The Difference Between Computability In Theory And Practice
Computability theory examines the inherent capabilities and limitations of computation in the abstract. It asks foundational questions like: What functions can be computed at all? How efficiently can they be computed? What problems cannot be solved by any computer program? While computability theory focuses on computation in the Platonic realm of ideals, the practice of computation must contend with real-world resource constraints and the messy complexities of implementing algorithms and data structures with physical computing devices.
This article explores the relationship between the pure theory of computability and the gritty reality of practical computation. We juxtapose the elegant thought experiments of theorists like Alan Turing and Alonzo Church with the nuts-and-bolts challenges overcome by practicing computer scientists and engineers. Our journey traverses the realms of computational complexity, intractability, heuristics, and quantum computing. We reflect on why closing the gap between theory and practice remains an elusive but worthy challenge.
Theoretical Underpinnings of Computability
The foundations of computability theory were established in the 1930s by logicians exploring the scope and limits of mathematical provability. Independent works published by Alan Turing and Alonzo Church in 1936 set out two equivalent formalisms for defining computable functions that could be carried out via mechanical calculation. These became known as the Church-Turing thesis on computable functions.
Church-Turing Thesis on Computable Functions
The Church-Turing thesis states that a function on the natural numbers can be calculated by an effective method if and only if it is computable by a Turing machine. Turing machines provide a simple yet universal model of computation equivalent to other proposed formalisms like lambda calculus and general recursive functions. This establishes Turing machines as sufficient to characterize computability in the abstract.
Turing Machines as a Model of Computation
A Turing machine consists of a finite set of states, an infinite tape divided into cells, and a read-write head that scans the tape and transitions between states based on the machine’s transition function. It models sequential execution of algorithms by manipulating symbols on the tape. Though simple, Turing machines can simulate any real-world computer by encoding inputs as symbols.
Examples of Turing Machine Configurations
A busy beaver Turing machine refers to machines that output as many 1s as possible before halting when started on an all 0s tape. Small busy beaver machines demonstrate uncomputable functions by trying all possible machines of a certain size. A universal Turing machine can simulate an arbitrary Turing machine when provided an encoding of its transition table. Universal machines show that Turing machines can replicate any mechanical computation procedure.
Practical Implications
While computability theory precisely delineates the possible from the impossible, practitioners must determine what is efficiently computable within real-world constraints. Unlike Turing machines, real computers have finite memory and processors that execute operations in discrete time steps. Theoretical efficiency is defined asymptoticially with Big O notation, but practical efficiency depends on constants and real usage context.
Polynomial vs. Exponential Time Algorithms
Algorithms require computation time growing polynomially or exponentially as input size increases. Polynomial time algorithms scale well, like sorting n items in n log n steps. Exponential algorithms bog down quickly, like checking all 2^n subsets. Polynomial growth is generally feasible while exponential is impractical for large inputs. Quantum algorithms sometimes provide exponential speedup over classical methods.
NP-Completeness and Intractability
NP-complete problems like the traveling salesman involve checking a possible solution quickly but finding an optimal solution requires trying all permutations. No polynomial-time solutions are known despite much effort. Thousands of problems across science and engineering share this intractability property. Workarounds include approximation algorithms, heuristics, and restricting inputs.
Coping with Intractability in Practice
Various techniques make hard problems manageable in practice. Greedy algorithms find locally optimal choices to approximate solutions. Randomized algorithms use random bits to enable probabilistic analysis of performance. Parallel computing leverages multiple processors to reduce time. Hard limits on resource usage curtail excessive computation based on practical constraints.
Bridging the Theory-Practice Divide
While asymptotic analysis precisely characterizes computation costs for theoretical machines, real computers operate on concrete data with variability that dominates performance. Bridging theory and practice requires layers of analysis blending empirical measurements and statistical techniques with abstract models.
Average Case Analysis
Average case analysis complements worst case big O notation by studying algorithm cost distributions on input datasets typically encountered in applications. This provides tighter bounds on real performance. Analysis methods derive probability distribution models from problem structure cues.
Randomized Algorithms
Algorithms employing random number generators enable probabilistic analysis of properties like execution times or approximation bounds. Randomness allows robust guarantees unattainable with deterministic methods. Cryptographic applications rely on randomness. Analysis uses Markov chain and information theory mathematical tools.
Heuristics and Approximation Algorithms
Heuristics shortcut exhaustive searches by guessing answers likely to be good enough using insights into problem structure. Approximation algorithms efficiently find near-optimal solutions. Performance tradeoffs balance precision versus time. Machine learning can automate heuristic development. Locality-sensitive hashing facilitates quick similarity search in high dimensions.
Looking Ahead: Quantum Computing
Quantum physics allows computing mechanisms radically unlike classical methods. Quantum bits encode a superposition of 0 and 1 simultaneously. Quantum parallelism promises exponential speedups but remains volatile and error-prone currently. The coming decades will see rapid advances harnessing quantum effects more robustly.
Promises of Quantum Computing
Quantum algorithms manipulate probability waves representing many values at once unlike digital bits. Quantum Fourier transforms, Shor’s algorithm for factoring integers, and Grover’s algorithm for search offer potential exponential speed over classical techniques. Physicists work to scale up quantum computation.
Current State of Quantum Hardware
Today’s quantum computers have less than 100 qubits and can run limited demonstrations. Quantum noise from decoherence and gate errors causes results to collapse probabilistically. Leading tech companies race to build fault-tolerant universal quantum hardware with over 1000 qubits in this decade. Cloud access democratizes early quantum algorithm prototyping.
Potential Quantum Speedups
Exciting applications await pending quantum hardware maturation. Chemistry simulations could advance drug development. Optimized machine learning could transform big data analytics. Secure communication would benefit from quantum cryptography. Further discoveries in quantum algorithm design are sure to unlock new use cases. Partnerships amplify quantum development.
Conclusion: Working Together for Progress
Exploring connections between computability theory and practical computation sheds light in both directions – abstract knowledge informs concrete problem solving while real-world observations lead to advances in models and understanding. Communication across theory and practice remains vital to catalyze innovations in computing.
Importance of Cross-Pollination
Cross-pollination of ideas between mathematical theory and systems building practices will grow increasingly significant as both hardware capabilities and software complexities continue scaling exponentially. Paradigm shifts like quantum computing underscore the need to reevaluate theoretical assumptions against empirical findings.
Advancing Theory and Practice in Tandem
Advances in theory inform the development of new techniques for practical gain while innovations in practical domains inspire new theoretical questions to pursue. The interplay generates a virtuous cycle lifting both theory and practice in tandem. Realizing mutual benefit requires open interchange among disciplines.
