Intensionality, Definability and Computation Samson Abramsky Oxford University Department of Computer Science Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 1 / 48 Computability and Computer Science Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 2 / 48 Computability and Computer Science Computability theory is concerned with computability of extensional objects: numbers, sets, functions ... Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 2 / 48 Computability and Computer Science Computability theory is concerned with computability of extensional objects: numbers, sets, functions ... These objects are inherited from mathematics and logic. Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 2 / 48 Computability and Computer Science Computability theory is concerned with computability of extensional objects: numbers, sets, functions ... These objects are inherited from mathematics and logic. Computability requires additional structure: an algorithmic process for computing the set or function. Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 2 / 48 Computability and Computer Science Computability theory is concerned with computability of extensional objects: numbers, sets, functions ... These objects are inherited from mathematics and logic. Computability requires additional structure: an algorithmic process for computing the set or function. Relies on a characterization of algorithmic processes. Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 2 / 48 Computability and Computer Science Computability theory is concerned with computability of extensional objects: numbers, sets, functions ... These objects are inherited from mathematics and logic. Computability requires additional structure: an algorithmic process for computing the set or function. Relies on a characterization of algorithmic processes. Computer Science asks a broader question: What is a process? Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 2 / 48 Computability and Computer Science Computability theory is concerned with computability of extensional objects: numbers, sets, functions ... These objects are inherited from mathematics and logic. Computability requires additional structure: an algorithmic process for computing the set or function. Relies on a characterization of algorithmic processes. Computer Science asks a broader question: What is a process? No established mathematical theory of what processes are. Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 2 / 48 Why processes matter in Computer Science Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 3 / 48 Why processes matter in Computer Science The purpose of much of the software we routinely run is not to compute a function, but to exhibit some behaviour. Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 3 / 48 Why processes matter in Computer Science The purpose of much of the software we routinely run is not to compute a function, but to exhibit some behaviour. What function does the Internet compute? Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 3 / 48 Why processes matter in Computer Science The purpose of much of the software we routinely run is not to compute a function, but to exhibit some behaviour. What function does the Internet compute? Think of communication protocols, operating systems, browsers, iTunes, Facebook, Twitter, ... Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 3 / 48 Why processes matter in Computer Science The purpose of much of the software we routinely run is not to compute a function, but to exhibit some behaviour. What function does the Internet compute? Think of communication protocols, operating systems, browsers, iTunes, Facebook, Twitter, ... What is a process? When are two processes equivalent? Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 3 / 48 Why processes matter in Computer Science The purpose of much of the software we routinely run is not to compute a function, but to exhibit some behaviour. What function does the Internet compute? Think of communication protocols, operating systems, browsers, iTunes, Facebook, Twitter, ... What is a process? When are two processes equivalent? The situation is very different to that which we find in computability theory. Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 3 / 48 The non-confluence of notions Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 4 / 48 The non-confluence of notions In computability theory we have: Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 4 / 48 The non-confluence of notions In computability theory we have: A confluence of notions Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 4 / 48 The non-confluence of notions In computability theory we have: A confluence of notions A definitive calculus of functions: the λ-calculus. Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 4 / 48 The non-confluence of notions In computability theory we have: A confluence of notions A definitive calculus of functions: the λ-calculus. There has been active research on concurrency theory and processes for the past 5 decades in CS. Many important concepts and results have emerged. However, one cannot help noticing that: Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 4 / 48 The non-confluence of notions In computability theory we have: A confluence of notions A definitive calculus of functions: the λ-calculus. There has been active research on concurrency theory and processes for the past 5 decades in CS. Many important concepts and results have emerged. However, one cannot help noticing that: Hundreds of different process calculi, equivalences, logics have been proposed. Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 4 / 48 The non-confluence of notions In computability theory we have: A confluence of notions A definitive calculus of functions: the λ-calculus. There has been active research on concurrency theory and processes for the past 5 decades in CS. Many important concepts and results have emerged. However, one cannot help noticing that: Hundreds of different process calculi, equivalences, logics have been proposed. No λ-calculus for concurrency has emerged. Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 4 / 48 The non-confluence of notions In computability theory we have: A confluence of notions A definitive calculus of functions: the λ-calculus. There has been active research on concurrency theory and processes for the past 5 decades in CS. Many important concepts and results have emerged. However, one cannot help noticing that: Hundreds of different process calculi, equivalences, logics have been proposed. No λ-calculus for concurrency has emerged. There is no plausible Church-Turing thesis for processes. Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 4 / 48 The non-confluence of notions In computability theory we have: A confluence of notions A definitive calculus of functions: the λ-calculus. There has been active research on concurrency theory and processes for the past 5 decades in CS. Many important concepts and results have emerged. However, one cannot help noticing that: Hundreds of different process calculi, equivalences, logics have been proposed. No λ-calculus for concurrency has emerged. There is no plausible Church-Turing thesis for processes. The ‘next 700’ syndrome. Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 4 / 48 The non-confluence of notions In computability theory we have: A confluence of notions A definitive calculus of functions: the λ-calculus. There has been active research on concurrency theory and processes for the past 5 decades in CS. Many important concepts and results have emerged. However, one cannot help noticing that: Hundreds of different process calculi, equivalences, logics have been proposed. No λ-calculus for concurrency has emerged. There is no plausible Church-Turing thesis for processes. The ‘next 700’ syndrome. It’s a much harder problem! Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 4 / 48 Prospectus Samson Abramsky (Oxford University Department of ComputerIntensionality,Science) Definability and Computation 5 / 48 Prospectus We shall tell one success story: the characterization of sequential functional processes using game semantics, solving in best possible