A definitive reference
Building on our exploration of undecidability, this module delves into the fundamental limits of computation itself. We examine the boundaries of what is computable, even with unlimited resources, and investigate the profound implications of these limits across mathematics, computer science, and physics.
First, we explore theoretical boundaries from foundational mathematics, including Gödel's Incompleteness Theorems and Tarski's undefinability of truth. We then investigate non-computable functions like the Busy Beaver function, which grow faster than any computable function.
Finally, we examine practical implications of these limits in compiler optimization, formal methods, and artificial intelligence, before considering speculative models of hypercomputation that attempt to transcend these fundamental barriers and the connection to algorithmic information theory.
In 1931, Kurt Gödel published two groundbreaking results that fundamentally changed our understanding of formal systems and established inherent limitations on what can be proven within consistent mathematical frameworks.
A formal system F consists of:
A formal system is consistent if it is not possible to derive a contradiction (both a statement and its negation). It is complete if every statement in the language can either be proven or disproven within the system.
Any consistent formal system F that is sufficiently powerful to express elementary arithmetic contains statements that can neither be proven nor disproven within F.
Gödel's proof relied on a technique now known as "Gödelization" - encoding statements about a formal system as numbers within that same system. This allows self-reference, enabling the construction of a statement that essentially says "this statement is not provable".
The key steps of the proof are:
G that effectively states "this formula is not provable in F".F is consistent, then G is neither provable nor refutable in F.The construction relies on a "diagonalization" technique similar to Cantor's proof that the real numbers are uncountable and Turing's solution to the Halting Problem.
If a formal system F is consistent and can express elementary arithmetic, then F cannot prove its own consistency.
The Second Incompleteness Theorem follows from the First by formalizing the notion of consistency within the system itself. It dealt a significant blow to Hilbert's program, which sought to establish the consistency of mathematics using only "finitary" methods.
These theorems have profound computational implications:
In 1933, Alfred Tarski established another fundamental limitation relating to formal systems, focusing on the concept of truth itself.
A truth predicate for a language L is a predicate Tr such that for any sentence φ in L,Tr(⌜φ⌝) ↔ φ, where ⌜φ⌝ is a name or code for the sentence φ.
For any formal language L that contains basic arithmetic, there is no formula Tr(x) in L that defines the set of Gödel numbers of true sentences of L.
The proof proceeds by contradiction. Suppose there exists a truth predicate Tr(x) in L.
We can construct a self-referential sentence λ that asserts its own falsehood:λ ↔ ¬Tr(⌜λ⌝)
If λ is true, then Tr(⌜λ⌝) holds, which means ¬Tr(⌜λ⌝) holds (by definition of λ), a contradiction.
If λ is false, then ¬Tr(⌜λ⌝) holds, which means λ is true (by definition of λ), another contradiction.
Therefore, no such truth predicate can exist in L.
Tarski's theorem represents a limitation on self-reference in formal systems and has several important computational implications:
Tarski's solution to the paradox was the introduction of a hierarchy of languages, where the truth of statements in one language can only be defined in a metalanguage, whose truth in turn is defined in a meta-metalanguage, and so forth.
The Busy Beaver function, introduced by Tibor Radó in 1962, provides a concrete example of a well-defined function that grows faster than any computable function. It exemplifies the notion of non-computability in a vivid way.
Let Σ(n) be the maximum number of 1s that can be written by an n-state Turing machine that starts with a blank tape and eventually halts.
Similarly, let S(n) be the maximum number of steps that can be executed by an n-state Turing machine that starts with a blank tape and eventually halts.
The values of this function grow extremely quickly:
Σ(1) = 1 (trivial case)Σ(2) = 4Σ(3) = 6Σ(4) = 13Σ(5) = 4098Σ(6) is not known exactly but exceeds 10↑↑↑↑↑10 (using Knuth's up-arrow notation)The function Σ(n) is not computable. That is, there exists no algorithm that, given n as input, outputs Σ(n) for all n.
We proceed by contradiction. Suppose Σ(n) is computable by some algorithm A.
Then we could solve the Halting Problem as follows:
M with n states and input w, construct a machine M' that:M on wM halts on w, then M' writes as many 1s as possible and haltsk be the number of states in M'.Σ(k) using the assumed algorithm A.M' for at most Σ(k) steps.M' halts within Σ(k) steps, then M halts on w. Otherwise, M does not halt on w.This would give us an algorithm to solve the Halting Problem, which is known to be undecidable.
Therefore, our assumption that Σ(n) is computable must be false.
For any computable function f: ℕ → ℕ, there exists a constant c such that for all sufficiently large n,Σ(n) > f(n).
The Busy Beaver function provides key insights into computational limits:
The theoretical limits discussed above have direct implications for what compilers can and cannot do, especially regarding optimizations.
A program transformation T: P → P' is an optimization if:
P and P' are equivalent (for all inputs, both programs produce the same output)P' is "better" than P according to some metric (e.g., execution time, memory usage, code size)Let P be any non-trivial semantic property of programs (i.e., a property that depends only on the function computed by the program, not on the specific implementation). Then the problem of determining whether a program has property P is undecidable.
This theorem has profound implications for compiler optimization. Here are several undecidable problems that compilers face:
Despite these theoretical limitations, compiler writers have developed various approaches to build practical optimizers:
A notable achievement in this area is the CompCert project, which provides a formally verified C compiler. CompCert's optimizations come with mathematical proofs that the optimized code preserves the semantics of the original program. This represents a balance between theoretical limitations and practical compiler design.
Formal verification aims to mathematically prove that a system satisfies its specification. However, computational limits place significant constraints on what can be verified.
Formal verification is the process of checking whether a system S satisfies a formal specification φ, typically written as S ⊨ φ.
The fundamental verification challenges stem from undecidability results:
However, several approaches have been developed to make verification practical within these theoretical boundaries:
While general program verification is undecidable, it is semi-decidable: if a program P satisfies a specification φ, there exists a proof that can be verified algorithmically, even if finding such a proof is not algorithmic.
This semi-decidability is the basis for many practical verification systems that rely on human intuition to guide proof search, combined with automated tools to verify proof correctness.
Type systems and static analysis provide powerful tools for ensuring program correctness, but they too face fundamental computational limits.
A type system is a formal system of rules that assigns types to program constructs and uses these types to determine whether a program is well-formed according to a set of typing rules.
Type systems and static analysis face several undecidability challenges:
While type-checking (verifying that a program with explicit type annotations is well-typed) is decidable for many type systems, type inference (automatically determining types without annotations) becomes undecidable for sufficiently powerful type systems, including dependent types.
Programming language designers navigate these limitations through various approaches:
The Hindley-Milner type system, used in languages like ML and Haskell, represents a sweet spot that allows powerful polymorphic type inference while remaining decidable. More expressive systems, like dependent types in languages such as Coq and Agda, require more programmer guidance but can express richer properties.
Computational limits also shape what artificial intelligence systems can and cannot achieve, regardless of advances in hardware or algorithms.
Artificial Intelligence systems can be viewed as implementations of computable functions f: I → Omapping inputs from a domain I to outputs in a range O, where the function f is constrained by the limits of computability theory.
Key implications of computational limits for AI include:
All machine learning algorithms that search for an optimal mapping from inputs to outputs have identical expected performance when averaged over all possible problems. In other words, no machine learning algorithm can outperform random guessing when averaged across all possible target functions.
Despite these fundamental limits, AI research has made substantial progress by:
These computational limits also inform discussions about AI capabilities and risks. While AI systems face fundamental constraints on what they can compute, they can still reach or exceed human capabilities in many domains because humans face these same computational limits.
Hypercomputation refers to theoretical models of computation that can compute functions beyond those computable by a Turing machine. While mathematically intriguing, the physical realizability of these models remains highly controversial.
A hypercomputer is a theoretical computational model that can compute functions not computable by a Turing machine, such as solving the Halting Problem or computing Busy Beaver values.
Several hypercomputation models have been proposed:
Most hypercomputation models require at least one physically problematic feature: infinite precision, infinite energy, infinite space, or processes completed in zero time. Under currently accepted physical theories, such features appear to be physically impossible.
The common critique of hypercomputation revolves around several issues:
Despite these challenges, hypercomputation models remain valuable for:
Infinite-time Turing machines (ITTMs) extend the classical Turing machine model by allowing computations to continue for transfinite ordinal time.
An infinite-time Turing machine operates like a standard Turing machine but:
Infinite-time Turing machines can solve many problems that are uncomputable for standard Turing machines:
Despite their increased power, infinite-time Turing machines still face undecidable problems. Specifically, the halting problem for infinite-time Turing machines themselves is undecidable by infinite-time Turing machines.
Infinite-time computation leads to a rich hierarchy of computational power:
The theory of infinite-time computation provides a precise mathematical framework for understanding computability beyond the Turing barrier, even if the physical implementation of such machines remains speculative at best.
Blum-Shub-Smale (BSS) machines provide a theoretical model for computation over the real numbers, offering a different perspective on the boundaries of computation.
A BSS machine is a theoretical model of computation that:
BSS machines differ from Turing machines in several important ways:
BSS machines can compute some functions that are uncomputable by Turing machines, such as:
However, they cannot compute all functions. In particular, they cannot decide membership in arbitrary Halting Sets, and they have their own halting problem that is undecidable within the BSS model.
The physical realizability of BSS machines is highly questionable because:
Nevertheless, BSS machines provide valuable insights into:
The BSS model highlights how different computational paradigms lead to different notions of computability, and reminds us that our standard model of computation is just one possible formalization among many.
Algorithmic information theory, centered around Kolmogorov complexity, provides a deep connection between information, randomness, and computability.
The Kolmogorov complexity K(x) of a string x is the length of the shortest program that produces x when run on a universal Turing machine U:
K(x) = min{|p| : U(p) = x}
Where |p| denotes the length of program p.
Key properties of Kolmogorov complexity include:
K(x) is not computable; no algorithm can calculate it for all inputs.There is no algorithm that, given a string x, computes K(x) for all possible inputs x.
Assume, for contradiction, that K(x) is computable by some algorithm A.
Consider the following program P:
ns, use algorithm A to compute K(s)s such that K(s) ≥ nsLet c be the length of program P. For any n, P(n) outputs a string s withK(s) ≥ n.
But note that P(n) is itself a program of length c + log(n) (where log(n) is the number of bits needed to encode n) that produces s. Therefore:
K(s) ≤ c + log(n)
For large enough n, we have c + log(n) < n, which means:
K(s) ≤ c + log(n) < n ≤ K(s)
This is a contradiction. Therefore, K(x) cannot be computable.
Despite its non-computability, Kolmogorov complexity has profound applications:
Algorithmic Information Theory (AIT) extends Kolmogorov complexity to develop a comprehensive theory of information, induction, and probability based on computation.
The universal probability of a string x, denoted m(x), is defined as:
m(x) = Σp:U(p)=x 2-|p|
Where the sum is over all programs p that produce x when run on a universal prefix-free Turing machine U.
Key principles and results from Algorithmic Information Theory include:
-log m(x) = K(x) + O(1)
This theory has profound implications for understanding the limits of knowledge and prediction:
Despite these limitations, AIT provides the theoretically optimal framework for prediction and induction, serving as an idealized standard against which practical methods can be evaluated.
Kolmogorov complexity provides a precise mathematical definition of randomness based on incompressibility, connecting computability theory to the foundations of probability.
A finite string x is c-incompressible if K(x) ≥ |x| - c.
An infinite sequence ω is Martin-Löf random if there exists a constant c such that for all prefixes ωn of length n:
K(ωn) ≥ n - c
Key insights about algorithmic randomness include:
n are incompressible.For any length n and any constant c, the probability that a randomly selected string x of length n has K(x) < n - c is less than 2-c+1.
The incompressibility method, based on these principles, has become a powerful mathematical proof technique:
Interestingly, despite being central to the definition of randomness, Kolmogorov complexity itself is non-computable. This leads to a profound insight: we can define randomness precisely, but we cannot reliably detect it algorithmically. This represents another manifestation of computational limits, connecting questions of information, randomness, and computability.