ISSN 1055-1344, Siberian Advances in Mathematics, 2018, Vol. 28, No. 2, pp. 79–100.
Allerton Press, Inc., 2018.
Original Russian Text
A.S. Gerasimov, 2017, published in Matematicheskie Trudy, 2017, Vol. 20, No. 2, pp. 3–34.
Inﬁnite-Valued First-Order Łukasiewicz Logic: Hypersequent Calculi
Without Structural Rules and Proof Search for Sentences
in the Prenex Form
A. S. Gerasimov
39-35 Ozerkovaya St., Saint Petersburg, 198516 Russia.
Received March 21, 2016
Abstract—The rational ﬁrst-order Pavelka logic is an expansion of the inﬁnite-valued ﬁrst-order
Łukasiewicz logic Ł∀ by truth constants. For this logic, we introduce a cumulative hypersequent
Ł∀ and a noncumulative hypersequent calculus G
Ł∀ without structural inference rules.
We compare these calculi with the Baaz–Metcalfe hypersequent calculus GŁ∀ with structural rules.
Inparticular,weshowthateveryGŁ∀-provable sentence is G
Ł∀-provable and a Ł∀-sentence
in the prenex form is GŁ∀-provable if and only if it is G
Ł∀-provable. For a tableau version of
the calculus G
Ł∀, we describe a family of proof search algorithms that allow us to construct a proof
of each G
Ł∀-provable sentence in the prenex form.
Keywords: fuzzy logic, inﬁnite-valued ﬁrst-order Łukasiewicz logic, rational ﬁrst-order
Pavelka logic, hypersequent calculus, proof search algorithm.
The inﬁnite-valued ﬁrst-order Łukasiewicz logic Ł∀ and the rational ﬁrst-order Pavelka logic RPL∀
(which is an expansion of Ł∀ by truth constants) are among the most important fuzzy logics that are
used for formalization of fuzzy reasoning [5; 12]. However, methods for automated proof search for Ł∀
and RPL∀ are underdeveloped. In fact, only the following calculi are known for these logics.
(1) Hilbert type calculi (see, for example, ) which do not ﬁt for bottom-up proof search.
(2) The Gentzen type hypersequent calculus GŁ∀ (see ) and its slight modiﬁcation for Ł∀
(see ). These are calculi with structural rules, which makes bottom-up proof search too much non-
(3) The Gentzen type sequent calculus F∀ for the logic of fuzzy inequalities (see [8, 9]) that
extends RPL∀. This is a calculus without structural rules and its axioms are recognized with the use of
methods of linear programming.
In , this calculus was transformed into the tableau calculus T
with meta-variables and a weakened restriction on one of quantiﬁer inference rules. Among other results
of that article, we mention embedding of the set of all GŁ∀-provable Ł∀-sentences into the set of
F∀-provable F∀-sentences and an algorithm for solving the problem on closability of a T
tableau together with the proof of NP-completeness of this problem. However, proving Ł∀-sentences
and RPL∀-sentences in T
F∀ ﬁrst required translation into an F∀-sentence; moreover, proof search
In the present article, we solve several problems formulated in . Combining approaches (2)
and (3), we obtain hypersequent calculi for RPL∀ (and for Ł∀) without structural rules. For a tableau
Calculi for the inﬁnite-valued propositional Łukasiewicz logic possessing similar properties are considered in  and [17,
Ch. 7]. The idea to recognize axioms by methods of linear programming used in [8, 9] ﬁrst appeared in .