Observational equivalence of 3rd-order Idealized Algol is decidable

We prove that observational equivalence of 3rd-order finitary Idealized Algol (IA) is decidable using Game Semantics. By modelling state explicitly in our games, we show that the denotation of a term M of this fragment of IA (built up from finite base types) is a compactly innocent strategy-with-state i.e. the strategy is generated by a finite view function f/sub M/. Given any such f/sub M/, we construct a real-time deterministic pushdown automata (DPDA) that recognizes the complete plays of the knowing-strategy denotation of M. Since such plays characterize observational equivalence, and there is an algorithm for deciding whether any two DPDAs recognize the same language, we obtain a procedure for deciding observational equivalence of 3rd-order finitary IA. This algorithmic representation of program meanings, which is compositional, provides a foundation for model-checking a wide range of behavioural properties of IA and other cognate programming languages. Another result concerns 2nd-order IA with recursion: we show that observational equivalence for this fragment is undecidable.

[1]  Radha Jagadeesan,et al.  Full Abstraction for PCF , 1994, Inf. Comput..

[2]  Colin Stirling,et al.  Decidability of DPDA equivalence , 2001, Theor. Comput. Sci..

[3]  Neil D. Jones,et al.  The Complexity of Finite Memory Programs with Recursion , 1976 .

[4]  Dan R. Ghica.  A Regular-Language Model for Hoare-Style Correctness Statements , 2001 .

[5]  Samson Abramsky,et al.  Games for Recursive Types , 1994, Theory and Formal Methods.

[6]  Martin Hyland.  Semantics and Logics of Computation: Game Semantics , 1997 .

[7]  Dan R. Ghica.  Regular-Language Semantics for a Call-by-Value Programming Language , 2001, MFPS.

[8]  C.-H. Luke Ong,et al.  On Full Abstraction for PCF: I, II, and III , 2000, Inf. Comput..

[9]  Peter W. O'Hearn,et al.  ALGOL-like Languages, Volume 1 , 1997 .

[10]  Marvin Minsky,et al.  Computation : finite and infinite machines , 2016 .

[11]  Samson Abramsky,et al.  Linearity, Sharing and State: a fully abstract game semantics for Idealized Algol with active expressions , 1996, Electron. Notes Theor. Comput. Sci..

[12]  S. Abramsky.  Game Semantics , 1999 .

[13]  John C. Reynolds,et al.  The essence of ALGOL , 1997 .

[14]  Dan R. Ghica,et al.  Reasoning about Idealized ALGOL Using Regular Languages , 2000, ICALP.

[15]  Géraud Sénizergues,et al.  Complete formal systems for equivalence problems , 2000, Theor. Comput. Sci..

[16]  Peter W. O'Hearn,et al.  Algol-like Languages , 1997, Progress in Theoretical Computer Science.

[17]  Michio Oyamaguchi,et al.  The Equivalence Problem for Real-Time Strict Deterministic Languages , 1980, Inf. Control..

[18]  Colin Stirling,et al.  Deciding DPDA Equivalence Is Primitive Recursive , 2002, ICALP.

[19]  Andrew M. Pitts,et al.  Operational Semantics and Program Equivalence , 2000, APPSEM.

[20]  Ralph Loader.  Finitary PCF is not decidable , 2001, Theor. Comput. Sci..