Programming Computable Functions

The Programming language for Computable Functions, or PCF, is a typed functional language introduced by Gordon Plotkin in 1977. It is based on the Logic of Computable Functions (LCF) by Dana Scott. It can be considered as a simplified version of modern typed functional languages such as ML.

A fully abstract model for PCF was first given by Milner (1977). However, since Milner's model was essentially based on the syntax of PCF it was considered less than satisfactory (Ong, 1995). The first two fully abstract models not employing syntax were formulated during the 1990s. These models are based on game semantics (Hyland and Ong, 2000; Abramsky, Jagadeesan, and Malacaria, 2000) and Kripke logical relations (O'Hearn and Riecke, 1995). For a time it was felt that neither of these models was completely satisfactory, since they were not effectively presentable. However, Ralph Loader demonstrated that no effectively presentable fully abstract model could exist, since the question of program equivalence in the finitary fragment of PCF is not decidable.

Syntax

The types of PCF are inductively defined as

• nat is a type
• For types σ and τ, there is a type σ → τ

A context is a list of pairs x : σ, where x is a variable name and σ is a type, such that no variable name is duplicated. One then defines typing judgments of terms-in-context in the usual way for the following syntactical constructs:

• Variables (if x : σ is part of a context Γ, then Γ ⊢ x : σ)
• Application (of a term of type σ → τ to a term of type σ)
• λ-abstraction
• The Y fixed point combinator (making terms of type σ out of terms of type σ → σ)
• The successor (succ) and predecessor (pred) operations on nat and the constant 0
• The conditional if with the typing rule:

${\displaystyle {\frac {\Gamma \;\vdash \;t\;:{\textbf {nat}},\quad \quad \Gamma \;\vdash \;s_{0}\;:\sigma ,\quad \quad \Gamma \;\vdash \;s_{1}\;:\sigma }{\Gamma \;\vdash \;{\textbf {if}}(t,s_{0},s_{1})\;:\sigma }}}$ 

(nats will be interpreted as booleans here with a convention like zero denoting truth, and any other number denoting falsity)

External links

• Introduction to RealPCF
• Lexer and Parser for PCF written in SML

Sources

• Abramsky, S., Jagadeesan, R., and Malacaria, P. (2000). "Full Abstraction for PCF". Information and Computation. 163 (2): 409–470. doi:10.1006/inco.2000.2930.
• Hyland, J. M. E. and Ong, C.-H. L. (2000). "On Full Abstraction for PCF". Information and Computation. 163 (2): 285–408. doi:10.1006/inco.2000.2917.
• O'Hearn, P. W. and Riecke, J. G (1995). "Kripke Logical Relations and PCF". Information and Computation. 120 (1): 107–116. doi:10.1006/inco.1995.1103.
• Loader, R. (2001). "Finitary PCF is not decidable". Theoretical Computer Science. 266 (1–2): 341–364. doi:10.1016/S0304-3975(00)00194-8.
• Ong, C.-H. L. (1995). "Correspondence between Operational and Denotational Semantics: The Full Abstraction Problem for PCF". In Abramsky, S., Gabbay, D., and Maibau, T. S. E. (ed.). Handbook of Logic in Computer Science. Oxford University Press. pp. 269–356.
• Plotkin, G. D. (1977). "LCF considered as a programming language". Theoretical Computer Science. 5: 223–255. doi:10.1016/0304-3975(77)90044-5.