Михаил Рыбаков - Academia.edu (original) (raw)

Papers by Михаил Рыбаков

Research paper thumbnail of Variations on the Kripke Trick

Studia logica, Mar 6, 2024

In the early 1960s, to prove undecidability of monadic fragments of sublogics of the predicate mo... more In the early 1960s, to prove undecidability of monadic fragments of sublogics of the predicate modal logic QS5 that include the classical predicate logic QCl, Saul Kripke showed how a classical atomic formula with a binary predicate letter can be simulated by a monadic modal formula. We consider adaptations of Kripke's simulation, which we call the Kripke trick, to various modal and superintuitionistic predicate logics not considered by Kripke. We also discuss settings where the Kripke trick does not work and where, as a result, decidability of monadic modal predicate logics can be obtained.

Research paper thumbnail of Complexity and expressivity of propositional dynamic logics with finitely many variables

Logic Journal of the IGPL, Jun 5, 2018

We investigate the complexity of satisfiability for finite-variable fragments of propositional dy... more We investigate the complexity of satisfiability for finite-variable fragments of propositional dynamic logics. We consider three formalisms belonging to three representative complexity classes, broadly understood,-regular PDL, which is EXPTIME-complete, PDL with intersection, which is 2EXPTIMEcomplete, and PDL with parallel composition, which is undecidable. We show that, for each of these logics, the complexity of satisfiability remains unchanged even if we only allow as inputs formulas built solely out of propositional constants, i.e. without propositional variables. Moreover, we show that this is a consequence of the richness of the expressive power of variable-free fragments: for all the logics we consider, such fragments are as semantically expressive as entire logics. We conjecture that this is representative of PDL-style, as well as closely related, logics.

Research paper thumbnail of Complexity function and complexity of validity of modal and superintuitionistic propositional logics

Journal of Logic and Computation, Jan 3, 2023

Research paper thumbnail of Undecidability of First-Order Modal and Intuitionistic Logics with Two Variables and One Monadic Predicate Letter

Studia Logica, Jul 17, 2018

We prove that the positive fragment of first-order intuitionistic logic in the language with two ... more We prove that the positive fragment of first-order intuitionistic logic in the language with two individual variables and a single monadic predicate letter, without functional symbols, constants, and equality, is undecidable. This holds true regardless of whether we consider semantics with expanding or constant domains. We then generalise this result to intervals [QBL, QKC] and [QBL, QFL], where QKC is the logic of the weak law of the excluded middle and QBL and QFL are first-order counterparts of Visser's basic and formal logics, respectively. We also show that, for most "natural" first-order modal logics, the two-variable fragment with a single monadic predicate letter, without functional symbols, constants, and equality, is undecidable, regardless of whether we consider semantics with expanding or constant domains. These include all sublogics of QKTB, QGL, and QGrz-among them, QK, QT, QKB, QD, QK4, and QS4.

Research paper thumbnail of A Recursively Enumerable Kripke Complete First-Order Logic Not Complete with Respect to a First-Order Definable Class of Frames

Advances in Modal Logic, 2018

It is well-known that every quantified modal logic complete with respect to a firstorder definabl... more It is well-known that every quantified modal logic complete with respect to a firstorder definable class of Kripke frames is recursively enumerable. Numerous examples are also known of "natural" quantified modal logics complete with respect to a class of frames defined by an essentially second-order condition which are not recursively enumerable. It is not, however, known if these examples are instances of a pattern, i.e., whether every recursively enumerable, Kripke complete quantified modal logic can be characterized by a first-order definable class of frames. While the question remains open for normal logics, we show that, in the context of quasi-normal logics, this is not so, by exhibiting an example of a recursively enumerable, Kripke complete quasi-normal logic that is not complete with respect to any first-order definable class of (pointed) frames.

Research paper thumbnail of Complexity of finite-variable fragments of propositional temporal and modal logics of computation

Theoretical Computer Science, Aug 1, 2022

Research paper thumbnail of On complexity of propositional linear-time temporal logic with finitely many variables

It is known [4] that both satisfiability and model-checking problems for propositional Linear-tim... more It is known [4] that both satisfiability and model-checking problems for propositional Linear-time Temporal Logic, LTL, with only a single propositional variable in the language are PSPACE-complete, which coincides with the complexity of these problems for LTL with an arbitrary number of propositional variables [14]. In the present paper, we show that the same result can be obtained by modifying the original proof of PSPACE-hardness for LTL from [14]; i.e., we show how to modify the construction from [14] to model the computations of polynomially-space bound Turing machines using only formulas of one variable. We believe that our alternative proof of the results from [4] gives additional insight into the semantic and computational properties of LTL.

Research paper thumbnail of Recursive enumerability and elementary frame definability in predicate modal logic

Journal of Logic and Computation, Dec 20, 2019

We investigate the relationship between recursive enumerability and elementary frame definability... more We investigate the relationship between recursive enumerability and elementary frame definability in first-order predicate modal logic. On the one hand, it is wellknown that every first-order predicate modal logic complete with respect to an elementary class of Kripke frames, i.e., a class of frames definable by a classical first-order formula, is recursively enumerable. On the other, numerous examples are known of predicate modal logics, based on "natural" propositional modal logics with essentially second-order Kripke semantics, that are either not recursively enumerable or Kripke incomplete. This raises the question of whether every Kripke complete, recursively enumerable predicate modal logic can be characterized by an elementary class of Kripke frames. We answer this question in the negative, by constructing a normal predicate modal logic which is Kripke complete, recursively enumerable, but not complete with respect to an elementary class of frames. We also present an example of a normal predicate modal logic that is recursively enumerable, Kripke complete, and not complete with respect to an elementary class of rooted frames, but is complete with respect to an elementary class of frames that are not rooted.

Research paper thumbnail of Algorithmic properties of first-order modal logics of finite Kripke frames in restricted languages

Journal of Logic and Computation, Aug 26, 2020

We study algorithmic properties of first-order predicate monomodal logics of the frames N, < and ... more We study algorithmic properties of first-order predicate monomodal logics of the frames N, < and N, in languages with restrictions on the number of individual variables as well as the number and arity of predicate letters. The languages we consider have no constants, function symbols, or the equality symbol. We show that satisfiability for the logic of N, < is Σ 1 1-hard in languages with two individual variables and two monadic predicate letters. We also show that satisfiability for the logic of N, is Σ 1 1-hard in languages with two individual variables, two monadic, and one 0-ary predicate letter. Thus, these logics are Π 1 1-hard, and therefore not recursively enumerable, in languages with the aforementioned restrictions. Similar results are obtained for the class of first-order predicate monomodal logics of frames N, R , where R is a binary relation between < and .

Research paper thumbnail of Kripke trick and decidability of monadic fragments of modal and superintuitionistic logics

arXiv (Cornell University), Jul 6, 2023

Трюк Крипке позволяет моделировать бинарную предикатную букву в классических формулах модальными ... more Трюк Крипке позволяет моделировать бинарную предикатную букву в классических формулах модальными формулами с двумя унарными предикатными буквами. Рассматриваются вариации трюка Крипке и возможности его применения в модальных и суперинтуиционистских предикатных логиках. Кроме того, обсуждаются ситуации, когда применить трюк Крипке невозможно.

Research paper thumbnail of Algorithmic properties of QK4.3 and QS4.3

arXiv (Cornell University), Jul 6, 2023

We prove that predicate modal logics QK4.3 and QS4.3 are undecidable more precisely, Σ 0 1-comple... more We prove that predicate modal logics QK4.3 and QS4.3 are undecidable more precisely, Σ 0 1-complete-in languages with two individual variables, one modandic predicate letter, and one proposition letter.

Research paper thumbnail of Бинарный предикат, транзитивное замыкание, две-три переменные: сыграем в домино?

Logičeskie issledovaniâ, May 27, 2023

Research paper thumbnail of Complexity of intuitionistic and Visser's basic and formal logics in finitely many variables

Advances in Modal Logic, 2006

Research paper thumbnail of Complexity of finite-variable fragments of propositional modal logics of symmetric frames

Logic Journal of the IGPL, Jul 2, 2018

While finite-variable fragments of the propositional modal logic S5-complete with respect to refl... more While finite-variable fragments of the propositional modal logic S5-complete with respect to reflexive, symmetric, and transitive frames-are polynomialtime decidable, the restriction to finite-variable formulas for logics of reflexive and transitive frames yields fragments that remain "intractable." The role of the symmetry condition in this context has not been investigated. We show that symmetry either by itself or in combination with reflexivity produces logics that behave just like logics of reflexive and transitive frames, i.e., their finite-variable fragments remain intractable, namely PSPACE-hard. This raises the question of where exactly the borderline lies between modal logics whose finite-variable fragments are tractable and the rest.

Research paper thumbnail of Алгоритмическая неразрешимость проблемы первопорядковой определимости формул логики ветвящегося времени

Программные продукты и системы, Aug 31, 2018

Research paper thumbnail of Correction to: Undecidability of First-Order Modal and Intuitionistic Logics with Two Variables and One Monadic Predicate Letter

Studia Logica, Dec 23, 2021

The original version of the paper contains three errors. The first concerns the way the results o... more The original version of the paper contains three errors. The first concerns the way the results of Kontchakov, Kurucz, and Zakharyschev [12] are used to obtain Theorem 2.6. The second concerns the definition of the intuitionistic Kripke frame F and a-suitable models based on F used in the proof of Lemma 3.6. The third concern the definition of formulas in Section 2.3. To correct the errors, the following changes need to be made to the original version of the paper: The discussion in the first three paragraphs of Section 2.2 should be disregarded; Section 2.2 should be read starting from the sentence "Let ϕ be a (closed) formula containing monadic predicate letters P 1 ,. .. , P n ." on page 700. Lemma 2.5 should be restated to apply only to L-suitable (where L ∈ {QK, QGL, QGrz}) formulas, defined as follows: ψ is L-suitable if it contains only monadic predicate letters and either ψ is not L-satisfiable or ψ is satisfiable in an L-model M with the downward inheritance property for monadic predicate letters: M |= ♦P (x) → P (x), for every monadic letter P. In the proof of Theorem 2.6, the translation e should be applied not to the formulas ξ T , but to slightly modified formulas, defined as follows: first, replace in the formula χ T [12, p. 433] every occurrence of D(x) by ¬D(x); then, in thus obtained formula, replace every subformula ψ by (∀x Q(x) → ψ); last, in thus obtained formula replace succ H (x, y) by ♦(¬Q H 1 (x) ∧ ¬Q H 2 (y)) and succ V (x, y) by ♦(¬Q V 1 (x)∧¬Q V 2 (y)). The resultant formula is L-suitable (where L ∈ {QK, QGL, QGrz}); this permits application of the corrected Lemma 2.5.

Research paper thumbnail of Algorithmical properties of quasinormal modal logics with linear finite model property

Вестник Тверского государственного университета, Sep 1, 2019

Research paper thumbnail of Axiomatizability of non-normal and quasi-normal modal predicate logics of first-order definable classes of Kripke frames

Вестник Тверского государственного университета, May 10, 2018

Research paper thumbnail of Predicate counterparts of modal logics of provability: High undecidability and Kripke incompleteness

Logic Journal of the IGPL, Feb 28, 2023

In this paper, the predicate counterparts, defined both axiomatically and semantically by means o... more In this paper, the predicate counterparts, defined both axiomatically and semantically by means of Kripke frames, of the modal propositional logics textbfGL\textbf {GL}textbfGL, textbfGrz\textbf {Grz}textbfGrz, textbfwGrz\textbf {wGrz}textbfwGrz and their extensions are considered. It is proved that the set of semantical consequences on Kripke frames of every logic between textbfQwGrz\textbf {QwGrz}textbfQwGrz and textbfQGL.3\textbf {QGL.3}textbfQGL.3 or between textbfQwGrz\textbf {QwGrz}textbfQwGrz and textbfQGrz.3\textbf {QGrz.3}textbfQGrz.3 is Pi1_1\Pi ^1_1Pi11-hard even in languages with three (sometimes, two) individual variables, two (sometimes, one) unary predicate letters, and a single proposition letter. As a corollary, it is proved that infinite families of modal predicate axiomatic systems, based on the classical first-order logic and the modal propositional logics textbfGL\textbf {GL}textbfGL, textbfGrz\textbf {Grz}textbfGrz, textbfwGrz\textbf {wGrz}textbfwGrz are not Kripke complete. Both Pi11\Pi ^1_1Pi1_1-hardness and Kripke incompleteness results of the paper do not depend on whether the logics contain the Barcan formula.

Research paper thumbnail of Undecidability of the Logic of Partial Quasiary Predicates

Logic Journal of the IGPL, May 20, 2021

We obtain an effective embedding of the classical predicate logic into the logic of partial quasi... more We obtain an effective embedding of the classical predicate logic into the logic of partial quasiary predicates. The embedding has the property that an image of a non-theorem of the classical logic is refutable in a model of the logic of partial quasiary predicates that has the same cardinality as the classical countermodel of the non-theorem. Therefore, we also obtain an embedding of the classical predicate logic of finite models into the logic of partial quasiary predicates over finite structures. As a consequence, we prove that the logic of partial quasiary predicates is undecidable—more precisely, varSigma0_1\varSigma ^0_1varSigma01-complete—over arbitrary structures and not recursively enumerable—more precisely, varPi01\varPi ^0_1varPi0_1-complete—over finite structures.

Research paper thumbnail of Variations on the Kripke Trick

Studia logica, Mar 6, 2024

In the early 1960s, to prove undecidability of monadic fragments of sublogics of the predicate mo... more In the early 1960s, to prove undecidability of monadic fragments of sublogics of the predicate modal logic QS5 that include the classical predicate logic QCl, Saul Kripke showed how a classical atomic formula with a binary predicate letter can be simulated by a monadic modal formula. We consider adaptations of Kripke's simulation, which we call the Kripke trick, to various modal and superintuitionistic predicate logics not considered by Kripke. We also discuss settings where the Kripke trick does not work and where, as a result, decidability of monadic modal predicate logics can be obtained.

Research paper thumbnail of Complexity and expressivity of propositional dynamic logics with finitely many variables

Logic Journal of the IGPL, Jun 5, 2018

We investigate the complexity of satisfiability for finite-variable fragments of propositional dy... more We investigate the complexity of satisfiability for finite-variable fragments of propositional dynamic logics. We consider three formalisms belonging to three representative complexity classes, broadly understood,-regular PDL, which is EXPTIME-complete, PDL with intersection, which is 2EXPTIMEcomplete, and PDL with parallel composition, which is undecidable. We show that, for each of these logics, the complexity of satisfiability remains unchanged even if we only allow as inputs formulas built solely out of propositional constants, i.e. without propositional variables. Moreover, we show that this is a consequence of the richness of the expressive power of variable-free fragments: for all the logics we consider, such fragments are as semantically expressive as entire logics. We conjecture that this is representative of PDL-style, as well as closely related, logics.

Research paper thumbnail of Complexity function and complexity of validity of modal and superintuitionistic propositional logics

Journal of Logic and Computation, Jan 3, 2023

Research paper thumbnail of Undecidability of First-Order Modal and Intuitionistic Logics with Two Variables and One Monadic Predicate Letter

Studia Logica, Jul 17, 2018

We prove that the positive fragment of first-order intuitionistic logic in the language with two ... more We prove that the positive fragment of first-order intuitionistic logic in the language with two individual variables and a single monadic predicate letter, without functional symbols, constants, and equality, is undecidable. This holds true regardless of whether we consider semantics with expanding or constant domains. We then generalise this result to intervals [QBL, QKC] and [QBL, QFL], where QKC is the logic of the weak law of the excluded middle and QBL and QFL are first-order counterparts of Visser's basic and formal logics, respectively. We also show that, for most "natural" first-order modal logics, the two-variable fragment with a single monadic predicate letter, without functional symbols, constants, and equality, is undecidable, regardless of whether we consider semantics with expanding or constant domains. These include all sublogics of QKTB, QGL, and QGrz-among them, QK, QT, QKB, QD, QK4, and QS4.

Research paper thumbnail of A Recursively Enumerable Kripke Complete First-Order Logic Not Complete with Respect to a First-Order Definable Class of Frames

Advances in Modal Logic, 2018

It is well-known that every quantified modal logic complete with respect to a firstorder definabl... more It is well-known that every quantified modal logic complete with respect to a firstorder definable class of Kripke frames is recursively enumerable. Numerous examples are also known of "natural" quantified modal logics complete with respect to a class of frames defined by an essentially second-order condition which are not recursively enumerable. It is not, however, known if these examples are instances of a pattern, i.e., whether every recursively enumerable, Kripke complete quantified modal logic can be characterized by a first-order definable class of frames. While the question remains open for normal logics, we show that, in the context of quasi-normal logics, this is not so, by exhibiting an example of a recursively enumerable, Kripke complete quasi-normal logic that is not complete with respect to any first-order definable class of (pointed) frames.

Research paper thumbnail of Complexity of finite-variable fragments of propositional temporal and modal logics of computation

Theoretical Computer Science, Aug 1, 2022

Research paper thumbnail of On complexity of propositional linear-time temporal logic with finitely many variables

It is known [4] that both satisfiability and model-checking problems for propositional Linear-tim... more It is known [4] that both satisfiability and model-checking problems for propositional Linear-time Temporal Logic, LTL, with only a single propositional variable in the language are PSPACE-complete, which coincides with the complexity of these problems for LTL with an arbitrary number of propositional variables [14]. In the present paper, we show that the same result can be obtained by modifying the original proof of PSPACE-hardness for LTL from [14]; i.e., we show how to modify the construction from [14] to model the computations of polynomially-space bound Turing machines using only formulas of one variable. We believe that our alternative proof of the results from [4] gives additional insight into the semantic and computational properties of LTL.

Research paper thumbnail of Recursive enumerability and elementary frame definability in predicate modal logic

Journal of Logic and Computation, Dec 20, 2019

We investigate the relationship between recursive enumerability and elementary frame definability... more We investigate the relationship between recursive enumerability and elementary frame definability in first-order predicate modal logic. On the one hand, it is wellknown that every first-order predicate modal logic complete with respect to an elementary class of Kripke frames, i.e., a class of frames definable by a classical first-order formula, is recursively enumerable. On the other, numerous examples are known of predicate modal logics, based on "natural" propositional modal logics with essentially second-order Kripke semantics, that are either not recursively enumerable or Kripke incomplete. This raises the question of whether every Kripke complete, recursively enumerable predicate modal logic can be characterized by an elementary class of Kripke frames. We answer this question in the negative, by constructing a normal predicate modal logic which is Kripke complete, recursively enumerable, but not complete with respect to an elementary class of frames. We also present an example of a normal predicate modal logic that is recursively enumerable, Kripke complete, and not complete with respect to an elementary class of rooted frames, but is complete with respect to an elementary class of frames that are not rooted.

Research paper thumbnail of Algorithmic properties of first-order modal logics of finite Kripke frames in restricted languages

Journal of Logic and Computation, Aug 26, 2020

We study algorithmic properties of first-order predicate monomodal logics of the frames N, < and ... more We study algorithmic properties of first-order predicate monomodal logics of the frames N, < and N, in languages with restrictions on the number of individual variables as well as the number and arity of predicate letters. The languages we consider have no constants, function symbols, or the equality symbol. We show that satisfiability for the logic of N, < is Σ 1 1-hard in languages with two individual variables and two monadic predicate letters. We also show that satisfiability for the logic of N, is Σ 1 1-hard in languages with two individual variables, two monadic, and one 0-ary predicate letter. Thus, these logics are Π 1 1-hard, and therefore not recursively enumerable, in languages with the aforementioned restrictions. Similar results are obtained for the class of first-order predicate monomodal logics of frames N, R , where R is a binary relation between < and .

Research paper thumbnail of Kripke trick and decidability of monadic fragments of modal and superintuitionistic logics

arXiv (Cornell University), Jul 6, 2023

Трюк Крипке позволяет моделировать бинарную предикатную букву в классических формулах модальными ... more Трюк Крипке позволяет моделировать бинарную предикатную букву в классических формулах модальными формулами с двумя унарными предикатными буквами. Рассматриваются вариации трюка Крипке и возможности его применения в модальных и суперинтуиционистских предикатных логиках. Кроме того, обсуждаются ситуации, когда применить трюк Крипке невозможно.

Research paper thumbnail of Algorithmic properties of QK4.3 and QS4.3

arXiv (Cornell University), Jul 6, 2023

We prove that predicate modal logics QK4.3 and QS4.3 are undecidable more precisely, Σ 0 1-comple... more We prove that predicate modal logics QK4.3 and QS4.3 are undecidable more precisely, Σ 0 1-complete-in languages with two individual variables, one modandic predicate letter, and one proposition letter.

Research paper thumbnail of Бинарный предикат, транзитивное замыкание, две-три переменные: сыграем в домино?

Logičeskie issledovaniâ, May 27, 2023

Research paper thumbnail of Complexity of intuitionistic and Visser's basic and formal logics in finitely many variables

Advances in Modal Logic, 2006

Research paper thumbnail of Complexity of finite-variable fragments of propositional modal logics of symmetric frames

Logic Journal of the IGPL, Jul 2, 2018

While finite-variable fragments of the propositional modal logic S5-complete with respect to refl... more While finite-variable fragments of the propositional modal logic S5-complete with respect to reflexive, symmetric, and transitive frames-are polynomialtime decidable, the restriction to finite-variable formulas for logics of reflexive and transitive frames yields fragments that remain "intractable." The role of the symmetry condition in this context has not been investigated. We show that symmetry either by itself or in combination with reflexivity produces logics that behave just like logics of reflexive and transitive frames, i.e., their finite-variable fragments remain intractable, namely PSPACE-hard. This raises the question of where exactly the borderline lies between modal logics whose finite-variable fragments are tractable and the rest.

Research paper thumbnail of Алгоритмическая неразрешимость проблемы первопорядковой определимости формул логики ветвящегося времени

Программные продукты и системы, Aug 31, 2018

Research paper thumbnail of Correction to: Undecidability of First-Order Modal and Intuitionistic Logics with Two Variables and One Monadic Predicate Letter

Studia Logica, Dec 23, 2021

The original version of the paper contains three errors. The first concerns the way the results o... more The original version of the paper contains three errors. The first concerns the way the results of Kontchakov, Kurucz, and Zakharyschev [12] are used to obtain Theorem 2.6. The second concerns the definition of the intuitionistic Kripke frame F and a-suitable models based on F used in the proof of Lemma 3.6. The third concern the definition of formulas in Section 2.3. To correct the errors, the following changes need to be made to the original version of the paper: The discussion in the first three paragraphs of Section 2.2 should be disregarded; Section 2.2 should be read starting from the sentence "Let ϕ be a (closed) formula containing monadic predicate letters P 1 ,. .. , P n ." on page 700. Lemma 2.5 should be restated to apply only to L-suitable (where L ∈ {QK, QGL, QGrz}) formulas, defined as follows: ψ is L-suitable if it contains only monadic predicate letters and either ψ is not L-satisfiable or ψ is satisfiable in an L-model M with the downward inheritance property for monadic predicate letters: M |= ♦P (x) → P (x), for every monadic letter P. In the proof of Theorem 2.6, the translation e should be applied not to the formulas ξ T , but to slightly modified formulas, defined as follows: first, replace in the formula χ T [12, p. 433] every occurrence of D(x) by ¬D(x); then, in thus obtained formula, replace every subformula ψ by (∀x Q(x) → ψ); last, in thus obtained formula replace succ H (x, y) by ♦(¬Q H 1 (x) ∧ ¬Q H 2 (y)) and succ V (x, y) by ♦(¬Q V 1 (x)∧¬Q V 2 (y)). The resultant formula is L-suitable (where L ∈ {QK, QGL, QGrz}); this permits application of the corrected Lemma 2.5.

Research paper thumbnail of Algorithmical properties of quasinormal modal logics with linear finite model property

Вестник Тверского государственного университета, Sep 1, 2019

Research paper thumbnail of Axiomatizability of non-normal and quasi-normal modal predicate logics of first-order definable classes of Kripke frames

Вестник Тверского государственного университета, May 10, 2018

Research paper thumbnail of Predicate counterparts of modal logics of provability: High undecidability and Kripke incompleteness

Logic Journal of the IGPL, Feb 28, 2023

In this paper, the predicate counterparts, defined both axiomatically and semantically by means o... more In this paper, the predicate counterparts, defined both axiomatically and semantically by means of Kripke frames, of the modal propositional logics textbfGL\textbf {GL}textbfGL, textbfGrz\textbf {Grz}textbfGrz, textbfwGrz\textbf {wGrz}textbfwGrz and their extensions are considered. It is proved that the set of semantical consequences on Kripke frames of every logic between textbfQwGrz\textbf {QwGrz}textbfQwGrz and textbfQGL.3\textbf {QGL.3}textbfQGL.3 or between textbfQwGrz\textbf {QwGrz}textbfQwGrz and textbfQGrz.3\textbf {QGrz.3}textbfQGrz.3 is Pi1_1\Pi ^1_1Pi11-hard even in languages with three (sometimes, two) individual variables, two (sometimes, one) unary predicate letters, and a single proposition letter. As a corollary, it is proved that infinite families of modal predicate axiomatic systems, based on the classical first-order logic and the modal propositional logics textbfGL\textbf {GL}textbfGL, textbfGrz\textbf {Grz}textbfGrz, textbfwGrz\textbf {wGrz}textbfwGrz are not Kripke complete. Both Pi11\Pi ^1_1Pi1_1-hardness and Kripke incompleteness results of the paper do not depend on whether the logics contain the Barcan formula.

Research paper thumbnail of Undecidability of the Logic of Partial Quasiary Predicates

Logic Journal of the IGPL, May 20, 2021

We obtain an effective embedding of the classical predicate logic into the logic of partial quasi... more We obtain an effective embedding of the classical predicate logic into the logic of partial quasiary predicates. The embedding has the property that an image of a non-theorem of the classical logic is refutable in a model of the logic of partial quasiary predicates that has the same cardinality as the classical countermodel of the non-theorem. Therefore, we also obtain an embedding of the classical predicate logic of finite models into the logic of partial quasiary predicates over finite structures. As a consequence, we prove that the logic of partial quasiary predicates is undecidable—more precisely, varSigma0_1\varSigma ^0_1varSigma01-complete—over arbitrary structures and not recursively enumerable—more precisely, varPi01\varPi ^0_1varPi0_1-complete—over finite structures.