Weak Completeness Theorem for Propositional Linear Time Temporal Logic

Open access

Summary

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.

[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.

[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.

[17] Mariusz Giero. The derivations of temporal logic formulas. Formalized Mathematics, 20(3):215-219, 2012, doi: 10.2478/v10037-012-0025-x.

[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.

[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.

[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.

[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.

Formalized Mathematics

(a computer assisted approach)

Journal Information

SCImago Journal Rank (SJR) 2017: 0.119
Source Normalized Impact per Paper (SNIP) 2017: 0.237



Target Group

researchers in the fields of formal methods and computer-checked mathematics

Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 112 112 15
PDF Downloads 14 14 1