# Weak Completeness Theorem for Propositional Linear Time Temporal Logic

Formalized Mathematics (2012)

- Volume: 20, Issue: 3, page 227-234
- ISSN: 1426-2630

## Access Full Article

top## Abstract

top## How to cite

topMariusz Giero. "Weak Completeness Theorem for Propositional Linear Time Temporal Logic." Formalized Mathematics 20.3 (2012): 227-234. <http://eudml.org/doc/268035>.

@article{MariuszGiero2012,

abstract = {We prove weak (finite set of premises) completeness theorem for extended propositional linear time temporal logic with irreflexive version of until-operator. We base it on the proof of completeness for basic propositional linear time temporal logic given in [20] which roughly follows the idea of the Henkin-Hasenjaeger method for classical logic. We show that a temporal model exists for every formula which negation is not derivable (Satisfiability Theorem). The contrapositive of that theorem leads to derivability of every valid formula. We build a tree of consistent and complete PNPs which is used to construct the model.},

author = {Mariusz Giero},

journal = {Formalized Mathematics},

keywords = {weak completeness theorem; completeness theorem; temporal logic; temporal model; satisfiability theorem},

language = {eng},

number = {3},

pages = {227-234},

title = {Weak Completeness Theorem for Propositional Linear Time Temporal Logic},

url = {http://eudml.org/doc/268035},

volume = {20},

year = {2012},

}

TY - JOUR

AU - Mariusz Giero

TI - Weak Completeness Theorem for Propositional Linear Time Temporal Logic

JO - Formalized Mathematics

PY - 2012

VL - 20

IS - 3

SP - 227

EP - 234

AB - We prove weak (finite set of premises) completeness theorem for extended propositional linear time temporal logic with irreflexive version of until-operator. We base it on the proof of completeness for basic propositional linear time temporal logic given in [20] which roughly follows the idea of the Henkin-Hasenjaeger method for classical logic. We show that a temporal model exists for every formula which negation is not derivable (Satisfiability Theorem). The contrapositive of that theorem leads to derivability of every valid formula. We build a tree of consistent and complete PNPs which is used to construct the model.

LA - eng

KW - weak completeness theorem; completeness theorem; temporal logic; temporal model; satisfiability theorem

UR - http://eudml.org/doc/268035

ER -

## References

top- [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
- [2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. Zbl06213858
- [3] Grzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421-427, 1990.
- [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
- [5] Grzegorz Bancerek. K¨onig’s lemma. Formalized Mathematics, 2(3):397-402, 1991.
- [6] Grzegorz Bancerek. Joining of decorated trees. Formalized Mathematics, 4(1):77-82, 1993.
- [7] Grzegorz Bancerek. Subtrees. Formalized Mathematics, 5(2):185-190, 1996.
- [8] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
- [9] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.
- [10] Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. FormalizedMathematics, 1(3):529-536, 1990.
- [11] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
- [12] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
- [13] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
- [14] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
- [15] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
- [16] Mariusz Giero. The axiomatization of propositional linear time temporal logic. FormalizedMathematics, 19(2):113-119, 2011, doi: 10.2478/v10037-011-0018-1.[Crossref] Zbl1276.03018
- [17] Mariusz Giero. The derivations of temporal logic formulas. Formalized Mathematics, 20(3):215-219, 2012, doi: 10.2478/v10037-012-0025-x.[Crossref] Zbl1285.03009
- [18] Mariusz Giero. The properties of sets of temporal logic subformulas. Formalized Mathematics, 20(3):221-226, 2012, doi: 10.2478/v10037-012-0026-9.[Crossref] Zbl1285.03010
- [19] Adam Grabowski. Hilbert positive propositional calculus. Formalized Mathematics, 8(1):69-72, 1999.
- [20] Fred Kr¨oger and Stephan Merz. Temporal Logic and State Systems. Springer-Verlag, 2008. Zbl1169.03001
- [21] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.
- [22] Karol Pak. Continuity of barycentric coordinates in Euclidean topological spaces. FormalizedMathematics, 19(3):139-144, 2011, doi: 10.2478/v10037-011-0022-5.[Crossref] Zbl1276.57020
- [23] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.
- [24] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.
- [25] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.
- [26] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.
- [27] Andrzej Trybulec. Defining by structural induction in the positive propositional language. Formalized Mathematics, 8(1):133-137, 1999.
- [28] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
- [29] Edmund Woronowicz. Many argument relations. Formalized Mathematics, 1(4):733-737, 1990.
- [30] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
- [31] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.