## Searching for Invariants using Temporal Resolution (2002)

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Venue: | Proceedings of LPAR 2002 |

Citations: | 2 - 2 self |

### BibTeX

@INPROCEEDINGS{Brotherston02searchingfor,

author = {James Brotherston and Anatoli Degtyarev and Michael Fisher and Alexei Lisitsa},

title = {Searching for Invariants using Temporal Resolution},

booktitle = {Proceedings of LPAR 2002},

year = {2002},

publisher = {Springer Verlag}

}

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### Abstract

Abstract. In this paper, we show how the clausal temporal resolution technique developed for temporal logic provides an effective method for searching for invariants, and so is suitable for mechanising a wide class of temporal problems. We demonstrate that this scheme of searching for invariants can be also applied to a class of multi-predicate induction problems represented by mutually recursive definitions. Completeness of the approach, examples of the application of the scheme, and overview of the implementation are described. 1

### Citations

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Citation Context ...fied. 5 Implementation The method described in this paper has been implemented as a part of a prototype prover for temporal specifications in the λClam envinronment [RSG98]. λClam is a proof planning =-=[Bun88]-=- system, implemented in Teyjus λProlog, a higher-order typed logic programming language. A proof plan is a representation of a proof at some level of abstraction (usually above the level of basic infe... |

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Citation Context ...poral specification. After that an analogue of the temporal resolution rule [DF00,DFK02] is applied.At both stages we work with generalisations of step rules, namely with merged step rules based on T =-=[FDP01]-=- of the form n� i=1 pi ⇒ ❣ n � ri where (pi ⇒ ❣ ri) ∈T for all 1 ≤ i ≤ n, and n ≥ 0. Ifn =0the degenerate merged rule true ⇒ ❣ true is produced. Clearly, every merged step rule based on T is a logical... |

87 | Decidable fragments of first-order temporal logics. Annals of pure and applied logic 106:85–134
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Citation Context ...ses our arguments are heuristic since both sequents lie outside of any known complete fragment of FOLTL. Recently, the interesting monodic fragment of first-order temporal logic has been investigated =-=[HWZ00]-=-. This has a quite transparent (and intuitive) syntactic definition and a finite Hilbert-like inference system [WZ01]. In [DF01] a clausal temporal resolution procedure has been developed covering a s... |

79 | leanT A P : Lean tableau-based deduction
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- 1995
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Citation Context ...l translations of temporal formulae. For first-order (non-temporal) proving required within the prover an atomic method proof tableau reimplementing the simple, but convenient LeanTap tableaux prover =-=[BP95]-=- in λProlog, is used. The kernel of the system is an atomic method mutual induction, implementing an invariant scheme more general than one discussed above and applicable not only to monodic specifica... |

59 | System description: proof planning in higher-order logic with Lambda-Clam
- Richardson, Smaill, et al.
- 1998
(Show Context)
Citation Context ...e conditions of Theorem 5 are satisfied. 5 Implementation The method described in this paper has been implemented as a part of a prototype prover for temporal specifications in the λClam envinronment =-=[RSG98]-=-. λClam is a proof planning [Bun88] system, implemented in Teyjus λProlog, a higher-order typed logic programming language. A proof plan is a representation of a proof at some level of abstraction (us... |

48 | Monodic fragments of first-order temporal logics: 2000–2001 A.D
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- 2001
(Show Context)
Citation Context ... interesting monodic fragment of first-order temporal logic has been investigated [HWZ00]. This has a quite transparent (and intuitive) syntactic definition and a finite Hilbert-like inference system =-=[WZ01]-=-. In [DF01] a clausal temporal resolution procedure has been developed covering a special subclass of the monodic fragment, namely the subclass of ground eventuality monodic problems. In this paper we... |

45 | A Normal Form for Temporal Logic and its Application in TheoremProving and Execution - Fisher |

42 | The automation of proof by mathematical induction
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Citation Context ... straightforward application of usual (one-step) induction. The first example can be tackled by two-step induction, but in general the task of finding an appropriate induction scheme is a work of art =-=[Bun01]-=-.s92 J. Brotherston et al. 4 First-Order Invariant Scheme We now consider a more complex invariant scheme corresponding to a fragment of firstorder temporal logic. A first-order temporal specification... |

36 | The Classical Decision Problems
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Citation Context ... a constant distribution. Note 1. The notion of the colour scheme came from the well known method within the decidability proof for the monadic class in classical first-order logic (see, for example, =-=[BGG97]-=-). In our case Γ is the quotient domain (a subset of all possible equivalence classes of predicate values), θ is a propositional valuation, and ρ is a standard interpretation of constants in the domai... |

27 | Towards first-order temporal resolution
- DEGTYAREV, FISHER
- 2001
(Show Context)
Citation Context ...g monodic fragment of first-order temporal logic has been investigated [HWZ00]. This has a quite transparent (and intuitive) syntactic definition and a finite Hilbert-like inference system [WZ01]. In =-=[DF01]-=- a clausal temporal resolution procedure has been developed covering a special subclass of the monodic fragment, namely the subclass of ground eventuality monodic problems. In this paper we apply this... |

14 | Equality and monodic first-order temporal logic
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- 2002
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Citation Context ...uction problem becomes not only undecidable but not even partially decidable. (Simulating Minsky mashines by formulae of two-variable monadic monodic first-order temporal logic with equality given in =-=[DFL02]-=- can be transformed into simulating them by non-monodic ground induction problems.) Without loss of generality we suppose that there are no two distinct temporal step rules with the same left-hand sid... |

13 | A simplified clausal resolution procedure for propositional linear-time temporal logic, in: Automated Reasoning with Analytic Tableaux and Related Methods, volume 2381 - Degtyarev, Fisher, et al. |

13 |
Gentzen-systems for propositional temporal logics
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(Show Context)
Citation Context ... systems for FOLTL contain the ω-type infinitary rule 1 of inference [Kaw87]: Γ → ∆,ψ; Γ → ∆, ❣ ψ; ... Γ → ∆, ❣n ψ; ... Γ → ∆, ψ (→ ω) However, in some cases (in particular, in the propositional case =-=[Pae88]-=-), instead of the ω-type rule (→ ω) the following finitary rule can be used: Γ → ∆,I; I → ❣ I; I → ψ Γ → ∆, ψ (→ ) This rule corresponds to the induction axiom within temporal logic: ψ ∧ (ψ ⊃ ❣ ψ) ⇒ ψ... |

11 |
Concerning the semantic consequence relation in first-order temporal logic
- Szalas
- 1986
(Show Context)
Citation Context ...emporal logic followed by application of a clausal temporal resolution method. It has been known for some time that first-order temporal logic over the Natural numbers (FOLTL, in short) is incomplete =-=[Sza86]-=-; that is, there exists no finitistic inference system which is sound and complete for this logic or, equivalently, the set of valid formulae of the logic is not recursively enumerable. The complete G... |

6 |
Sequential calculus for a first order infinitary temporal logic. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 33
- Kawai
- 1987
(Show Context)
Citation Context ... this logic or, equivalently, the set of valid formulae of the logic is not recursively enumerable. The complete Gentzen-like proof systems for FOLTL contain the ω-type infinitary rule 1 of inference =-=[Kaw87]-=-: Γ → ∆,ψ; Γ → ∆, ❣ ψ; ... Γ → ∆, ❣n ψ; ... Γ → ∆, ψ (→ ω) However, in some cases (in particular, in the propositional case [Pae88]), instead of the ω-type rule (→ ω) the following finitary rule can b... |

3 | Automatic derivation and application of induction schemes for mutually recursive functions
- Boulton, Slind
- 2000
(Show Context)
Citation Context ...umbers with recursive predicate definitions. Such recursion is difficult for many systems to work with effectively, often leading to quite complex and non-trivial induction schemes (see, for example, =-=[BS00]-=- where the use of mutually recursive definitions has been investigated and several heuristic multi-predicate induction schemes have been developed in order to make implementations of such definitions ... |

2 | Propositional temporal resolution revised - Degtyarev, Fisher - 2000 |

1 |
Searching for Invariants using Temporal Resolution
- Lisitsa
- 2002
(Show Context)
Citation Context ...inally, in §6, we provide concluding remarks. Some technical proofs in §4 are ommited due to lack of space and can be found in the full version of this paper, which is available as a technical report =-=[BDFL02]-=-.s88 J. Brotherston et al. 2 Preliminaries We consider a first-order temporal logic over the Natural numbers TL(N) via a firstorder temporal language TL. The language TL is constructed in a standard w... |

1 | Simple decision procedures for non-monodic decidable fragments of FOLTL - Degtyarev, Fisher, et al. - 2002 |

1 | A decidable deductive procedure for a restricted FTL - Pliuskevicius - 2000 |