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1
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The Axiomatization of Propositional Logic

100%
Giero M.
Formalized Mathematics
|
2016
|
tom 24
|
nr 4
281-290
EN
This article introduces propositional logic as a formal system ([14], [10], [11]). The formulae of the language are as follows φ ::= ⊥ | p | φ → φ. Other connectives are introduced as abbrevations. The notions of model and satisfaction in model are defined. The axioms are all the formulae of the following schemes α ⇒ (β ⇒ α), (α ⇒ (β ⇒ γ)) ⇒ ((α ⇒ β) ⇒ (α ⇒ γ)), (¬β ⇒ ¬α) ⇒ ((¬β ⇒ α) ⇒ β). Modus ponens is the only derivation rule. The soundness theorem and the strong completeness theorem are proved. The proof of the completeness theorem is carried out by a counter-model existence method. In order to prove the completeness theorem, Lindenbaum’s Lemma is proved. Some most widely used tautologies are presented.
2
Content available remote

The Derivations of Temporal Logic Formulas

100%
Giero M.
Formalized Mathematics
|
2012
|
tom 20
|
nr 3
215-219
EN
This is a preliminary article to prove the completeness theorem of an extension of basic propositional temporal logic. We base it on the proof of completeness for basic propositional temporal logic given in [12]. We introduce n-ary connectives and prove their properties. We derive temporal logic formulas.
3
Content available remote

Weak Completeness Theorem for Propositional Linear Time Temporal Logic

100%
Giero M.
Formalized Mathematics
|
2012
|
tom 20
|
nr 3
227-234
EN
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.
4
Content available remote

The Axiomatization of Propositional Linear Time Temporal Logic

100%
Giero M.
Formalized Mathematics
|
2011
|
tom 19
|
nr 2
113-119
EN
The article introduces propositional linear time temporal logic as a formal system. Axioms and rules of derivation are defined. Soundness Theorem and Deduction Theorem are proved [9].
5
Content available remote

The Properties of Sets of Temporal Logic Subformulas

100%
Giero M.
Formalized Mathematics
|
2012
|
tom 20
|
nr 3
221-226
EN
This is a second preliminary article to prove the completeness theorem of an extension of basic propositional temporal logic. We base it on the proof of completeness for basic propositional temporal logic given in [17]. We introduce two modified definitions of a subformula. In the former one we treat until-formula as indivisible. In the latter one, we extend the set of subformulas of until-formulas by a special disjunctive formula. This is needed to construct a temporal model. We also define an ordered positive-negative pair of finite sequences of formulas (PNP). PNPs represent states of a temporal model.
6
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Propositional Linear Temporal Logic with Initial Validity Semantics1

100%
Giero M.
Formalized Mathematics
|
2015
|
tom 23
|
nr 4
379-386
EN
In the article [10] a formal system for Propositional Linear Temporal Logic (in short LTLB) with normal semantics is introduced. The language of this logic consists of “until” operator in a very strict version. The very strict “until” operator enables to express all other temporal operators. In this article we construct a formal system for LTLB with the initial semantics [12]. Initial semantics means that we define the validity of the formula in a model as satisfaction in the initial state of model while normal semantics means that we define the validity as satisfaction in all states of model. We prove the Deduction Theorem, and the soundness and completeness of the introduced formal system. We also prove some theorems to compare both formal systems, i.e., the one introduced in the article [10] and the one introduced in this article. Formal systems for temporal logics are applied in the verification of computer programs. In order to carry out the verification one has to derive an appropriate formula within a selected formal system. The formal systems introduced in [10] and in this article can be used to carry out such verifications in Mizar [4].
7
Content available remote

The Formalization of Decision-Free Petri Net

51%
Shah P. K., Kawamoto P. N., Giero M.
Formalized Mathematics
|
2014
|
tom 22
|
nr 1
29-35
EN
In this article we formalize the definition of Decision-Free Petri Net (DFPN) presented in [19]. Then we formalize the concept of directed path and directed circuit nets in Petri nets to prove properties of DFPN. We also present the definition of firing transitions and transition sequences with natural numbers marking that always check whether transition is enabled or not and after firing it only removes the available tokens (i.e., it does not remove from zero number of tokens). At the end of this article, we show that the total number of tokens in a circuit of decision-free Petri net always remains the same after firing any sequences of the transition.
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