## Observational equivalence of 3rd-order Idealized Algol is decidable (2002)

Venue: | In Proceedings of LICS’02. IEEE |

Citations: | 11 - 2 self |

### BibTeX

@INPROCEEDINGS{Ong02observationalequivalence,

author = {C. -h. L. Ong},

title = {Observational equivalence of 3rd-order Idealized Algol is decidable},

booktitle = {In Proceedings of LICS’02. IEEE},

year = {2002},

pages = {245--256},

publisher = {Press}

}

### OpenURL

### Abstract

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 fM . Given any such fM , 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 3rdorder 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

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Citation Context ...X = a goto L, and halt, where L 0 is a label. A Queue program is a finite sequence of the form 1 : I 1 ; 2 : I 2 ; n : I n , where each I i is an instruction. By simulating Post's Tag Systems [12]=-= in Queue,-=- it is known that the problem QUEUE-HALTING: "Given a Queue program, will it halt eventually?" is undecidable. Given a Queue program P , we construct an IA program MP of the fragment in ques... |

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Citation Context ...here is at most one pending occurrence of a question at any time, so that each move has a unique justifier (and so pointers may be ignored). 2 Idealized Algol Reynolds' Idealized Algol (IA for short) =-=[16-=-] is a compact language that elegantly combines state-based procedural and higher-order functional programming. The basis of IA is a simply-typed call-by-name -calculus in which the standard construct... |

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Citation Context ...rogramming languages, in which data types and control flow are important. Overview. A notable feature of Abramsky and McCusker 's fully abstract knowing (or history-sensitive) strategy game semantics =-=[2]-=- for IA is the implicit nature of its model of state. Knowing strategies that denote finitary IA terms are in general infinite sets of justified sequences (i.e. sequences of moves, each of which is eq... |

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Citation Context ...y it would be interesting to identify fragments of Reduced ML (callby -value PCF augmented by statically scoped, dynamically allocated, mutable state, with equality test for references, as studied in =-=[15]-=-) that have decidable observational equivalence. Our preferred approach is by Algorithmic Game Semantics, but to our knowledge, the prior (for this approach) problem of the existence of a fully abstra... |

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Citation Context ...ion and program analysis. The important first steps have been taken by Ghica OUCL, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom. www.comlab.ox.ac.uk/oucl/work/luke.ong/ and McCusker [=-=5]-=-; they show that the 2nd-order finitary (i.e. recursion-free) fragment of the fully abstract game semantics of Reynolds' language, Idealized Algol (IA), can be represented in a remarkably simple form ... |

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Citation Context ...in C, by the Switching Condition). Termination of the algorithm amounts to showing that there is no infinite short-sighted interaction sequences. The proof of the last fact uses a lemma due to Hyland =-=[6]-=-: For any infinite state-less play w over a finitary arena, either the set of P-views spanned by w (i.e. f psq : s 6 w g) is infinite, or the set of O-views spanned by w is infinite. 4 Innocent repres... |

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Citation Context ...e show that observational equivalence for this fragment is undecidable (Theorem 5.5). Related work. Perhaps the most important result that we should mention here is Loader's undecidability theorem in =-=[9]-=-. He proves that observational equivalence of 4th-order finitary PCF is undecidable. Observational equivalence of (active) IA does not extend that of PCF conservatively; indeed it is not known whether... |

24 | On Full Abstraction for PCF: I. Models, observables and the full abstraction problem, II. Dialogue games and innocent strategies, III. A fully abstract and universal game model
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Citation Context ...effect of restraining the growth of states as the play unfolds. We show that [[ ` M : A ]], the strategy-with-state that denotes a 3rd-order IA term-in-context, is compactly innocentsin the sense (of =-=[7-=-]) that the strategy is generated by a finite view function fM . Moreover [[ ` M : A ]] is closely related to the knowing-strategy denotation, written [[ ` M : A ]] : we have (Theorem 4.3(i)) erase [... |

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(Show Context)
Citation Context ... problem was only solved recently, also positively, by Senizergues [17] (solution announced in 1997). A simpler proof and a primitive recursive complexity bound were subsequently obtained by Stirling =-=[18, 19]-=-. Construction of the DPDA P f . Let f be a finite view function that generates an innocent strategy-with-state over a 3rd-order arena A. We shall construct a real-time DPDA P f that recognizes the co... |

15 | Regular-language semantics for a call-by-value programming language
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Citation Context ... observational equivalence of 4th or higher-order IA is decidable. On the (Algorithmic) Game Semantics front, Ghica has extended his earlier work with McCusker to a call-by-value language with arrays =-=[4]-=-, and to model checking Hoare-style program correctness assertions [3]. In a foundational direction, Abramsky [1] has extended the regular language representation to games in whose plays there is at m... |

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Citation Context ...ve procedure for deciding whether any two DPDAs recognize the same language?") was first posed in 1966. Restricted to the real-time case, the problem was solved positively by Oyamaguchi et al in =-=1980 [14]-=-. The general decidability problem was only solved recently, also positively, by Senizergues [17] (solution announced in 1997). A simpler proof and a primitive recursive complexity bound were subseque... |

10 | A regular-language model for Hoare-style correctness statements - Ghica - 2001 |

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Citation Context ...structor mkvar. Using the model he shows that adding mkvar is conservative for observational equivalence but not for observational preorder. 3 Game semantics We shall consider arenas as introduced in =-=[7, 10]-=-, and use square and round parentheses in bold type as meta-variables for moves as follows: O-question P-answer P-question O-answer [ ] ( ) We assume the reader is familiar with the following standard... |

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Citation Context ...seskip (ii) Parametricity: For p : com ) exp 2 ) com (where exp 2 is the type of binary-valued expressions), we have newx := 1 inp (x := !x) (!x > 0) newy := tt in p (y := noty) (!y) Remark 2.2. In [=-=11]-=- McCusker has adapted the knowingstrategy semantics of IA to give a characterization of IA without the bad-variable constructor mkvar. Using the model he shows that adding mkvar is conservative for ob... |

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Citation Context ...ility). Assuming that base types are finite, observational preorder of 2nd-order IA with recursion (thus admitting recursively-defined first-order functions) is undecidable. The proof uses an idea in =-=[8-=-]. Fix a finite alphabet , and consider a programming system called Queue, which has a single memory cell X , a queue (unbounded, first-in-firstout) data structure, and four instructions, namely, enqu... |

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Citation Context ... 1966. Restricted to the real-time case, the problem was solved positively by Oyamaguchi et al in 1980 [14]. The general decidability problem was only solved recently, also positively, by Senizergues =-=[17]-=- (solution announced in 1997). A simpler proof and a primitive recursive complexity bound were subsequently obtained by Stirling [18, 19]. Construction of the DPDA P f . Let f be a finite view functio... |

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(Show Context)
Citation Context ...ca has extended his earlier work with McCusker to a call-by-value language with arrays [4], and to model checking Hoare-style program correctness assertions [3]. In a foundational direction, Abramsky =-=[1]-=- has extended the regular language representation to games in whose plays there is at most one pending occurrence of a question at any time, so that each move has a unique justifier (and so pointers m... |

1 |
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(Show Context)
Citation Context ... problem was only solved recently, also positively, by Senizergues [17] (solution announced in 1997). A simpler proof and a primitive recursive complexity bound were subsequently obtained by Stirling =-=[18, 19]-=-. Construction of the DPDA P f . Let f be a finite view function that generates an innocent strategy-with-state over a 3rd-order arena A. We shall construct a real-time DPDA P f that recognizes the co... |