Church–Turing thesis

Church–Turing thesis. In the fields of mathematical logic and theoretical computer science, the Church–Turing thesis is a fundamental hypothesis about the nature of mechanical computation. It posits that any function which can be effectively calculated by an algorithm is computable by a Turing machine, and equivalently, by lambda calculus or any other model shown to be computationally equivalent. While not a theorem that can be formally proven, the thesis is widely accepted and forms the conceptual foundation for the modern theory of computability.

Statement of the thesis

The thesis asserts the equivalence between the intuitive, informal notion of an "effectively calculable" function and the mathematically precise concept of a function computable by a Turing machine. This means any procedure that a human computer could carry out by following a finite set of deterministic, mechanical rules can be simulated by the abstract machine proposed by Alan Turing. An alternative, equivalent formulation by Alonzo Church states that effectively calculable functions are precisely those definable in his system of lambda calculus. The logical equivalence of these two models, proven in the work of Stephen Kleene and others, provided strong evidence for the thesis. This conceptual bridge between informal intuition and formal mathematics resolved foundational questions raised by David Hilbert and Kurt Gödel regarding the limits of formal systems.

Historical context and formulation

The thesis emerged in the 1930s from independent work by Alonzo Church and his student Alan Turing, who were addressing the Entscheidungsproblem (decision problem) posed within Hilbert's program. Church first published his assertion in 1936, defining effective calculability as lambda definability. Shortly thereafter, Alan Turing introduced his abstract Turing machine model in his seminal paper "On Computable Numbers, with an Application to the Entscheidungsproblem". Turing demonstrated that his machine could compute any lambda-definable function, and Church acknowledged that Turing's formulation was more convincing. The collaboration and correspondence between figures like Kurt Gödel, John von Neumann, and Emil Post helped solidify the thesis's acceptance. The convergence of these distinct approaches from Princeton University and Cambridge University provided a robust historical foundation for modern computability theory.

Equivalent models of computation

Numerous formal systems have been shown to be computationally equivalent to the Turing machine, bolstering the empirical evidence for the thesis. These include Church's lambda calculus, Emil Post's Post–Turing machine and canonical systems, and Kleene's general recursive functions. Later developments, such as the random-access machine (RAM) model and standard imperative programming languages like C or Java, are also considered equivalent in computational power. The concept of Turing completeness for a system is defined by this equivalence. Even novel models like conway's game of life and certain cellular automaton rules have been proven to be universal, capable of simulating a Turing machine. This robust equivalence class underscores the thesis's claim that these models capture the same fundamental notion of algorithmic process.

Philosophical implications and interpretations

The thesis has profound implications for the philosophy of mathematics and philosophy of mind. It provides a precise standard for what constitutes a mechanical procedure, influencing debates in the foundations of mathematics following the Gödel's incompleteness theorems. In the philosophy of mind, it is central to discussions of computational theory of mind and strong AI, as argued by thinkers like Hilary Putnam and Jerry Fodor. The thesis also raises questions about the nature of human computation and whether mental processes are Turing-computable. Some interpretations, like Church–Turing–Deutsch principle, extend the thesis to physical systems, suggesting any finite physical system can be simulated by a universal Turing machine. These debates intersect with work by Roger Penrose and his critiques in books like The Emperor's New Mind.

Extensions and related theses

Several extensions and related hypotheses have been proposed to address the limits of classical computation. The Church–Turing–Deutsch principle extends the thesis to the physical realm, stating that every finite, physical process can be simulated by a universal Turing machine. In the realm of hypercomputation, models like the oracle machine proposed by Alan Turing and infinite-time Turing machines explore computations beyond the standard thesis. The physical Church–Turing thesis considers whether all physically realizable processes are Turing-computable, a question relevant to quantum computing and theories involving black hole physics. Contemporary research in complexity theory, such as the P versus NP problem, operates within the framework established by the thesis but focuses on resources like time and space rather than absolute computability.

Category:Computability theory Category:Mathematical logic Category:Theoretical computer science Category:Philosophy of mathematics