as an oracle, we can compute some solution of P . A Muchnik degree is an equivalence class of mass problems under mutual Muchnik reducibility. The Muchnik degree of P is denoted deg w ( P ). The partial ordering of all Muchnik degrees is denoted D w . 3
Turing degrees versus Muchnik degrees (continued). Recall D T = the partial ordering of all Turing degrees, and D w = the partial ordering of all Muchnik degrees. Identifying deg T ( X ) with deg w ( { X } ), we have an order-preserving embedding deg T ( X ) �→ deg w ( { X } ) : D T ֒ → D w . This induces an order-reversing one-to-one correspondence between Muchnik degrees and upwardly closed sets of Turing degrees. The upwardly closed set corresponding to p ∈ D w is { a ∈ D T | p ≤ a } . Thus we may identify D w = � D T = the completion of D T . In particular, D w is a complete and completely distributive lattice. D T is not even a lattice. However, D T is an upper semilattice. Namely, for all X, Y ∈ N N the Turing degree deg T ( X ⊕ Y ) = sup( a , b ) is the supremum (= l.u.b.) of deg T ( X ) = a and deg T ( Y ) = b . Also, D T has a bottom element, namely 0 = deg T (0). Our embedding of D T into D w preserves these features. 4
The completion of a partial ordering. Our identification of D w as the completion of D T is an instance of a general construction. Let K be any partial ordering , i.e., partially ordered set. Let � K be the set of upwardly closed subsets of K , partially ordered by reverse inclusion, i.e., U ≤ V if and only if U ⊇ V . Then � K is a complete and completely distributive lattice, called the completion of K . Identifying a ∈ K with the upwardly closed set U a = { x ∈ K | x ≥ a } , we see that K is a subordering of � K , namely, a ≤ b if and only if U a ≤ U b . For P ⊆ N N let P ∗ = { Y | ( ∃ X ∈ P ) ( X ≤ T Y ) } = the Turing upward closure of P . It is easy to check that P ≤ w Q if and only if P ∗ ⊇ Q ∗ . Thus D w = � D T = the completion of D T , and Muchnik degrees are identified with upwardly closed sets of Turing degrees. 5
The Muchnik topos. We may view D T as a topological space in which the open sets are the upwardly closed subsets of D T . Recall also that we have identified the upwardly closed subsets of D T with the Muchnik degrees. Therefore, by McKinsey/Tarski 1944, the Muchnik lattice D w is a topological model of intuitionistic propositional calculus. For any topological space T , a sheaf over T consists of a topological space X together with a local homeomorphism p : X → T . A sheaf morphism from a sheaf p : X → T to another sheaf q : Y → T is a continuous function f : X → Y such that p ( x ) = q ( f ( x )) for all x ∈ X . Let Sh( T ) = the category of sheaves and sheaf morphisms over T . By Fourman/Scott 1979, Sh( T ) is a topos and a model of intuitionistic higher-order logic. In this model, the truth values are open subsets of T . Applying the above construction to the topological space D T , we obtain Sh( D T ) = the Muchnik topos . In this model of intuitionistic mathematics, the truth values are the Muchnik degrees. We offer Sh( D T ) as a rigorous implementation of Kolmogorov’s 1932 non-rigorous interpretation of intuitionistic mathematics as a “calculus of problems.” 6
The real number system(s) in the Muchnik topos. Consider the topological space R C = R × D T with basic open sets { x } × U where x ∈ R and U ⊆ D T is upwardly closed. There is a projection map p : R C → D T given by p ( x, a ) = a . Thus R C is a sheaf over D T representing the Cauchy/Dedekind real number system. An interesting subsheaf of R C is R M = { ( x, a ) ∈ R C | deg T ( x ) ≤ a } , the sheaf of Muchnik reals , which supports an analog of computable analysis. Intuitively, a Cauchy/Dedekind real can exist anywhere within the Turing degrees, but a Muchnik real can exist only where we have enough Turing oracle power to compute it. Theorem (Basu/Simpson 2014) . Let x, y, z be variables ranging over Muchnik reals, let w be a variable ranging over functions from Muchnik reals to Muchnik reals, and let Φ( x, y ) be a formula in which w and z do not occur. Then, the Muchnik topos Sh( D T ) satisfies a Choice and Bounding Principle ( ∀ x ∃ y Φ( x, y )) ⇒ ( ∃ w ∃ z ∀ x ( wx ≤ T x ⊕ z and Φ( x, wx ))). Corollary of the proof. If Sh( D T ) satisfies ∀ x ∃ y Φ( x, y ), then Sh( D T ) satisfies ∃ w ∀ x ( wx ≤ T x and Φ( x, wx )). 7
Summary of main points in this survey. 1. D T = the semilattice of Turing degrees. 2. D w = � D T = the lattice of Muchnik degrees. 3. There is a natural embedding of D T into its completion D w . 4. In D T the only known specific, natural, degrees are among 0 , 0 ′ , 0 ′′ , . . . , 0 ( α ) , 0 ( α +1) , . . . . 5. In D w there are many other specific, natural degrees including r α ’s and b α ’s. 6. E T = the semilattice of recursively enumerable Turing degrees. 1 sets in { 0 , 1 } N . 7. E w = the lattice of Muchnik degrees of nonempty Π 0 8. There is a natural embedding of E T into E w . 9. The Splitting and Density Theorems hold for E T and for E w . 10. There is a strong analogy between E T and E w . 11. In E T the only known specific, natural degrees are 0 and 0 ′ . 12. In E w there are many specific, natural degrees including 0 , 1 , r 1 = k 1 , k = d , k REC = d REC , k f , d h , d slow , inf( r 2 , 1 ) , inf( b α , 1 ) where α < ω CK . 1 So far we have covered points 1 through 3. We now turn to examples. 8
Some specific, natural, Turing degrees. Given a decision problem X ∈ N N , let X ′ ∈ N N encode the halting problem relative to X , i.e., with X used as a Turing oracle. If a = deg T ( X ), let a ′ = deg T ( X ′ ). It can be shown that a ′ is independent of the choice of X such that deg T ( X ) = a . The operator a �→ a ′ : D T → D T is called the jump operator . Generalizing Turing’s proof of unsolvability of the halting problem, we have a < a ′ . In other words, the decision problem X ′ is “more unsolvable than” the decision problem X . Inductively we define a (0) = a and a ( n +1) = ( a ( n ) ) ′ for all n ∈ N . Extending this induction into the transfinite, we can define a ( α ) where α ranges over a large initial segment of the ordinal numbers. The naturalness of this transfinite induction is proved in a series of theorems due to Spector, Sacks, Jockusch/Simpson, and Hodes. In particular, we have a transfinite sequence of Turing degrees 0 < 0 ′ < 0 ′′ < · · · < 0 ( α ) < 0 ( α +1) < · · · . Apart from these, no specific natural Turing degrees are known!!! 9
A picture of D T , the upper semilattice of Turing degrees. ... 0 (α+1) 0 (α) ... 0’’’ 0’’ 0’ 0 Apart from the Turing degrees 0 < 0 ′ < 0 ′′ < · · · < 0 ( α ) < 0 ( α +1) < · · · , no specific, natural Turing degrees are known. 10
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