Between quantum logic and concurrency

We start from two closure operators defined on the elements of a special kind of partially ordered sets, called causal nets. Causal nets are used to model histories of concurrent processes, recording occurrences of local states and of events. If every maximal chain (line) of such a partially ordered set meets every maximal antichain (cut), then the two closure operators coincide, and generate a complete orthomodular lattice. In this paper we recall that, for any closed set in this lattice, every line meets either it or its orthocomplement in the lattice, and show that to any line, a two-valued state on the lattice can be associated. Starting from this result, we delineate a logical language whose formulas are interpreted over closed sets of a causal net, where every line induces an assignment of truth values to formulas. The resulting logic is non-classical; we show that maximal antichains in a causal net are associated to Boolean (hence "classical") substructures of the overall quantum logic.Comment: In Proceedings QPL 2012, arXiv:1407.842

Orthomodular Lattices Induced by the Concurrency Relation

n/

Substructurality and residuation in logic and algebra

A very and natural way of introducing a logic is by using a sequent calculus, or Gentzen system. These systems are determined by specifying a set of axioms and a set of rules. Axioms are then starting points from which we can derive new consequences by using the rules. Hilbert systems consist also on a set of axioms and a set of rules that are used to deduce consequences. The main difference is that, whereas the axioms in Hilbert systems are formulas, and the rules allow to deduce certain formulas from other sets of formulas, in the case of Gentzen systems the axioms are sequents and the rules indicate which sequents can be inferred from other sets of sequents. By a sequent we understand a pair hG, Si, where G and S are finite sequences of formulas. We denote the sequent hG, Si by G . S.1 The sequent G . S intends to formalize – at least in its origin – the concept “the conjunction of all the formulas of G implies the disjunction of all the formulas of S.” The notion of a sequent calculus was invented by G. Gentzen in order to give axiomatizations for Classical and Intuitionistic Propositional Logics. And the rules he gave in both cases can be grouped in different categories: because of its character, the Cut rule deserves a special category for itself; then we have the rules of introduction and elimination of each one of the connectives, both on the left and on the right – of the symbol . –; and finally a set of rules that do not involve any particular connective. These rules are necessary in Classical and Intuitionistic logics because in these logics 1Traditional notations for sequents are G ) S and G ` S, but since both the symbols ) and ` have many other meanings, we prefer to denote sequents by using the less overloaded symbol ., which can also be found in literature with this use. the order in which we are given the premises, or if we have them repeated, is irrelevant, and we do not loose consequences if we extend the set of hypotheses. But there are other logics that do not satisfy all these rules: for instance, relevance logics and linear logic. At first, these logics were studied separately, and different theories were developed for their investigation. But later on, researches arrived to the conclusion that all of them share a common feature, which became more apparent after the work of W. Blok and D. Pigozzi. It was discovered that (pointed) residuated lattices – or FL algebras – are the algebraic counterpart of substructural logics. In the XIX century, Boole noticed a close connection between “the laws of thought,” as he put it, and algebra. After him, other mathematicians put together all the pieces and described a sort of algebras, named Boole algebras after him, and shed light on the connection anticipated by Boole: Boole algebras are the “natural” semantics for Classical Propositional Logic. More connections were discovered between other logics and other sorts of algebras: for instance, Heyting algebras are the “natural” semantics for Intuitionistic Propositional Logic, and MV algebras for Łukasievicz Multivalued Logic. But it was not until 1989, when Blok and Pigozzi published their book Algebraizable Logics, that for the first time the connections between these logics and classes of algebras were finally described with absolute precision. According to their definitions, these classes of algebras are the equivalent algebraic sematics of the corresponding logics. That is, these classes of algebras are the algebraic counterparts of the corresponding logics. Their ideas paved the way to a new branch of mathematics called Abstract Algebraic Logic, which investigates the connections between logics and classes of algebras, and the so-called bridge theorems: that is, theorems that establish bridges between some property of one realm (logic or algebra) with another property of the other realm. The core of the connection between substructural logics and residuated lattices is that in all these logics, some theorem of the following form could always be proven. Thus, we could think that the metalogical symbol ’,’ is acting as a real connective. More precisely, we could introduce a new connective , called fusion, and impose the following rule. Given an algebraic model with a lattice reduct, it is usually the case that the meet and join operations serve as the interpretations of the conjunction and disjunction connectives. What should be then the interpretation of the fusion? Usually, the elements of the lattice are thought as different degrees of truth, and “a . b is provable” is interpreted as “for every assignment, the degree of truth of a is less than that of b.” Under this natural interpretation, the condition (1) becomes: That is, the fusion is interpreted as a residuated operation on the lattice. Being the algebraic semantics of substructural logics and containing many interesting subvarieties such as Heyting algebras, MV algebras, and lattice-ordered groups, to name a few, the variety of residuated lattices is of utmost importance to the studies of Logic and Algebra, hence our interest. In this dissertation we carry out some investigations on different problems concerning residuated lattices. In what follows we give a brief description of the contents and organization of this dissertation. Every chapter – except for the first one, which is devoted to setting the preliminaries – starts with an introduction in which the reader will find a lengthier explanation of the subject of the chapter, the way the material is organized, and references. We start by compiling in Chapter 1 all the essential well-known results about residuated lattices that we will need in the subsequent chapters. We present here the definitions of those concepts that are not specific to some particular chapter, but general. We define the variety of residuated lattices, and some of its more significant subvarieties. We also introduce nuclei, and nucleus retracts. As it is widely known, the lattice of normal convex subalgebras of a residuated lattice is isomorphic to its congruence lattice, and hence its importance. But it turns out that also the lattice of convex (not necessarily normal) subalgebras is of great significance, specially in the case of e-cyclic residuated lattices. Many of its properties depend on the fact that it is a pseudo-complemented lattice. Actually, it is a Heyting algebra. For instance, polars are special sets usually defined in terms of a certain notion of orthogonality; in the case of e-cyclic residuated lattices, polars are the pseudo-complements of the convex subalgebras. We end the chapter by briefly explaining the notions of semilinearity and projectability for residuated lattices. In the 1960’s, P. F. Conrad and other authors set in motion a general program for the investigation of lattice-ordered groups, aimed at elucidating some order-theoretic properties of these algebras by inquiring into the structure of their lattices of convex `-subgroups. This approach can be naturally extended to residuated lattices and their convex subalgebras. We devote Chapters 2 and 3 to two different problems that can be framed within Conrad’s program for residuated lattices. More specifically, in Chapter 2 we revisit the Galatos-Tsinakis categorical equivalence between integral GMV algebras and negative cones of `-groups with a nucleus, showing that it restricts to an equivalence of the full subcategories whose objects are the projectable members of these classes. Afterwards, we introduce the notion of Gödel GMV algebras, which are expansions of projectable integral GMV algebras by a binary term that realizes a positive Gödel implication in every such algebra. We see that Gödel GMV algebras and projectable integral GMV algebras are essentially the same thing. Analogously, Gödel negative cones are those Gödel GMV algebras whose residuated lattice reducts are negative cones of `-groups. Thus, we turn projectable integral GMV algebras and negative cones of projectable `-groups into varieties by including this implication in their signature. We prove that there is an adjunction between the categories whose objects are the members of these varieties and whose morphisms are required to preserve implications. We devote Chapter 3 to the study of certain kinds of completions of semilinear residuated lattices. We can find in the literature different notions of completions for residuated lattices, like for example Dedekind-McNeil completions, regular completions, complete ideal completions, . . . Very often it happens that for a certain algebra in a variety of residuated lattices, those completions exists but do not belong to the same variety. That is, varieties are not closed, in general, under the operations of taking these kinds of completions. But there are other notions of completions that might have better properties in this regard. Conrad and other authors proved the existence of lateral completions, projectable completions, and orthocompletions for representable `-groups, and moreover, that the varieties of representable `-groups are closed under these completions. Our goal in this chapter is to prove the existence of lateral completions, (strongly) projectable completions, and orthocompletions for semilinear e-cyclic residuated lattices, as they are a natural generalization of representable `-groups. We introduce all these concepts along the chapter, and prove first that every semilinear e-cyclic residuated lattice can be densely embedded into another residuated lattice which is latterly complete and strongly projectable. We obtain this lattice as a direct limit of a certain family of algebras obtained from the original lattice by taking quotients and products, so the direct limit stays in the same variety where the original algebra lives. Finally, we prove that for semilinear GMV algebras, we can find minimal dense extensions satisfying all the required properties. In Chapter 4 we study the failure of the Amalgamation Property on several varieties of residuated lattices. The Amalgamation Property is of particular interest in the study of residuated lattices due to its relation with various syntactic interpolation properties of substructural logics. There are no examples to date of non-commutative varieties of residuated lattices that satisfy the Amalgamation Property. The variety of semilinear Abstract 5 residuated lattices is a natural candidate for enjoying this property, since most varieties that have a manageable representation theory and satisfy the Amalgamation Property are semilinear. However, we prove that this is not the case, and in the process we establish that the same happens for the variety of semilinear cancellative residuated lattices, that is, it also lacks the Amalgamation Property. In addition, we prove that the variety whose members have a distributive lattice reduct and satisfy the identity x(y ^ z)w xyw ^ xzw also fails the Amalgamation Property. In Chapter 5 we show how some well-known results of the theory of automata, in particular those related to regular languages, can be viewed within a wider framework. In order to do so, we introduce the concept of module over a residuated lattice, and show that modules over a fixed residuated lattice – that is, partially ordered sets acted upon by a residuated lattice – provide a suitable algebraic framework for extending the concept of a recognizable language as defined by Kleene. More specifically, we introduce the notion of a recognizable element of a residuated lattice by a finite module and provide a characterization of such an element in the spirit of Myhill’s characterization of recognizable languages. Further, we investigate the structure of the set of recognizagle elements of a residuated lattice, and also provide sufficient conditions for a recognizable element to be recognized by a Boolean module. We summarize in Chapter 6 the main results of this dissertation and propose some of the problems that still remain open. We end this dissertation with an appendix on directoids. These structures were introduced independently three times, and their aim is to study directed ordered sets from an algebraic perspective. The structures that we have studied in this dissertations have an underlying order, but moreover they have a lattice reduct. That is not always the case for directed ordered sets. Hence the importance of the study of directoids. We prove some properties of directoids and their expansions by additional and complemented directoids. Among other results, we provide a shorter proof of the direct decomposition theorem for bounded involute directoids. We present a description of central elements of complemented directoids. And finally we show that the variety of directoids, as well as its expansions mentioned above, all have the strong amalgamation property

Substructurality and residuation in logic and algebra

A very and natural way of introducing a logic is by using a sequent calculus, or Gentzen system. These systems are determined by specifying a set of axioms and a set of rules. Axioms are then starting points from which we can derive new consequences by using the rules. Hilbert systems consist also on a set of axioms and a set of rules that are used to deduce consequences. The main difference is that, whereas the axioms in Hilbert systems are formulas, and the rules allow to deduce certain formulas from other sets of formulas, in the case of Gentzen systems the axioms are sequents and the rules indicate which sequents can be inferred from other sets of sequents. By a sequent we understand a pair hG, Si, where G and S are finite sequences of formulas. We denote the sequent hG, Si by G . S.1 The sequent G . S intends to formalize – at least in its origin – the concept “the conjunction of all the formulas of G implies the disjunction of all the formulas of S.” The notion of a sequent calculus was invented by G. Gentzen in order to give axiomatizations for Classical and Intuitionistic Propositional Logics. And the rules he gave in both cases can be grouped in different categories: because of its character, the Cut rule deserves a special category for itself; then we have the rules of introduction and elimination of each one of the connectives, both on the left and on the right – of the symbol . –; and finally a set of rules that do not involve any particular connective. These rules are necessary in Classical and Intuitionistic logics because in these logics 1Traditional notations for sequents are G ) S and G ` S, but since both the symbols ) and ` have many other meanings, we prefer to denote sequents by using the less overloaded symbol ., which can also be found in literature with this use. the order in which we are given the premises, or if we have them repeated, is irrelevant, and we do not loose consequences if we extend the set of hypotheses. But there are other logics that do not satisfy all these rules: for instance, relevance logics and linear logic. At first, these logics were studied separately, and different theories were developed for their investigation. But later on, researches arrived to the conclusion that all of them share a common feature, which became more apparent after the work of W. Blok and D. Pigozzi. It was discovered that (pointed) residuated lattices – or FL algebras – are the algebraic counterpart of substructural logics. In the XIX century, Boole noticed a close connection between “the laws of thought,” as he put it, and algebra. After him, other mathematicians put together all the pieces and described a sort of algebras, named Boole algebras after him, and shed light on the connection anticipated by Boole: Boole algebras are the “natural” semantics for Classical Propositional Logic. More connections were discovered between other logics and other sorts of algebras: for instance, Heyting algebras are the “natural” semantics for Intuitionistic Propositional Logic, and MV algebras for Łukasievicz Multivalued Logic. But it was not until 1989, when Blok and Pigozzi published their book Algebraizable Logics, that for the first time the connections between these logics and classes of algebras were finally described with absolute precision. According to their definitions, these classes of algebras are the equivalent algebraic sematics of the corresponding logics. That is, these classes of algebras are the algebraic counterparts of the corresponding logics. Their ideas paved the way to a new branch of mathematics called Abstract Algebraic Logic, which investigates the connections between logics and classes of algebras, and the so-called bridge theorems: that is, theorems that establish bridges between some property of one realm (logic or algebra) with another property of the other realm. The core of the connection between substructural logics and residuated lattices is that in all these logics, some theorem of the following form could always be proven. Thus, we could think that the metalogical symbol ’,’ is acting as a real connective. More precisely, we could introduce a new connective , called fusion, and impose the following rule. Given an algebraic model with a lattice reduct, it is usually the case that the meet and join operations serve as the interpretations of the conjunction and disjunction connectives. What should be then the interpretation of the fusion? Usually, the elements of the lattice are thought as different degrees of truth, and “a . b is provable” is interpreted as “for every assignment, the degree of truth of a is less than that of b.” Under this natural interpretation, the condition (1) becomes: That is, the fusion is interpreted as a residuated operation on the lattice. Being the algebraic semantics of substructural logics and containing many interesting subvarieties such as Heyting algebras, MV algebras, and lattice-ordered groups, to name a few, the variety of residuated lattices is of utmost importance to the studies of Logic and Algebra, hence our interest. In this dissertation we carry out some investigations on different problems concerning residuated lattices. In what follows we give a brief description of the contents and organization of this dissertation. Every chapter – except for the first one, which is devoted to setting the preliminaries – starts with an introduction in which the reader will find a lengthier explanation of the subject of the chapter, the way the material is organized, and references. We start by compiling in Chapter 1 all the essential well-known results about residuated lattices that we will need in the subsequent chapters. We present here the definitions of those concepts that are not specific to some particular chapter, but general. We define the variety of residuated lattices, and some of its more significant subvarieties. We also introduce nuclei, and nucleus retracts. As it is widely known, the lattice of normal convex subalgebras of a residuated lattice is isomorphic to its congruence lattice, and hence its importance. But it turns out that also the lattice of convex (not necessarily normal) subalgebras is of great significance, specially in the case of e-cyclic residuated lattices. Many of its properties depend on the fact that it is a pseudo-complemented lattice. Actually, it is a Heyting algebra. For instance, polars are special sets usually defined in terms of a certain notion of orthogonality; in the case of e-cyclic residuated lattices, polars are the pseudo-complements of the convex subalgebras. We end the chapter by briefly explaining the notions of semilinearity and projectability for residuated lattices. In the 1960’s, P. F. Conrad and other authors set in motion a general program for the investigation of lattice-ordered groups, aimed at elucidating some order-theoretic properties of these algebras by inquiring into the structure of their lattices of convex `-subgroups. This approach can be naturally extended to residuated lattices and their convex subalgebras. We devote Chapters 2 and 3 to two different problems that can be framed within Conrad’s program for residuated lattices. More specifically, in Chapter 2 we revisit the Galatos-Tsinakis categorical equivalence between integral GMV algebras and negative cones of `-groups with a nucleus, showing that it restricts to an equivalence of the full subcategories whose objects are the projectable members of these classes. Afterwards, we introduce the notion of Gödel GMV algebras, which are expansions of projectable integral GMV algebras by a binary term that realizes a positive Gödel implication in every such algebra. We see that Gödel GMV algebras and projectable integral GMV algebras are essentially the same thing. Analogously, Gödel negative cones are those Gödel GMV algebras whose residuated lattice reducts are negative cones of `-groups. Thus, we turn projectable integral GMV algebras and negative cones of projectable `-groups into varieties by including this implication in their signature. We prove that there is an adjunction between the categories whose objects are the members of these varieties and whose morphisms are required to preserve implications. We devote Chapter 3 to the study of certain kinds of completions of semilinear residuated lattices. We can find in the literature different notions of completions for residuated lattices, like for example Dedekind-McNeil completions, regular completions, complete ideal completions, . . . Very often it happens that for a certain algebra in a variety of residuated lattices, those completions exists but do not belong to the same variety. That is, varieties are not closed, in general, under the operations of taking these kinds of completions. But there are other notions of completions that might have better properties in this regard. Conrad and other authors proved the existence of lateral completions, projectable completions, and orthocompletions for representable `-groups, and moreover, that the varieties of representable `-groups are closed under these completions. Our goal in this chapter is to prove the existence of lateral completions, (strongly) projectable completions, and orthocompletions for semilinear e-cyclic residuated lattices, as they are a natural generalization of representable `-groups. We introduce all these concepts along the chapter, and prove first that every semilinear e-cyclic residuated lattice can be densely embedded into another residuated lattice which is latterly complete and strongly projectable. We obtain this lattice as a direct limit of a certain family of algebras obtained from the original lattice by taking quotients and products, so the direct limit stays in the same variety where the original algebra lives. Finally, we prove that for semilinear GMV algebras, we can find minimal dense extensions satisfying all the required properties. In Chapter 4 we study the failure of the Amalgamation Property on several varieties of residuated lattices. The Amalgamation Property is of particular interest in the study of residuated lattices due to its relation with various syntactic interpolation properties of substructural logics. There are no examples to date of non-commutative varieties of residuated lattices that satisfy the Amalgamation Property. The variety of semilinear Abstract 5 residuated lattices is a natural candidate for enjoying this property, since most varieties that have a manageable representation theory and satisfy the Amalgamation Property are semilinear. However, we prove that this is not the case, and in the process we establish that the same happens for the variety of semilinear cancellative residuated lattices, that is, it also lacks the Amalgamation Property. In addition, we prove that the variety whose members have a distributive lattice reduct and satisfy the identity x(y ^ z)w xyw ^ xzw also fails the Amalgamation Property. In Chapter 5 we show how some well-known results of the theory of automata, in particular those related to regular languages, can be viewed within a wider framework. In order to do so, we introduce the concept of module over a residuated lattice, and show that modules over a fixed residuated lattice – that is, partially ordered sets acted upon by a residuated lattice – provide a suitable algebraic framework for extending the concept of a recognizable language as defined by Kleene. More specifically, we introduce the notion of a recognizable element of a residuated lattice by a finite module and provide a characterization of such an element in the spirit of Myhill’s characterization of recognizable languages. Further, we investigate the structure of the set of recognizagle elements of a residuated lattice, and also provide sufficient conditions for a recognizable element to be recognized by a Boolean module. We summarize in Chapter 6 the main results of this dissertation and propose some of the problems that still remain open. We end this dissertation with an appendix on directoids. These structures were introduced independently three times, and their aim is to study directed ordered sets from an algebraic perspective. The structures that we have studied in this dissertations have an underlying order, but moreover they have a lattice reduct. That is not always the case for directed ordered sets. Hence the importance of the study of directoids. We prove some properties of directoids and their expansions by additional and complemented directoids. Among other results, we provide a shorter proof of the direct decomposition theorem for bounded involute directoids. We present a description of central elements of complemented directoids. And finally we show that the variety of directoids, as well as its expansions mentioned above, all have the strong amalgamation property

Weighted Tree Automata -- May it be a little more?

This is a book on weighted tree automata. We present the basic definitions and some of the important results in a coherent form with full proofs. The concept of weighted tree automata is part of Automata Theory and it touches the area of Universal Algebra. It originated from two sources: weighted string automata and finite-state tree automata

Foundations of Quantum Theory: From Classical Concepts to Operator Algebras

Quantum physics; Mathematical physics; Matrix theory; Algebr

Entanglement and Information in Algebraic Quantum Theories

The algebraic approach to physical theories provides a general framework encompassing both classical and quantum mechanics. Accordingly, by looking at the behaviour of the relevant algebras of observables one can investigate structural and conceptual differences between the theories. Interesting foundational questions can be formulated algebraically and their answers are then given in a mathematically compelling way. My dissertation focuses on some philosophical issues concerning entanglement and quantum information as they arise in the algebraic approach. The se two concepts are connected in that one can exploit the non-local character of quantum theory to construct protocols of information theory which are not realized in the classical world. I first introduce the basic concepts of the algebraic formalism by reviewing von Neumann's work on the mathematical foundations of quantum theories. After presenting the reasons why von Neumann abandoned the standard Hilbert space formulation in favour of the algebraic approach, I show how his axiomatic program remained a mathematical "utopia" in mathematical physics. The Bayesian interpretation of quantum mechanics is grounded in information-theoretical considerations. I take on some specific problems concerning the extension of Bayesian statistical inference in infinite dimensional Hilbert space. I demonstrate that the failure of a stability condition, formulated as a rationality constraint for classical Bayesian conditional probabilities, does not undermine the Bayesian interpretation of quantum probabilities. I then provide a solution to the problem of Bayesian noncommutative statistical inference in general probability spaces. Furthermore, I propose a derivation of the a priori probability state in quantum mechanics from symmetry considerations. Finally, Algebraic Quantum Field Theory offers a rigorous axiomatization of quantum field theory, namely the synthesis of quantum mechanics and special relativity. In such a framework one can raise the question of whether or not quantum correlations are made stronger by adding relativistic constraints. I argue that entanglement is more robust in the relativistic context than in ordinary quantum theory. In particular, I show how to generalize the claim that entanglement across space-like separated regions of Minkowski spacetime would persist, no matter how one acts locally