An Algebraic Proof of the Disjunction Property
|
|
- Chester Charles
- 5 years ago
- Views:
Transcription
1 An Algebraic Proof of the Disjunction Property Rostislav Horčík joint work with Kazushige Terui Institute of Computer Science Academy of Sciences of the Czech Republic Algebra & Coalgebra meet Proof Theory Bern 2011 Rostislav Horčík (ICS) ALCOP / 23
2 Outline 1 Motivation Rostislav Horčík (ICS) ALCOP / 23
3 Outline 1 Motivation 2 Substructural logics Rostislav Horčík (ICS) ALCOP / 23
4 Outline 1 Motivation 2 Substructural logics 3 Disjunction property (DP) Rostislav Horčík (ICS) ALCOP / 23
5 Outline 1 Motivation 2 Substructural logics 3 Disjunction property (DP) 4 Why is the DP interesting? Rostislav Horčík (ICS) ALCOP / 23
6 Outline 1 Motivation 2 Substructural logics 3 Disjunction property (DP) 4 Why is the DP interesting? 5 Rostislav Horčík (ICS) ALCOP / 23
7 Complexity issues Motivation Proving a complexity of the decision problem of a (substructural) logic is often a tedious task usually based on a detailed inspection of the underlying (sequent) calculus. Rostislav Horčík (ICS) ALCOP / 23
8 Complexity issues Motivation Proving a complexity of the decision problem of a (substructural) logic is often a tedious task usually based on a detailed inspection of the underlying (sequent) calculus. Such results often rely on proof theoretic methods and presuppose that the logic under consideration possesses a good sequent calculus for which cut-elimination holds. Rostislav Horčík (ICS) ALCOP / 23
9 Complexity issues Motivation Proving a complexity of the decision problem of a (substructural) logic is often a tedious task usually based on a detailed inspection of the underlying (sequent) calculus. Such results often rely on proof theoretic methods and presuppose that the logic under consideration possesses a good sequent calculus for which cut-elimination holds. A typical example is MALL which is known to be PSPACE-complete (Lincoln et al.). The proof is very long and technical. Rostislav Horčík (ICS) ALCOP / 23
10 Complexity issues Motivation Proving a complexity of the decision problem of a (substructural) logic is often a tedious task usually based on a detailed inspection of the underlying (sequent) calculus. Such results often rely on proof theoretic methods and presuppose that the logic under consideration possesses a good sequent calculus for which cut-elimination holds. A typical example is MALL which is known to be PSPACE-complete (Lincoln et al.). The proof is very long and technical. Can we have uniform methods which work for wider classes of substructural logics? Rostislav Horčík (ICS) ALCOP / 23
11 Base logic Substructural logics Our base logic is the full Lambek calculus FL (multiplicative-additive fragment of noncommutative intuitionistic linear logic). Rostislav Horčík (ICS) ALCOP / 23
12 Base logic Substructural logics Our base logic is the full Lambek calculus FL (multiplicative-additive fragment of noncommutative intuitionistic linear logic). Multiplicative connectives:, \, /, 1, 0, Rostislav Horčík (ICS) ALCOP / 23
13 Base logic Substructural logics Our base logic is the full Lambek calculus FL (multiplicative-additive fragment of noncommutative intuitionistic linear logic). Multiplicative connectives:, \, /, 1, 0, Additive connectives:,,,. Rostislav Horčík (ICS) ALCOP / 23
14 Base logic Substructural logics Our base logic is the full Lambek calculus FL (multiplicative-additive fragment of noncommutative intuitionistic linear logic). Multiplicative connectives:, \, /, 1, 0, Additive connectives:,,,. FL is given by a single-conclusion sequent calculus: α α 1 0 Γ α Π, α, Σ ϕ (cut) Π, Γ, Σ ϕ Γ, α, Σ ϕ Γ, β, Σ ϕ ( ) Γ, α β, Σ ϕ Γ ϕ Γ ϕ ψ ( ) Γ ψ Γ ϕ ψ ( ). Rostislav Horčík (ICS) ALCOP / 23
15 Substructural logics Substructural logics Definition A substructural logic is an extension of FL by a set of rules (axioms) closed under substitutions having the form: Γ 1 ϕ 1 Γ n ϕ n Γ 0 ϕ 0 Rostislav Horčík (ICS) ALCOP / 23
16 Substructural logics Substructural logics Definition A substructural logic is an extension of FL by a set of rules (axioms) closed under substitutions having the form: Γ 1 ϕ 1 Γ n ϕ n Γ 0 ϕ 0 Example Γ, α, α, ϕ (c) Γ, α, ϕ Γ, α, β, ϕ (e) Γ, β, α, ϕ Γ, ϕ (i) Γ, α, ϕ Γ Γ α (o) MALL = InFL e = FL+(e)+(α \ 0) \ 0 α, Int = FL+(e)+(c)+(i)+(o). Rostislav Horčík (ICS) ALCOP / 23
17 Substructural logics Algebraic semantics Definition An FL-algebra is an algebra A = A,,,, /, \, 0, 1, where A,, is a lattice, A,, 1 is a monoid, 0 is an arbitrary element and the following condition holds: x y z iff x z/y iff y x \ z. Rostislav Horčík (ICS) ALCOP / 23
18 Substructural logics Algebraic semantics Definition An FL-algebra is an algebra A = A,,,, /, \, 0, 1, where A,, is a lattice, A,, 1 is a monoid, 0 is an arbitrary element and the following condition holds: x y z iff x z/y iff y x \ z. Fact The class of FL-algebras form a variety (i.e., an equational class). Rostislav Horčík (ICS) ALCOP / 23
19 Algebraizability Substructural logics FL is algebraizable and its equivalent algebraic semantics is the variety of FL-algebras. Rostislav Horčík (ICS) ALCOP / 23
20 Algebraizability Substructural logics FL is algebraizable and its equivalent algebraic semantics is the variety of FL-algebras. Thus there is a dual isomorphism Q between the lattice of substructural logics and the sub-quasivariety lattice of FL-algebras. Rostislav Horčík (ICS) ALCOP / 23
21 Algebraizability Substructural logics FL is algebraizable and its equivalent algebraic semantics is the variety of FL-algebras. Thus there is a dual isomorphism Q between the lattice of substructural logics and the sub-quasivariety lattice of FL-algebras. Let L be a substructural logic. Then we have the following equivalences: L ϕ iff = Q(L) 1 = 1 ϕ [1 ϕ]. = Q(L) ϕ = ψ iff L (ϕ \ ψ) (ψ \ ϕ). Rostislav Horčík (ICS) ALCOP / 23
22 Algebraizability Substructural logics FL is algebraizable and its equivalent algebraic semantics is the variety of FL-algebras. Thus there is a dual isomorphism Q between the lattice of substructural logics and the sub-quasivariety lattice of FL-algebras. Let L be a substructural logic. Then we have the following equivalences: L ϕ iff = Q(L) 1 = 1 ϕ [1 ϕ]. = Q(L) ϕ = ψ iff L (ϕ \ ψ) (ψ \ ϕ). By complexity of a substructural logic L we mean the complexity of its set of theorems. Due to algebraizability it is the same as the complexity of the equational theory for Q(L). Rostislav Horčík (ICS) ALCOP / 23
23 Substructural logics Correspondence between logic and algebra Logic Algebra logic FL variety FL axiom ϕ identity 1 ϕ inference rule (r) quasi-identity (r ) axiomatic extension L of FL subvariety V(L) of FL rule extension L of FL subquasivariety Q(L) of FL consistent nontrivial A logic L is consistent if there ϕ such that L ϕ. Rostislav Horčík (ICS) ALCOP / 23
24 Disjunction property (DP) Disjunction Property Definition Let L be a substructural logic. Then L satisfies the disjunction property (DP) if for all formulas ϕ, ψ L ϕ ψ implies L ϕ or L ψ. Rostislav Horčík (ICS) ALCOP / 23
25 Disjunction property (DP) Disjunction Property Definition Let L be a substructural logic. Then L satisfies the disjunction property (DP) if for all formulas ϕ, ψ L ϕ ψ implies L ϕ or L ψ. Analogously, we say the a quasivariety K of FL-algebras has the DP if = K 1 ϕ ψ implies = K 1 ϕ or = K 1 ψ. Rostislav Horčík (ICS) ALCOP / 23
26 Why is the DP interesting? DP and complexity Theorem (Chagrov, Zakharyaschev) Every consistent superintuitionistic logic having the DP is PSPACE-hard. Rostislav Horčík (ICS) ALCOP / 23
27 Why is the DP interesting? DP and complexity Theorem (Chagrov, Zakharyaschev) Every consistent superintuitionistic logic having the DP is PSPACE-hard. Theorem Every consistent substructural logic having the DP is PSPACE-hard. Rostislav Horčík (ICS) ALCOP / 23
28 Why is the DP interesting? DP and complexity Theorem (Chagrov, Zakharyaschev) Every consistent superintuitionistic logic having the DP is PSPACE-hard. Theorem Every consistent substructural logic having the DP is PSPACE-hard. Proof. By reduction from the set of true quantified Boolean formulas. Rostislav Horčík (ICS) ALCOP / 23
29 Why is the DP interesting? DP and complexity Theorem (Chagrov, Zakharyaschev) Every consistent superintuitionistic logic having the DP is PSPACE-hard. Theorem Every consistent substructural logic having the DP is PSPACE-hard. Proof. By reduction from the set of true quantified Boolean formulas. Remark One cannot use the coding of quantifiers from MALL. It does not work for some logics having the DP, e.g. FL + αβ αγ α(β γ). Rostislav Horčík (ICS) ALCOP / 23
30 Proof-theoretic proof of DP Rostislav Horčík (ICS) ALCOP / 23
31 Proof-theoretic proof of DP If our logic L enjoys the cut-elimination then one can use the following. Rostislav Horčík (ICS) ALCOP / 23
32 Proof-theoretic proof of DP If our logic L enjoys the cut-elimination then one can use the following. The very last rule in every cut-free proof of ϕ ψ has to be ( ). Thus either ϕ or ψ is provable. Rostislav Horčík (ICS) ALCOP / 23
33 Proof-theoretic proof of DP If our logic L enjoys the cut-elimination then one can use the following. The very last rule in every cut-free proof of ϕ ψ has to be ( ). Thus either ϕ or ψ is provable. What can we do if our logic does not have a cut-free presentation? E.g. if L is the extension of FL by α \ αβ β and βα/α β. Rostislav Horčík (ICS) ALCOP / 23
34 Algebraic characterization of the DP Definition An FL-algebra A is called well-connected if for all x, y A, x y 1 implies x 1 or y 1. Rostislav Horčík (ICS) ALCOP / 23
35 Algebraic characterization of the DP Definition An FL-algebra A is called well-connected if for all x, y A, x y 1 implies x 1 or y 1. Theorem Let L be a substructural logic. Then L has the DP iff the following condition holds: (*) for every A Q(L) there is a well-connected FL-algebra C Q(L) such that A is a homomorphic image of C. Rostislav Horčík (ICS) ALCOP / 23
36 Algebraic characterization of the DP Definition An FL-algebra A is called well-connected if for all x, y A, x y 1 implies x 1 or y 1. Theorem Let L be a substructural logic. Then L has the DP iff the following condition holds: (*) for every A Q(L) there is a well-connected FL-algebra C Q(L) such that A is a homomorphic image of C. Using this theorem we would like to find a large class of quasi-varieties of FL-algebras having the DP. Rostislav Horčík (ICS) ALCOP / 23
37 l-monoidal quasi-identities (rules) Definition An l-monoidal quasi-identity is a quasi-identity t 1 u 1 and... and t n u n = t 0 u 0, where t i is in the language {,,, 1} and u i is either 0 or in the language {,,, 1}. Rostislav Horčík (ICS) ALCOP / 23
38 l-monoidal quasi-identities (rules) Definition An l-monoidal quasi-identity is a quasi-identity t 1 u 1 and... and t n u n = t 0 u 0, where t i is in the language {,,, 1} and u i is either 0 or in the language {,,, 1}. Accordingly, an l-monoidal rule is a rule Γ 1 ϕ 1 Γ n ϕ n Γ 0 ϕ 0 where Γ i is a sequence of formulas in the language {,,, 1} and ϕ i is either empty or a formula in the language {,,, 1}. Rostislav Horčík (ICS) ALCOP / 23
39 Useful algebra Fix a quasivariety K of FL-algebras defined by l-monoidal quasi-identities. Rostislav Horčík (ICS) ALCOP / 23
40 Useful algebra Fix a quasivariety K of FL-algebras defined by l-monoidal quasi-identities. Lemma For any nontrivial algebra A K, there is an integral FL-algebra B K which has a unique subcover of 1. Rostislav Horčík (ICS) ALCOP / 23
41 Useful algebra Fix a quasivariety K of FL-algebras defined by l-monoidal quasi-identities. Lemma For any nontrivial algebra A K, there is an integral FL-algebra B K which has a unique subcover of 1. Proof. Let a A such that a < 1 and B = {a n n 0}. The submonoid B gives rise to an FL-algebra B by setting x y = {z B xz y} 0 B = 1 or a (depending whether = A 1 0 or not). Rostislav Horčík (ICS) ALCOP / 23
42 Construction of a well-connected algebra 1 A 1 B s A B Rostislav Horčík (ICS) ALCOP / 23
43 Construction of a well-connected algebra 1 A, 1 B 1 A, s A B Rostislav Horčík (ICS) ALCOP / 23
44 Construction of a well-connected algebra 1 A, 1 B 1 A, s σ[a B] Rostislav Horčík (ICS) ALCOP / 23
45 Construction of a well-connected algebra A B belongs to K. Rostislav Horčík (ICS) ALCOP / 23
46 Construction of a well-connected algebra A B belongs to K. σ[a B] is forms a subalgebra of A B with respect to the language {,,, 1, 0} Rostislav Horčík (ICS) ALCOP / 23
47 Construction of a well-connected algebra A B belongs to K. σ[a B] is forms a subalgebra of A B with respect to the language {,,, 1, 0} σ[a B] is an image of an interior operator σ. Rostislav Horčík (ICS) ALCOP / 23
48 Construction of a well-connected algebra A B belongs to K. σ[a B] is forms a subalgebra of A B with respect to the language {,,, 1, 0} σ[a B] is an image of an interior operator σ. Moreover, we have σ(x)σ(y) σ(xy), i.e., σ is a conucleus. Rostislav Horčík (ICS) ALCOP / 23
49 Construction of a well-connected algebra A B belongs to K. σ[a B] is forms a subalgebra of A B with respect to the language {,,, 1, 0} σ[a B] is an image of an interior operator σ. Moreover, we have σ(x)σ(y) σ(xy), i.e., σ is a conucleus. Thus σ[a B] is an FL-algebra (x \ σ y = σ(x \ y)). Rostislav Horčík (ICS) ALCOP / 23
50 Construction of a well-connected algebra A B belongs to K. σ[a B] is forms a subalgebra of A B with respect to the language {,,, 1, 0} σ[a B] is an image of an interior operator σ. Moreover, we have σ(x)σ(y) σ(xy), i.e., σ is a conucleus. Thus σ[a B] is an FL-algebra (x \ σ y = σ(x \ y)). σ[a B] is well-connected. Rostislav Horčík (ICS) ALCOP / 23
51 Construction of a well-connected algebra A B belongs to K. σ[a B] is forms a subalgebra of A B with respect to the language {,,, 1, 0} σ[a B] is an image of an interior operator σ. Moreover, we have σ(x)σ(y) σ(xy), i.e., σ is a conucleus. Thus σ[a B] is an FL-algebra (x \ σ y = σ(x \ y)). σ[a B] is well-connected. A is a homomorphic image of σ[a B]. Rostislav Horčík (ICS) ALCOP / 23
52 DP for l-monoidal extensions Theorem Every quasivariety of FL-algebras defined by l-monoidal quasi-identities has the DP. Every extension of FL by l-monoidal rules has the DP. Rostislav Horčík (ICS) ALCOP / 23
53 DP for l-monoidal extensions Theorem Every quasivariety of FL-algebras defined by l-monoidal quasi-identities has the DP. Every extension of FL by l-monoidal rules has the DP. Example Every extension of FL by structural rules (e), (c), (i), (o) enjoys the DP. Rostislav Horčík (ICS) ALCOP / 23
54 DP for l-monoidal extensions Theorem Every quasivariety of FL-algebras defined by l-monoidal quasi-identities has the DP. Every extension of FL by l-monoidal rules has the DP. Example Every extension of FL by structural rules (e), (c), (i), (o) enjoys the DP. The extension of FL by the rule ϕ ψ ϕ has the DP. It defines a proper subquasivariety of FL Rostislav Horčík (ICS) ALCOP / 23
55 M 2 -axioms Note that xy/y = x = y \ yx are equivalent to xz = yz x = y and zx = zy x = y. Rostislav Horčík (ICS) ALCOP / 23
56 M 2 -axioms Note that xy/y = x = y \ yx are equivalent to xz = yz x = y and zx = zy x = y. Definition (Class M 2 ) Let V be a set of variables. Given a set T of terms, let T be its closure under the operations {,,, 1}. Rostislav Horčík (ICS) ALCOP / 23
57 M 2 -axioms Note that xy/y = x = y \ yx are equivalent to xz = yz x = y and zx = zy x = y. Definition (Class M 2 ) Let V be a set of variables. Given a set T of terms, let T be its closure under the operations {,,, 1}. Likewise, let T be its closure under the following rules: 0 T, V T ; if t, u T then t u T ; if t T and u T, then t \ u, u/t T. Rostislav Horčík (ICS) ALCOP / 23
58 M 2 -axioms Note that xy/y = x = y \ yx are equivalent to xz = yz x = y and zx = zy x = y. Definition (Class M 2 ) Let V be a set of variables. Given a set T of terms, let T be its closure under the operations {,,, 1}. Likewise, let T be its closure under the following rules: 0 T, V T ; if t, u T then t u T ; if t T and u T, then t \ u, u/t T. We define M 1 = V and M 2 = M 1. An identity t u belongs to M 2 if t M 1 and u M 2. Analogously, α β M 2 if α M 1 and β M 2. Rostislav Horčík (ICS) ALCOP / 23
59 Examples of M 2 -axioms Axiom Name αβ βα exchange (e) α 1 integrality, left weakening (i) 0 α right weakening (o) α αα contraction (c) α n α m knotted axioms (n, m 0) α (α \ 0) no-contradiction αβ/β α, α \ αβ β cancellativity α (β γ) (α β) (α γ) distributivity ((α β) γ) β (α β) (γ β) modularity αβ αγ α(β γ) (, )-distributivity α (βγ) (α β)(α γ) (, )-distributivity Rostislav Horčík (ICS) ALCOP / 23
60 DP for extensions by M 2 -axioms Theorem Every identity in M 2 is equivalent in FL to a set of l-monoidal quasi-identities. Corollary Every extension of FL by M 2 -axioms has the DP. Rostislav Horčík (ICS) ALCOP / 23
61 Involutive substructural logics Negations: ϕ = ϕ \ 0, ϕ = 0/ϕ. Rostislav Horčík (ICS) ALCOP / 23
62 Involutive substructural logics Negations: ϕ = ϕ \ 0, ϕ = 0/ϕ. Double negation elimination laws (DN): ϕ ϕ, ϕ ϕ. Rostislav Horčík (ICS) ALCOP / 23
63 Involutive substructural logics Negations: ϕ = ϕ \ 0, ϕ = 0/ϕ. Double negation elimination laws (DN): ϕ ϕ, ϕ ϕ. Let L be a substructural logic. Then InL is L+(DN). Rostislav Horčík (ICS) ALCOP / 23
64 Involutive substructural logics Negations: ϕ = ϕ \ 0, ϕ = 0/ϕ. Double negation elimination laws (DN): ϕ ϕ, ϕ ϕ. Let L be a substructural logic. Then InL is L+(DN). Theorem Every extension of InFL and InFL e (MALL) by inference rules in the language {,, 1} has the DP. Rostislav Horčík (ICS) ALCOP / 23
65 Involutive substructural logics Negations: ϕ = ϕ \ 0, ϕ = 0/ϕ. Double negation elimination laws (DN): ϕ ϕ, ϕ ϕ. Let L be a substructural logic. Then InL is L+(DN). Theorem Every extension of InFL and InFL e (MALL) by inference rules in the language {,, 1} has the DP. Example The distributive extension of InFL e has the DP. Thus the relevance logic RW has the DP. Rostislav Horčík (ICS) ALCOP / 23
66 Construction for involutive logics 1 A, 1 1 A, 1/2 0 A, 1/2 0 A, 0 Rostislav Horčík (ICS) ALCOP / 23
67 Conclusions Theorem Let L be a consistent substructural logic. The decision problem for L is conp-hard. If L further satisfies the DP, then it is PSPACE-hard. Rostislav Horčík (ICS) ALCOP / 23
68 Conclusions Theorem Let L be a consistent substructural logic. The decision problem for L is conp-hard. If L further satisfies the DP, then it is PSPACE-hard. Corollary Let L be a consistent extension of FL by l-monoidal inference rules and/or M 2 -axioms. Then the decision problem for L is PSPACE-hard. The same is true also for every consistent extension of InFL or InFL e by inference rules in the language {,, 1}. Rostislav Horčík (ICS) ALCOP / 23
69 Conclusions Theorem Let L be a consistent substructural logic. The decision problem for L is conp-hard. If L further satisfies the DP, then it is PSPACE-hard. Corollary Let L be a consistent extension of FL by l-monoidal inference rules and/or M 2 -axioms. Then the decision problem for L is PSPACE-hard. The same is true also for every consistent extension of InFL or InFL e by inference rules in the language {,, 1}. The DP is a sufficient condition for PSPACE-hardness but not a necessary one. A counterexample is LQ obtained by extending intuitionistic logic with the law α α. Rostislav Horčík (ICS) ALCOP / 23
Disjunction Property and Complexity of Substructural Logics
Disjunction Property and Complexity of Substructural Logics Rostislav Horčík Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodárenskou věží 2, 182 07 Prague 8, Czech Republic
More informationThe quest for the basic fuzzy logic
Petr Cintula 1 Rostislav Horčík 1 Carles Noguera 1,2 1 Institute of Computer Science, Academy of Sciences of the Czech Republic Pod vodárenskou věží 2, 182 07 Prague, Czech Republic 2 Institute of Information
More informationSubstructural Logics and Residuated Lattices an Introduction
Hiroakira Ono Substructural Logics and Residuated Lattices an Introduction Abstract. This is an introductory survey of substructural logics and of residuated lattices which are algebraic structures for
More informationHow and when axioms can be transformed into good structural rules
How and when axioms can be transformed into good structural rules Kazushige Terui National Institute of Informatics, Tokyo Laboratoire d Informatique de Paris Nord (Joint work with Agata Ciabattoni and
More informationAn Overview of Residuated Kleene Algebras and Lattices Peter Jipsen Chapman University, California. 2. Background: Semirings and Kleene algebras
An Overview of Residuated Kleene Algebras and Lattices Peter Jipsen Chapman University, California 1. Residuated Lattices with iteration 2. Background: Semirings and Kleene algebras 3. A Gentzen system
More informationRESIDUATED FRAMES WITH APPLICATIONS TO DECIDABILITY
RESIDUATED FRAMES WITH APPLICATIONS TO DECIDABILITY NIKOLAOS GALATOS AND PETER JIPSEN Abstract. Residuated frames provide relational semantics for substructural logics and are a natural generalization
More informationAlgebraization, parametrized local deduction theorem and interpolation for substructural logics over FL.
Nikolaos Galatos Hiroakira Ono Algebraization, parametrized local deduction theorem and interpolation for substructural logics over FL. Dedicated to the memory of Willem Johannes Blok Abstract. Substructural
More informationRELATION ALGEBRAS AS EXPANDED FL-ALGEBRAS
RELATION ALGEBRAS AS EXPANDED FL-ALGEBRAS NIKOLAOS GALATOS AND PETER JIPSEN Abstract. This paper studies generalizations of relation algebras to residuated lattices with a unary De Morgan operation. Several
More informationFrom Residuated Lattices to Boolean Algebras with Operators
From Residuated Lattices to Boolean Algebras with Operators Peter Jipsen School of Computational Sciences and Center of Excellence in Computation, Algebra and Topology (CECAT) Chapman University October
More informationSemantics: Residuated Frames
Semantics: Residuated Frames Peter Jipsen, Chapman University, Orange, California, USA joint work with Nikolaos Galatos, University of Denver, Colorado, USA ICLA, January, 2009 P Jipsen (Chapman), N Galatos
More informationRelational semantics for a fragment of linear logic
Relational semantics for a fragment of linear logic Dion Coumans March 4, 2011 Abstract Relational semantics, given by Kripke frames, play an essential role in the study of modal and intuitionistic logic.
More informationAdmissible Rules of (Fragments of) R-Mingle. Admissible Rules of (Fragments of) R-Mingle. Laura Janina Schnüriger
Admissible Rules of (Fragments of) R-Mingle Admissible Rules of (Fragments of) R-Mingle joint work with George Metcalfe Universität Bern Novi Sad 5 June 2015 Table of contents 1. What and why? 1.1 What
More informationInterpolation and Beth Definability over the Minimal Logic
Interpolation and Beth Definability over the Minimal Logic Larisa Maksimova 1 Sobolev Institute of Mathematics Siberian Branch of Russian Academy of Sciences Novosibirsk 630090, Russia Abstract Extensions
More informationREPRESENTATION THEOREMS FOR IMPLICATION STRUCTURES
Wojciech Buszkowski REPRESENTATION THEOREMS FOR IMPLICATION STRUCTURES Professor Rasiowa [HR49] considers implication algebras (A,, V ) such that is a binary operation on the universe A and V A. In particular,
More informationCUT ELIMINATION AND STRONG SEPARATION FOR SUBSTRUCTURAL LOGICS: AN ALGEBRAIC APPROACH.
CUT ELIMINATION AND STRONG SEPARATION FOR SUBSTRUCTURAL LOGICS: AN ALGEBRAIC APPROACH. NIOLAOS GALATOS AND HIROAIRA ONO Abstract. We develop a general algebraic and proof-theoretic study of substructural
More informationUninorm Based Logic As An Extension of Substructural Logics FL e
Uninorm Based Logic As An Extension of Substructural Logics FL e Osamu WATARI Hokkaido Automotive Engineering College Sapporo 062-0922, JAPAN watari@haec.ac.jp Mayuka F. KAWAGUCHI Division of Computer
More informationSIMPLE LOGICS FOR BASIC ALGEBRAS
Bulletin of the Section of Logic Volume 44:3/4 (2015), pp. 95 110 http://dx.doi.org/10.18778/0138-0680.44.3.4.01 Jānis Cīrulis SIMPLE LOGICS FOR BASIC ALGEBRAS Abstract An MV-algebra is an algebra (A,,,
More informationStructural completeness for discriminator varieties
Structural completeness for discriminator varieties Micha l Stronkowski Warsaw University of Technology Praha, June 2014 Deductive systems Sent - set of propositional sentences Ax - axioms ( Sent) + inference
More informationHypersequent Calculi for some Intermediate Logics with Bounded Kripke Models
Hypersequent Calculi for some Intermediate Logics with Bounded Kripke Models Agata Ciabattoni Mauro Ferrari Abstract In this paper we define cut-free hypersequent calculi for some intermediate logics semantically
More informationDistributive residuated frames and generalized bunched implication algebras
Distributive residuated frames and generalized bunched implication algebras Nikolaos Galatos and Peter Jipsen Abstract. We show that all extensions of the (non-associative) Gentzen system for distributive
More information(Algebraic) Proof Theory for Substructural Logics and Applications
(Algebraic) Proof Theory for Substructural Logics and Applications Agata Ciabattoni Vienna University of Technology agata@logic.at (Algebraic) Proof Theory for Substructural Logics and Applications p.1/59
More informationInterpolation and FEP for Logics of Residuated Algebras
Interpolation and FEP for Logics of Residuated Algebras Wojciech Buszkowski Adam Mickiewicz University in Poznań University of Warmia and Mazury in Olsztyn Abstract A residuated algebra (RA) is a generalization
More informationAlgebraic Gentzen systems and ordered structures. Peter Jipsen, Chapman University, California
Algebraic Gentzen systems and ordered structures Peter Jipsen, Chapman University, California Equational logic Gentzen systems Algebraic Gentzen systems Kleene algebras and GBL-algebras Finite GBL-algebras
More informationDisplay calculi in non-classical logics
Display calculi in non-classical logics Revantha Ramanayake Vienna University of Technology (TU Wien) Prague seminar of substructural logics March 28 29, 2014 Revantha Ramanayake (TU Wien) Display calculi
More informationNon-classical Logics: Theory, Applications and Tools
Non-classical Logics: Theory, Applications and Tools Agata Ciabattoni Vienna University of Technology (TUV) Joint work with (TUV): M. Baaz, P. Baldi, B. Lellmann, R. Ramanayake,... N. Galatos (US), G.
More informationSLANEY S LOGIC F IS CONSTRUCTIVE LOGIC WITH STRONG NEGATION
Bulletin of the Section of Logic Volume 39:3/4 2010, pp. 161 173 M. Spinks and R. Veroff SLANEY S LOGIC F IS CONSTRUCTIVE LOGIC WITH STRONG NEGATION Abstract In [19] Slaney et al. introduced a little known
More informationAlgebraic Logic. Hiroakira Ono Research Center for Integrated Science Japan Advanced Institute of Science and Technology
Algebraic Logic Hiroakira Ono Research Center for Integrated Science Japan Advanced Institute of Science and Technology ono@jaist.ac.jp 1 Introduction Algebraic methods have been important tools in the
More informationPart II. Logic and Set Theory. Year
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 60 Paper 4, Section II 16G State and prove the ǫ-recursion Theorem. [You may assume the Principle of ǫ- Induction.]
More informationCayley s and Holland s Theorems for Idempotent Semirings and Their Applications to Residuated Lattices
Cayley s and Holland s Theorems for Idempotent Semirings and Their Applications to Residuated Lattices Nikolaos Galatos Department of Mathematics University of Denver ngalatos@du.edu Rostislav Horčík Institute
More informationThe lattice of varieties generated by small residuated lattices
The lattice of varieties generated by small residuated lattices Peter Jipsen School of Computational Sciences and Center of Excellence in Computation, Algebra and Topology (CECAT) Chapman University LATD,
More informationA GENTZEN SYSTEM EQUIVALENT TO THE BCK-LOGIC
Romà J. Adillon Ventura Verdú A GENTZEN SYSTEM EQUIVALENT TO THE BCK-LOGIC Abstract The sequent calculus L BCK is obtained by deleting the contraction rule and the introduction rules of the connectives
More informationA CATEGORY EQUIVALENCE FOR ODD SUGIHARA MONOIDS AND ITS APPLICATIONS
A CATEGORY EQUIVALENCE FOR ODD SUGIHARA MONOIDS AND ITS APPLICATIONS N. GALATOS AND J.G. RAFTERY Abstract. An odd Sugihara monoid is a residuated distributive latticeordered commutative idempotent monoid
More informationImplicational classes ofde Morgan Boolean algebras
Discrete Mathematics 232 (2001) 59 66 www.elsevier.com/locate/disc Implicational classes ofde Morgan Boolean algebras Alexej P. Pynko Department of Digital Automata Theory, V.M. Glushkov Institute of Cybernetics,
More informationThe lattice of varieties generated by residuated lattices of size up to 5
The lattice of varieties generated by residuated lattices of size up to 5 Peter Jipsen Chapman University Dedicated to Hiroakira Ono on the occasion of his 7th birthday Introduction Residuated lattices
More informationLinear Logic Pages. Yves Lafont (last correction in 2017)
Linear Logic Pages Yves Lafont 1999 (last correction in 2017) http://iml.univ-mrs.fr/~lafont/linear/ Sequent calculus Proofs Formulas - Sequents and rules - Basic equivalences and second order definability
More informationDynamic Epistemic Logic Displayed
1 / 43 Dynamic Epistemic Logic Displayed Giuseppe Greco & Alexander Kurz & Alessandra Palmigiano April 19, 2013 ALCOP 2 / 43 1 Motivation Proof-theory meets coalgebra 2 From global- to local-rules calculi
More informationPeriodic lattice-ordered pregroups are distributive
Periodic lattice-ordered pregroups are distributive Nikolaos Galatos and Peter Jipsen Abstract It is proved that any lattice-ordered pregroup that satisfies an identity of the form x lll = x rrr (for the
More informationAN ALTERNATIVE NATURAL DEDUCTION FOR THE INTUITIONISTIC PROPOSITIONAL LOGIC
Bulletin of the Section of Logic Volume 45/1 (2016), pp 33 51 http://dxdoiorg/1018778/0138-068045103 Mirjana Ilić 1 AN ALTERNATIVE NATURAL DEDUCTION FOR THE INTUITIONISTIC PROPOSITIONAL LOGIC Abstract
More informationMonadic GMV -algebras
Department of Algebra and Geometry Faculty of Sciences Palacký University of Olomouc Czech Republic TANCL 07, Oxford 2007 monadic structures = algebras with quantifiers = algebraic models for one-variable
More informationFROM AXIOMS TO STRUCTURAL RULES, THEN ADD QUANTIFIERS.
FROM AXIOMS TO STRUCTURAL RULES, THEN ADD QUANTIFIERS. REVANTHA RAMANAYAKE We survey recent developments in the program of generating proof calculi for large classes of axiomatic extensions of a non-classical
More informationFULL LAMBEK CALCULUS WITH CONTRACTION IS UNDECIDABLE
FULL LAMBEK CALCULUS WITH CONTRACTION IS UNDECIDABLE KAREL CHVALOVSKÝ AND ROSTISLAV HORČÍK Abstract. We prove that the set of formulae provable in the full Lambek calculus with the structural rule of contraction
More informationProof Theoretical Studies on Semilattice Relevant Logics
Proof Theoretical Studies on Semilattice Relevant Logics Ryo Kashima Department of Mathematical and Computing Sciences Tokyo Institute of Technology Ookayama, Meguro, Tokyo 152-8552, Japan. e-mail: kashima@is.titech.ac.jp
More informationMONADIC FRAGMENTS OF INTUITIONISTIC CONTROL LOGIC
Bulletin of the Section of Logic Volume 45:3/4 (2016), pp. 143 153 http://dx.doi.org/10.18778/0138-0680.45.3.4.01 Anna Glenszczyk MONADIC FRAGMENTS OF INTUITIONISTIC CONTROL LOGIC Abstract We investigate
More informationPreuves de logique linéaire sur machine, ENS-Lyon, Dec. 18, 2018
Université de Lorraine, LORIA, CNRS, Nancy, France Preuves de logique linéaire sur machine, ENS-Lyon, Dec. 18, 2018 Introduction Linear logic introduced by Girard both classical and intuitionistic separate
More informationInterpolation via translations
Interpolation via translations Walter Carnielli 2,3 João Rasga 1,3 Cristina Sernadas 1,3 1 DM, IST, TU Lisbon, Portugal 2 CLE and IFCH, UNICAMP, Brazil 3 SQIG - Instituto de Telecomunicações, Portugal
More informationA Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries
A Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries Johannes Marti and Riccardo Pinosio Draft from April 5, 2018 Abstract In this paper we present a duality between nonmonotonic
More informationModel Checking for Modal Intuitionistic Dependence Logic
1/71 Model Checking for Modal Intuitionistic Dependence Logic Fan Yang Department of Mathematics and Statistics University of Helsinki Logical Approaches to Barriers in Complexity II Cambridge, 26-30 March,
More informationOn the algebra of relevance logics
On the algebra of relevance logics by Johann Joubert Wannenburg Submitted in partial fulfilment of the requirements for the degree Master of Science in the Faculty of Natural & Agricultural Sciences University
More informationFrom Semirings to Residuated Kleene Lattices
Peter Jipsen From Semirings to Residuated Kleene Lattices Abstract. We consider various classes of algebras obtained by expanding idempotent semirings with meet, residuals and Kleene-. An investigation
More informationEmbedding theorems for normal divisible residuated lattices
Embedding theorems for normal divisible residuated lattices Chapman University Department of Mathematics and Computer Science Orange, California University of Siena Department of Mathematics and Computer
More informationOn Urquhart s C Logic
On Urquhart s C Logic Agata Ciabattoni Dipartimento di Informatica Via Comelico, 39 20135 Milano, Italy ciabatto@dsiunimiit Abstract In this paper we investigate the basic many-valued logics introduced
More informationThiago Nascimento, Programa de Pós-Graduação em Sistemas e Computação, Universidade Federal do Rio Grande do Norte, Natal, Brazil
Nelson s Logic S arxiv:803.085v [math.lo] 8 Mar 08 Thiago Nascimento, Programa de Pós-Graduação em Sistemas e Computação, Universidade Federal do Rio Grande do Norte, Natal, Brazil Umberto Rivieccio, Departamento
More informationLOGIC OF CLASSICAL REFUTABILITY AND CLASS OF EXTENSIONS OF MINIMAL LOGIC
Logic and Logical Philosophy Volume 9 (2001), 91 107 S. P. Odintsov LOGIC OF CLASSICAL REFUTABILITY AND CLASS OF EXTENSIONS OF MINIMAL LOGIC Introduction This article continues the investigation of paraconsistent
More informationModel Theory MARIA MANZANO. University of Salamanca, Spain. Translated by RUY J. G. B. DE QUEIROZ
Model Theory MARIA MANZANO University of Salamanca, Spain Translated by RUY J. G. B. DE QUEIROZ CLARENDON PRESS OXFORD 1999 Contents Glossary of symbols and abbreviations General introduction 1 xix 1 1.0
More informationModal systems based on many-valued logics
Modal systems based on many-valued logics F. Bou IIIA - CSIC Campus UAB s/n 08193, Bellaterra, Spain fbou@iiia.csic.es F. Esteva IIIA - CSIC Campus UAB s/n 08193, Bellaterra, Spain esteva@iiia.csic.es
More informationACLT: Algebra, Categories, Logic in Topology - Grothendieck's generalized topological spaces (toposes)
ACLT: Algebra, Categories, Logic in Topology - Grothendieck's generalized topological spaces (toposes) Steve Vickers CS Theory Group Birmingham 2. Theories and models Categorical approach to many-sorted
More informationAn Introduction to Modal Logic III
An Introduction to Modal Logic III Soundness of Normal Modal Logics Marco Cerami Palacký University in Olomouc Department of Computer Science Olomouc, Czech Republic Olomouc, October 24 th 2013 Marco Cerami
More informationIntroduction to Intuitionistic Logic
Introduction to Intuitionistic Logic August 31, 2016 We deal exclusively with propositional intuitionistic logic. The language is defined as follows. φ := p φ ψ φ ψ φ ψ φ := φ and φ ψ := (φ ψ) (ψ φ). A
More informationMV-algebras and fuzzy topologies: Stone duality extended
MV-algebras and fuzzy topologies: Stone duality extended Dipartimento di Matematica Università di Salerno, Italy Algebra and Coalgebra meet Proof Theory Universität Bern April 27 29, 2011 Outline 1 MV-algebras
More informationPropositional Calculus - Soundness & Completeness of H
Propositional Calculus - Soundness & Completeness of H Moonzoo Kim CS Dept. KAIST moonzoo@cs.kaist.ac.kr 1 Review Goal of logic To check whether given a formula Á is valid To prove a given formula Á `
More informationTeaching Natural Deduction as a Subversive Activity
Teaching Natural Deduction as a Subversive Activity James Caldwell Department of Computer Science University of Wyoming Laramie, WY Third International Congress on Tools for Teaching Logic 3 June 2011
More informationIDEMPOTENT RESIDUATED STRUCTURES: SOME CATEGORY EQUIVALENCES AND THEIR APPLICATIONS
IDEMPOTENT RESIDUATED STRUCTURES: SOME CATEGORY EQUIVALENCES AND THEIR APPLICATIONS N. GALATOS AND J.G. RAFTERY Abstract. This paper concerns residuated lattice-ordered idempotent commutative monoids that
More informationOutline. 1 Background and Aim. 2 Main results (in the paper) 3 More results (not in the paper) 4 Conclusion
Outline 1 Background and Aim 2 Main results (in the paper) 3 More results (not in the paper) 4 Conclusion De & Omori (Konstanz & Kyoto/JSPS) Classical and Empirical Negation in SJ AiML 2016, Sept. 2, 2016
More informationOn Jankov-de Jongh formulas
On Jankov-de Jongh formulas Nick Bezhanishvili Institute for Logic, Language and Computation University of Amsterdam http://www.phil.uu.nl/~bezhanishvili The Heyting day dedicated to Dick de Jongh and
More informationAutomated Support for the Investigation of Paraconsistent and Other Logics
Automated Support for the Investigation of Paraconsistent and Other Logics Agata Ciabattoni 1, Ori Lahav 2, Lara Spendier 1, and Anna Zamansky 1 1 Vienna University of Technology 2 Tel Aviv University
More informationMultiplicative Conjunction and an Algebraic. Meaning of Contraction and Weakening. A. Avron. School of Mathematical Sciences
Multiplicative Conjunction and an Algebraic Meaning of Contraction and Weakening A. Avron School of Mathematical Sciences Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv 69978, Israel Abstract
More informationLogics without the contraction rule residuated lattices. Citation Australasian Journal of Logic, 8(1):
JAIST Reposi https://dspace.j Title Logics without the contraction rule residuated lattices Author(s)Ono, Hiroakira Citation Australasian Journal of Logic, 8(1): Issue Date 2010-09-22 Type Journal Article
More informationKazimierz SWIRYDOWICZ UPPER PART OF THE LATTICE OF EXTENSIONS OF THE POSITIVE RELEVANT LOGIC R +
REPORTS ON MATHEMATICAL LOGIC 40 (2006), 3 13 Kazimierz SWIRYDOWICZ UPPER PART OF THE LATTICE OF EXTENSIONS OF THE POSITIVE RELEVANT LOGIC R + A b s t r a c t. In this paper it is proved that the interval
More informationSubtractive Logic. To appear in Theoretical Computer Science. Tristan Crolard May 3, 1999
Subtractive Logic To appear in Theoretical Computer Science Tristan Crolard crolard@ufr-info-p7.jussieu.fr May 3, 1999 Abstract This paper is the first part of a work whose purpose is to investigate duality
More informationVanderbilt University
Vanderbilt University Department of Mathematics Shanks Workshop on Ordered Groups in Logic Booklet of Abstracts Editor George Metcalfe March 21 22, 2009 Table of contents Semilinear logics Petr Cintula
More informationFuzzy Logics Interpreted as Logics of Resources
Fuzzy Logics Interpreted as Logics of Resources Libor Běhounek Girard s linear logic (1987) is often interpreted as the logic of resources, while formal fuzzy logics (see esp. Hájek, 1998) are usually
More informationTHE EQUATIONAL THEORIES OF REPRESENTABLE RESIDUATED SEMIGROUPS
TH QUATIONAL THORIS OF RPRSNTABL RSIDUATD SMIGROUPS SZABOLCS MIKULÁS Abstract. We show that the equational theory of representable lower semilattice-ordered residuated semigroups is finitely based. We
More informationChapter 11: Automated Proof Systems
Chapter 11: Automated Proof Systems SYSTEM RS OVERVIEW Hilbert style systems are easy to define and admit a simple proof of the Completeness Theorem but they are difficult to use. Automated systems are
More informationThe Blok-Ferreirim theorem for normal GBL-algebras and its application
The Blok-Ferreirim theorem for normal GBL-algebras and its application Chapman University Department of Mathematics and Computer Science Orange, California University of Siena Department of Mathematics
More informationThe model companion of the class of pseudo-complemented semilattices is finitely axiomatizable
The model companion of the class of pseudo-complemented semilattices is finitely axiomatizable (joint work with Regula Rupp and Jürg Schmid) Joel Adler joel.adler@phbern.ch Institut S1 Pädagogische Hochschule
More informationHypersequent and Labelled Calculi for Intermediate Logics
Hypersequent and Labelled Calculi for Intermediate Logics Agata Ciabattoni 1, Paolo Maffezioli 2, and Lara Spendier 1 1 Vienna University of Technology 2 University of Groningen Abstract. Hypersequent
More informationFuzzy filters and fuzzy prime filters of bounded Rl-monoids and pseudo BL-algebras
Fuzzy filters and fuzzy prime filters of bounded Rl-monoids and pseudo BL-algebras Jiří Rachůnek 1 Dana Šalounová2 1 Department of Algebra and Geometry, Faculty of Sciences, Palacký University, Tomkova
More informationAN INVESTIGATION OF RESIDUATED LATTICES WITH MODAL OPERATORS. William Young. Dissertation. Submitted to the Faculty of the
AN INVESTIGATION OF RESIDUATED LATTICES WITH MODAL OPERATORS By William Young Dissertation Submitted to the Faculty of the Graduate School of Vanderbilt University in partial fulfillment of the requirements
More informationExploring a Syntactic Notion of Modal Many-Valued Logics
Mathware & Soft Computing 15 (2008) 175-188 Exploring a Syntactic Notion of Modal Many-Valued Logics F. Bou, F. Esteva and L. Godo IIIA - CSIC 08193 Bellaterra, Spain {fbou,esteva,godo}@iiia.csic.es Abstract
More informationDe Jongh s characterization of intuitionistic propositional calculus
De Jongh s characterization of intuitionistic propositional calculus Nick Bezhanishvili Abstract In his PhD thesis [10] Dick de Jongh proved a syntactic characterization of intuitionistic propositional
More informationLogical Preliminaries
Logical Preliminaries Johannes C. Flieger Scheme UK March 2003 Abstract Survey of intuitionistic and classical propositional logic; introduction to the computational interpretation of intuitionistic logic
More informationThe Countable Henkin Principle
The Countable Henkin Principle Robert Goldblatt Abstract. This is a revised and extended version of an article which encapsulates a key aspect of the Henkin method in a general result about the existence
More informationcse371/mat371 LOGIC Professor Anita Wasilewska Fall 2018
cse371/mat371 LOGIC Professor Anita Wasilewska Fall 2018 Chapter 7 Introduction to Intuitionistic and Modal Logics CHAPTER 7 SLIDES Slides Set 1 Chapter 7 Introduction to Intuitionistic and Modal Logics
More informationUndefinability of standard sequent calculi for Paraconsistent logics
Undefinability of standard sequent calculi for Paraconsistent logics Michele Pra Baldi (with S.Bonzio) University of Padua m.prabaldi@gmail.com Topology, Algebra and Categories in Logic. Prague, June 6,
More informationOmitting Types in Fuzzy Predicate Logics
University of Ostrava Institute for Research and Applications of Fuzzy Modeling Omitting Types in Fuzzy Predicate Logics Vilém Novák and Petra Murinová Research report No. 126 2008 Submitted/to appear:
More informationAlgebras of Deductions in Category Theory. 1 Logical models from universal algebra
THIRD MATHEMATICAL CONFERENCE OF THE REPUBLIC OF SRPSKA Trebinje, 7 and 8 June 2013 Algebras of Deductions in Category Theory Kosta Dosen Faculty of Philosophy, University of Belgrade, and Mathematical
More informationKrivine s Intuitionistic Proof of Classical Completeness (for countable languages)
Krivine s Intuitionistic Proof of Classical Completeness (for countable languages) Berardi Stefano Valentini Silvio Dip. Informatica Dip. Mat. Pura ed Applicata Univ. Torino Univ. Padova c.so Svizzera
More informationRelation algebras as expanded FL-algebras
Algebra Univers. 69 (2013) 1 21 DOI 10.1007/s00012-012-0215-y Published online December 27, 2012 Springer Basel 2012 Algebra Universalis Relation algebras as expanded FL-algebras Nikolaos Galatos and Peter
More informationAdding truth-constants to logics of continuous t-norms: axiomatization and completeness results
Adding truth-constants to logics of continuous t-norms: axiomatization and completeness results Francesc Esteva, Lluís Godo, Carles Noguera Institut d Investigació en Intel ligència Artificial - CSIC Catalonia,
More informationMorita-equivalences for MV-algebras
Morita-equivalences for MV-algebras Olivia Caramello* University of Insubria Geometry and non-classical logics 5-8 September 2017 *Joint work with Anna Carla Russo O. Caramello Morita-equivalences for
More informationEquational Logic. Chapter Syntax Terms and Term Algebras
Chapter 2 Equational Logic 2.1 Syntax 2.1.1 Terms and Term Algebras The natural logic of algebra is equational logic, whose propositions are universally quantified identities between terms built up from
More informationVARIETIES OF DE MORGAN MONOIDS: COVERS OF ATOMS
VARIETIES OF DE MORGAN MONOIDS: COVERS OF ATOMS T. MORASCHINI, J.G. RAFTERY, AND J.J. WANNENBURG Abstract. The variety DMM of De Morgan monoids has just four minimal subvarieties. The join-irreducible
More informationNatural Deduction. Formal Methods in Verification of Computer Systems Jeremy Johnson
Natural Deduction Formal Methods in Verification of Computer Systems Jeremy Johnson Outline 1. An example 1. Validity by truth table 2. Validity by proof 2. What s a proof 1. Proof checker 3. Rules of
More informationThe overlap algebra of regular opens
The overlap algebra of regular opens Francesco Ciraulo Giovanni Sambin Abstract Overlap algebras are complete lattices enriched with an extra primitive relation, called overlap. The new notion of overlap
More informationExpanding the realm of systematic proof theory
Expanding the realm of systematic proof theory Agata Ciabattoni 1, Lutz Straßburger 2, and Kazushige Terui 3 1 Technische Universtät Wien, Austria 2 INRIA Saclay Île-de-France, France 3 RIMS, Kyoto University,
More informationBoolean Algebra CHAPTER 15
CHAPTER 15 Boolean Algebra 15.1 INTRODUCTION Both sets and propositions satisfy similar laws, which are listed in Tables 1-1 and 4-1 (in Chapters 1 and 4, respectively). These laws are used to define an
More information185.A09 Advanced Mathematical Logic
185.A09 Advanced Mathematical Logic www.volny.cz/behounek/logic/teaching/mathlog13 Libor Běhounek, behounek@cs.cas.cz Lecture #1, October 15, 2013 Organizational matters Study materials will be posted
More informationFuzzy Logics and Substructural Logics without Exchange
Fuzzy Logics and Substructural Logics without Exchange Mayuka F. KAWAGUCHI Division of Computer Science Hokkaido University Sapporo 060-0814, JAPAN mayuka@main.ist.hokudai.ac.jp Osamu WATARI Hokkaido Automotive
More informationSUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, III. THE CASE OF TOTALLY ORDERED SETS
SUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, III. THE CASE OF TOTALLY ORDERED SETS MARINA SEMENOVA AND FRIEDRICH WEHRUNG Abstract. For a partially ordered set P, let Co(P) denote the lattice of all order-convex
More informationA MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ
A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ NICOLAS FORD Abstract. The goal of this paper is to present a proof of the Nullstellensatz using tools from a branch of logic called model theory. In
More information