Boundary Algebra: A Simpler Approach to Boolean Algebra and the Sentential Connectives

Boundary algebra [BA] is a algebra of type , and a simplified notation for Spencer-Brown’s (1969) primary algebra. The syntax of the primary arithmetic [PA] consists of two atoms, () and the blank page, concatenation, and enclosure between ‘(‘ and ‘)’, denoting the primitive notion of distinction. Inserting letters denoting, indifferently, the presence or absence of () into a PA formula yields a BA formula. The BA axioms are A1: ()()= (), and A2: “(()) [abbreviated ‘⊥’] may be written or erased at will,” implying (⊥)=(). The repeated application of A1 and A2 simplifies any PA formula to either () or ⊥. The basis for BA is B1: abc=bca (concatenation commutes & associates); B2, ⊥a=a (BA has a lower bound, ⊥); B3, (a)a=() (BA is a complemented lattice); and B4, (ba)a=(b)a (implies that BA is a distributive lattice). BA has two intended models: (1) the Boolean algebra 2 with base set B={(),⊥}, such that () ⇔ 1 [dually 0], (a) ⇔ a′, and ab ⇔ a∪b [a∩b]; and (2) sentential logic, such that () ⇔ true [false], (a) ⇔ ~a, and ab ⇔ a∨b [a∧b]. BA is a self-dual notation, facilitates a calculational style of proof, and simplifies clausal reasoning and Quine’s truth value analysis. BA resembles C.S. Peirce’s graphical logic, the symbolic logics of Leibniz and W.E. Johnson, the 2 notation of Byrne (1946), and the Boolean term schemata of Quine (1982).Boundary algebra; boundary logic; primary algebra; primary arithmetic; Boolean algebra; calculation proof; G. Spencer-Brown; C.S. Peirce; existential graphs

Graph Interpolation Grammars: a Rule-based Approach to the Incremental Parsing of Natural Languages

Graph Interpolation Grammars are a declarative formalism with an operational semantics. Their goal is to emulate salient features of the human parser, and notably incrementality. The parsing process defined by GIGs incrementally builds a syntactic representation of a sentence as each successive lexeme is read. A GIG rule specifies a set of parse configurations that trigger its application and an operation to perform on a matching configuration. Rules are partly context-sensitive; furthermore, they are reversible, meaning that their operations can be undone, which allows the parsing process to be nondeterministic. These two factors confer enough expressive power to the formalism for parsing natural languages.Comment: 41 pages, Postscript onl

Sepia: a Framework for Natural Language Semantics

Source code and technical descriptionTo help explore linguistic semantics in the context of computational natural language understanding, Sepia provides a realization the central theoretical idea of categorial grammar: linking words and phrases to compositional lambda semantics. The Sepia framework provides a language in which to express complex transformations from text to data structures, and tools surrounding that language for parsing and machine learning. Lambda semantics are expressed as arbitrary Scheme programs, unlimited in the semantic representations they may build, and the rules for transformation are expressed in Combinatory Categorial Grammar, though the details of grammar formalism may be easily changed. This report explains the major design decisions, and is meant to teach the reader how to understand Sepia semantics and how to create lexical items for a new language understanding task

Hiding variables when decomposing specifications into GR(1) contracts

We propose a method for eliminating variables from component specifications during the decomposition of GR(1) properties into contracts. The variables that can be eliminated are identified by parameterizing the communication architecture to investigate the dependence of realizability on the availability of information. We prove that the selected variables can be hidden from other components, while still expressing the resulting specification as a game with full information with respect to the remaining variables. The values of other variables need not be known all the time, so we hide them for part of the time, thus reducing the amount of information that needs to be communicated between components. We improve on our previous results on algorithmic decomposition of GR(1) properties, and prove existence of decompositions in the full information case. We use semantic methods of computation based on binary decision diagrams. To recover the constructed specifications so that humans can read them, we implement exact symbolic minimal covering over the lattice of integer orthotopes, thus deriving minimal formulae in disjunctive normal form over integer variable intervals

A Bi-Polar Theory of Nominal and Clause Structure and Function

It is taken as axiomatic that grammar encodes meaning. Two key dimensions of meaning that get grammatically encoded are referential meaning and relational meaning. The key claim is that, in English, these two dimensions of meaning are typically encoded in distinct grammatical poles—a referential pole and a relational pole—with a specifier functioning as the locus of the referential pole and a head functioning as the locus of the relational pole. Specifiers and heads combine to form referring expressions corresponding to the syntactic notion of a maximal projection. Lexical items and expressions functioning as modifiers are preferentially attracted to one pole or the other. If the head of an expression describes a relation, one or more complements may be associated with the head. The four grammatical functions specifier, head, modifier and complement are generally adequate to represent much of the basic structure and function of nominals and clauses. These terms are borrowed from X-Bar Theory, but they are motivated on semantic grounds having to do with their grammatical function to encode referential and relational meaning