Closure under various completions are studied for the variety basic algebras and the variety of L... more Closure under various completions are studied for the variety basic algebras and the variety of Löb algebras. Basic algebras arise from (Visser's) basic logic, and the class of Löb algebras is an important subclass of basic algebras. Canonical extensions, ideal completions, MacNeille completions and complete ideal completions of these algebras are discussed.
For each ordinal a > 0, L(a) is the intermediate predicate logic characterized by the class of al... more For each ordinal a > 0, L(a) is the intermediate predicate logic characterized by the class of all Kripke frames with the poset a and with constant domain. This paper will be devoted to a study of logics of the form L(a). It will be shown that for each uncountable ordinal of the form a + r\ with a finite or a countable v\ (>0), there exists a countable ordinal of the form [1 + t\ such that L(tx + rj) = L(/J + tf). On the other hand, such a reduction of ordinals to countable ones is impossible for a logic L(a) if a is an uncountable regular ordinal. Moreover, it will be proved that the mapping L is injective if it is restricted to ordinals less than co w , i.e. a # ft implies L(a) # L(/J) for each ordinal a, fS < w". §1. Introduction. In the following, we will assume familiarity with the terminology and notation in [10]. But, for the reader's convenience, we will first give some basic definitions and explanations of the purpose of our study. We assume that our language contains neither constants nor function symbols. We say that a set L of first-order formulas is an intermediate predicate logic if 1) every formula provable in the intuitionistic predicate logic H belongs to L and every formula in L is provable in the classical predicate logic C, and 2) L is closed under modus ponens, substitution and generalization. Similarly, we can define an intermediate propositional logic. Of course, in this case, we must replace 2) by 2') L is closed under modus ponens and substitution. Next, we will give a Kripke-type semantics for intermediate logics. A pair (M, U) is a Kripke frame if M is a nonempty partially ordered set and U is a mapping from M to the power set of some set such that 1) U(a) # 0 for each ae M and 2) a < b implies U(a) £ V(b). We assume moreover that M has the smallest element. If U is a constant mapping such that U(a) = V for each a e M, we say that (M, U) is a Kripke frame with the constant domain V. Since we will restrict our attention only to Kripke
Logic in Asia: Studia logica library, Oct 17, 2015
This is a short survey of semantical study of cut elimination and subformula property in modal lo... more This is a short survey of semantical study of cut elimination and subformula property in modal logics. Cut elimination is a basic proof-theoretic notion in sequent systems, and subformula property is the most important consequence of cut elimination. A special feature of our presentation is its unified semantical approach to them based on Kripke models. Along the same lines as Takano’s works on subformula property, these properties, together with finite model property, will be discussed as modifications of standard construction of canonical Kripke models. These semantical approaches will be compared with algebraic approaches in modal logics, which often take the forms of various kinds of embedding theorems. In the last part of the paper, an attempt is made to clarify connections between semantical approach to cut elimination and algebraic one.
Gentzen-type sequent calculi usually contain three structural rules, i.e., exchange, contraction ... more Gentzen-type sequent calculi usually contain three structural rules, i.e., exchange, contraction and weakening rules. In recent years, however, there have been various studies on logics that have not included some or any of these structural rules. The motives or purposes of these studies have been so diverse that sometimes close connections between them have been overlooked. Here we will make a brief survey of recent results on these logics in an attempt to make these interrelationships clearer.
We will give a state-of-the-art survey of the study of substructural logics. Originally, substruc... more We will give a state-of-the-art survey of the study of substructural logics. Originally, substructural logics were introduced as logics which, when formulated as Gentzen-style systems, lack some of the three basic structural rules, i.e. contraction, weakening and exchange. For example, relevance logics and linear logic lack the weakening rule, many-valued logics, fuzzy logics and linear logic lack the contraction rule, and hence all of them can be regarded as substructural logics. These logics have been studied extensively and various theories have been developed for their investigation. However their study has been carried out independently, mainly due to the different motivations behind them, avoiding comparisons between different substructural logics. On the other hand, the general study of substructural logics has a comparative character, focusing on the absence or presence of structural rules. As such, at least in the initial stages of research, it was a study on how structural rules affect logical properties. This naturally led to a syntactic or proof-theoretic approach yielding deep results about properties of particular logics, provided that they can be formalized in systems, like cut-free Gentzen calculi. An obvious limitation of this study comes from the fact that not all logics have such a formulation. Semantical methods, in contrast, provide a powerful tool for analyzing substructural logics from a more uniform perspective. Both Kripke-style semantics and algebraic semantics for some particular subclasses of substructural logics, e.g. relevant logics, were already introduced in the 70s and 80s and were studied to a certain extent. An algebraic study of substructural logics for the last decade, based mainly on algebraic logic and universal algebra, has brought us a new perspective on substructural logics, which came from the observation that these logics all share the residuation property. Though this is usually not noticed, it is revealed explicitly in a sequent formulation by the use of extralogical symbols, denoted by commas. Precisely speaking, consider the following equivalence, concerning implication →, that holds in most of sequent systems: A formula γ follows from formulas α and β if and only if the implication α → γ follows from β alone. Here 'follows from' is given by the particular substructural logic, and is usually denoted by ⇒. Thus, the above equivalence can be restated as:
A b s t r a c t. The present paper is concerned with the cut elim- inability for some sequent sys... more A b s t r a c t. The present paper is concerned with the cut elim- inability for some sequent systems of noncommutative substructural logics, i.e. substructural logics without exchange rule. Sequent sys- tems of several extensions of noncommutative logics FL and LBB0I, which is sometimes called T! W, will be introduced. Then, the cut elimination theorem and the
A variant of the Robinson property (ROB*) is introduced. The property ROB* is equivalent to the u... more A variant of the Robinson property (ROB*) is introduced. The property ROB* is equivalent to the usual Robinson property and hence is equivalent also to the interpolation property, in any compact logic closed under the Boolean operations. On the other hand, it will be shown that ROB* is not always equal to the interpolation property, if a logic is not closed under them. Next, we will study ROB* and various interpolation properties for equational logics, which are typical examples of compact logics not closed under the Boolean operations. It will be shown that an equational logic has ROB* if and only if it has the amalgamation property for isomorphic embeddings. Recent results in this area are collected into a voluminous work, "Model-Theoretic Logics" [3], which we suggest for further information. In this paper, we will study the relationships between the interpolation property and the Robinson property, in a general setting. We don't assume that a logic is closed under the Boolean operations, i.e. closed under both conjunction and negation. It is well-known that for every compact logic 2g closed under the Boolean operations, ~ has the interpolation property if and only if it has the Robinson property. (See e.g. [3] Chapter II, Proposition 7.1.5. In [3], the proposition is shown under the stronger assumption that 5f is regular. But, by a careful observation of the proof, it can be seen that the closure under the Boolean operations is essential for the proof.) On the other hand, when a given logic lacks negation, the meaning of the Robinson property becomes unclear and inadequate, since it may happen that even the set of all sentences has a model. So, in the present paper, we will Presented by Bjarni Jonsson.
Journal of Computer and System Sciences, Jun 1, 1979
It is shown that the first-order arithmetic A[P(x), 2x, x-I-11 with two functions 2r, x + 1 and a... more It is shown that the first-order arithmetic A[P(x), 2x, x-I-11 with two functions 2r, x + 1 and a monadic predicate symbol P(x) is undecidable, by using a kind of twodimensional finite automata, called finite causal ws-systems. From this immediately follows R. M. Robinson's result, which says that the monadic second-order theory with two functions 2x, x + 1 is undecidable. This is also considered as an improvement on H. Putnam's result about the undecidability of the first-order arithmetic with addition and a monadic predicate symbol.
Aumann's Theorem on Rationality is a well-know but also contentious result in economics. It perta... more Aumann's Theorem on Rationality is a well-know but also contentious result in economics. It pertains to sequential game theory and says that "common knowledge of rationality leads to backwards-induction equilibria" .1 We present a formalist analysis of the result in a meta-theory with primitive support for proof and definition by induction. The analysis shows in part that validity of the result can be reduced to a so-called modal axiom T. Complementing the particular axiom T of Aumann's setup , we propose an alternative axiom that results in "decidable (local) rationality leads to backwards-induction equilibria". Aumann's result follows from ours but not vice versa and the two axioms T appear to be independent. Our development has been verified in full detail and fully transparently in the Coq proof assistant. 1The result is part of Aumann's research program on the benefits of sustained involvement in conflict resolution for which he was co-awarded the 2005 Nobel Memorial prize in economics.
World Scientific Publishing Co. Pte. Ltd. eBooks, 2006
Another Characterization of the Deduction-Detachment Theorem (S V Babyonyshev) On Behavior of 2-F... more Another Characterization of the Deduction-Detachment Theorem (S V Babyonyshev) On Behavior of 2-Formulas in Weakly o-Minimal Theories (B S Baizhanov & B Sh Kulpeshov) Arithmetic Turing Degrees and Categorical Theories of Computable Models (E Fokina) Negative Data in Learning Languages (S Jain & E Kinber) Effective Cardinals in the Nonstandard Universe (V Kanovei & M Reeken) Model-Theoretic Methods of Analysis of Computer Arithmetic (S P Kovalyov) The Functional Completeness of Le niewski's Systems (F Lepage) Hierarchies of Randomness Tests (J Reimann & F Stephan) Intransitive Linear Temporal Logic Based on Integer Numbers, Decidability, Admissible Logical Consecutions (V V Rybakov) The Logic of Prediction (E Vityaev) Conceptual Semantic Systems Theory and Applications (K E Wolff) Complexity Results on Minimal Unsatisfiable Formulas (X Zhao) and other papers.
Pictures of functions of one variable in the natural number theory are introduced as patterns ove... more Pictures of functions of one variable in the natural number theory are introduced as patterns over (0, 1) in the first quadrant. It is proved that the set of pictures cf at1 computable functions of one variable is obtained as the bottom planes of three-dimensional arrays accepted by some nondeterministic automaton on "'-tapes. To our surprise, it is also ihnwn that a set of pictures of functions containing a noncomputable function is deterministically obtained in a similar way. Further, it is proved that the set of initial segments of pictures of all computable functions and/or the noncomputable function is accepted hy a deterministic TL-ing rectangular array acceptor.
We will investigate th~ size of ~ Kripke model, in which a given unprov-~ble formul~ is refutable... more We will investigate th~ size of ~ Kripke model, in which a given unprov-~ble formul~ is refutable. For example, as R~siow~ and Sikorski showed in [5], if A is ~ formul~ not provable in the mod~l logic $4, then we need to check valuations into a Boolean algebra of size 2 ...
Closure under various completions are studied for the variety basic algebras and the variety of L... more Closure under various completions are studied for the variety basic algebras and the variety of Löb algebras. Basic algebras arise from (Visser's) basic logic, and the class of Löb algebras is an important subclass of basic algebras. Canonical extensions, ideal completions, MacNeille completions and complete ideal completions of these algebras are discussed.
For each ordinal a > 0, L(a) is the intermediate predicate logic characterized by the class of al... more For each ordinal a > 0, L(a) is the intermediate predicate logic characterized by the class of all Kripke frames with the poset a and with constant domain. This paper will be devoted to a study of logics of the form L(a). It will be shown that for each uncountable ordinal of the form a + r\ with a finite or a countable v\ (>0), there exists a countable ordinal of the form [1 + t\ such that L(tx + rj) = L(/J + tf). On the other hand, such a reduction of ordinals to countable ones is impossible for a logic L(a) if a is an uncountable regular ordinal. Moreover, it will be proved that the mapping L is injective if it is restricted to ordinals less than co w , i.e. a # ft implies L(a) # L(/J) for each ordinal a, fS < w". §1. Introduction. In the following, we will assume familiarity with the terminology and notation in [10]. But, for the reader's convenience, we will first give some basic definitions and explanations of the purpose of our study. We assume that our language contains neither constants nor function symbols. We say that a set L of first-order formulas is an intermediate predicate logic if 1) every formula provable in the intuitionistic predicate logic H belongs to L and every formula in L is provable in the classical predicate logic C, and 2) L is closed under modus ponens, substitution and generalization. Similarly, we can define an intermediate propositional logic. Of course, in this case, we must replace 2) by 2') L is closed under modus ponens and substitution. Next, we will give a Kripke-type semantics for intermediate logics. A pair (M, U) is a Kripke frame if M is a nonempty partially ordered set and U is a mapping from M to the power set of some set such that 1) U(a) # 0 for each ae M and 2) a < b implies U(a) £ V(b). We assume moreover that M has the smallest element. If U is a constant mapping such that U(a) = V for each a e M, we say that (M, U) is a Kripke frame with the constant domain V. Since we will restrict our attention only to Kripke
Logic in Asia: Studia logica library, Oct 17, 2015
This is a short survey of semantical study of cut elimination and subformula property in modal lo... more This is a short survey of semantical study of cut elimination and subformula property in modal logics. Cut elimination is a basic proof-theoretic notion in sequent systems, and subformula property is the most important consequence of cut elimination. A special feature of our presentation is its unified semantical approach to them based on Kripke models. Along the same lines as Takano’s works on subformula property, these properties, together with finite model property, will be discussed as modifications of standard construction of canonical Kripke models. These semantical approaches will be compared with algebraic approaches in modal logics, which often take the forms of various kinds of embedding theorems. In the last part of the paper, an attempt is made to clarify connections between semantical approach to cut elimination and algebraic one.
Gentzen-type sequent calculi usually contain three structural rules, i.e., exchange, contraction ... more Gentzen-type sequent calculi usually contain three structural rules, i.e., exchange, contraction and weakening rules. In recent years, however, there have been various studies on logics that have not included some or any of these structural rules. The motives or purposes of these studies have been so diverse that sometimes close connections between them have been overlooked. Here we will make a brief survey of recent results on these logics in an attempt to make these interrelationships clearer.
We will give a state-of-the-art survey of the study of substructural logics. Originally, substruc... more We will give a state-of-the-art survey of the study of substructural logics. Originally, substructural logics were introduced as logics which, when formulated as Gentzen-style systems, lack some of the three basic structural rules, i.e. contraction, weakening and exchange. For example, relevance logics and linear logic lack the weakening rule, many-valued logics, fuzzy logics and linear logic lack the contraction rule, and hence all of them can be regarded as substructural logics. These logics have been studied extensively and various theories have been developed for their investigation. However their study has been carried out independently, mainly due to the different motivations behind them, avoiding comparisons between different substructural logics. On the other hand, the general study of substructural logics has a comparative character, focusing on the absence or presence of structural rules. As such, at least in the initial stages of research, it was a study on how structural rules affect logical properties. This naturally led to a syntactic or proof-theoretic approach yielding deep results about properties of particular logics, provided that they can be formalized in systems, like cut-free Gentzen calculi. An obvious limitation of this study comes from the fact that not all logics have such a formulation. Semantical methods, in contrast, provide a powerful tool for analyzing substructural logics from a more uniform perspective. Both Kripke-style semantics and algebraic semantics for some particular subclasses of substructural logics, e.g. relevant logics, were already introduced in the 70s and 80s and were studied to a certain extent. An algebraic study of substructural logics for the last decade, based mainly on algebraic logic and universal algebra, has brought us a new perspective on substructural logics, which came from the observation that these logics all share the residuation property. Though this is usually not noticed, it is revealed explicitly in a sequent formulation by the use of extralogical symbols, denoted by commas. Precisely speaking, consider the following equivalence, concerning implication →, that holds in most of sequent systems: A formula γ follows from formulas α and β if and only if the implication α → γ follows from β alone. Here 'follows from' is given by the particular substructural logic, and is usually denoted by ⇒. Thus, the above equivalence can be restated as:
A b s t r a c t. The present paper is concerned with the cut elim- inability for some sequent sys... more A b s t r a c t. The present paper is concerned with the cut elim- inability for some sequent systems of noncommutative substructural logics, i.e. substructural logics without exchange rule. Sequent sys- tems of several extensions of noncommutative logics FL and LBB0I, which is sometimes called T! W, will be introduced. Then, the cut elimination theorem and the
A variant of the Robinson property (ROB*) is introduced. The property ROB* is equivalent to the u... more A variant of the Robinson property (ROB*) is introduced. The property ROB* is equivalent to the usual Robinson property and hence is equivalent also to the interpolation property, in any compact logic closed under the Boolean operations. On the other hand, it will be shown that ROB* is not always equal to the interpolation property, if a logic is not closed under them. Next, we will study ROB* and various interpolation properties for equational logics, which are typical examples of compact logics not closed under the Boolean operations. It will be shown that an equational logic has ROB* if and only if it has the amalgamation property for isomorphic embeddings. Recent results in this area are collected into a voluminous work, "Model-Theoretic Logics" [3], which we suggest for further information. In this paper, we will study the relationships between the interpolation property and the Robinson property, in a general setting. We don't assume that a logic is closed under the Boolean operations, i.e. closed under both conjunction and negation. It is well-known that for every compact logic 2g closed under the Boolean operations, ~ has the interpolation property if and only if it has the Robinson property. (See e.g. [3] Chapter II, Proposition 7.1.5. In [3], the proposition is shown under the stronger assumption that 5f is regular. But, by a careful observation of the proof, it can be seen that the closure under the Boolean operations is essential for the proof.) On the other hand, when a given logic lacks negation, the meaning of the Robinson property becomes unclear and inadequate, since it may happen that even the set of all sentences has a model. So, in the present paper, we will Presented by Bjarni Jonsson.
Journal of Computer and System Sciences, Jun 1, 1979
It is shown that the first-order arithmetic A[P(x), 2x, x-I-11 with two functions 2r, x + 1 and a... more It is shown that the first-order arithmetic A[P(x), 2x, x-I-11 with two functions 2r, x + 1 and a monadic predicate symbol P(x) is undecidable, by using a kind of twodimensional finite automata, called finite causal ws-systems. From this immediately follows R. M. Robinson's result, which says that the monadic second-order theory with two functions 2x, x + 1 is undecidable. This is also considered as an improvement on H. Putnam's result about the undecidability of the first-order arithmetic with addition and a monadic predicate symbol.
Aumann's Theorem on Rationality is a well-know but also contentious result in economics. It perta... more Aumann's Theorem on Rationality is a well-know but also contentious result in economics. It pertains to sequential game theory and says that "common knowledge of rationality leads to backwards-induction equilibria" .1 We present a formalist analysis of the result in a meta-theory with primitive support for proof and definition by induction. The analysis shows in part that validity of the result can be reduced to a so-called modal axiom T. Complementing the particular axiom T of Aumann's setup , we propose an alternative axiom that results in "decidable (local) rationality leads to backwards-induction equilibria". Aumann's result follows from ours but not vice versa and the two axioms T appear to be independent. Our development has been verified in full detail and fully transparently in the Coq proof assistant. 1The result is part of Aumann's research program on the benefits of sustained involvement in conflict resolution for which he was co-awarded the 2005 Nobel Memorial prize in economics.
World Scientific Publishing Co. Pte. Ltd. eBooks, 2006
Another Characterization of the Deduction-Detachment Theorem (S V Babyonyshev) On Behavior of 2-F... more Another Characterization of the Deduction-Detachment Theorem (S V Babyonyshev) On Behavior of 2-Formulas in Weakly o-Minimal Theories (B S Baizhanov & B Sh Kulpeshov) Arithmetic Turing Degrees and Categorical Theories of Computable Models (E Fokina) Negative Data in Learning Languages (S Jain & E Kinber) Effective Cardinals in the Nonstandard Universe (V Kanovei & M Reeken) Model-Theoretic Methods of Analysis of Computer Arithmetic (S P Kovalyov) The Functional Completeness of Le niewski's Systems (F Lepage) Hierarchies of Randomness Tests (J Reimann & F Stephan) Intransitive Linear Temporal Logic Based on Integer Numbers, Decidability, Admissible Logical Consecutions (V V Rybakov) The Logic of Prediction (E Vityaev) Conceptual Semantic Systems Theory and Applications (K E Wolff) Complexity Results on Minimal Unsatisfiable Formulas (X Zhao) and other papers.
Pictures of functions of one variable in the natural number theory are introduced as patterns ove... more Pictures of functions of one variable in the natural number theory are introduced as patterns over (0, 1) in the first quadrant. It is proved that the set of pictures cf at1 computable functions of one variable is obtained as the bottom planes of three-dimensional arrays accepted by some nondeterministic automaton on "'-tapes. To our surprise, it is also ihnwn that a set of pictures of functions containing a noncomputable function is deterministically obtained in a similar way. Further, it is proved that the set of initial segments of pictures of all computable functions and/or the noncomputable function is accepted hy a deterministic TL-ing rectangular array acceptor.
We will investigate th~ size of ~ Kripke model, in which a given unprov-~ble formul~ is refutable... more We will investigate th~ size of ~ Kripke model, in which a given unprov-~ble formul~ is refutable. For example, as R~siow~ and Sikorski showed in [5], if A is ~ formul~ not provable in the mod~l logic $4, then we need to check valuations into a Boolean algebra of size 2 ...
Uploads
Papers by Hiroakira Ono