Finite Systems Handling Language (YAFOLL message 1) (original) (raw)
Algebraic Logic and Computer Science
2011
Algebraic Logic and Computer Science Jacinta Poças, Manuel Martins, and Carlos Caleiro 1 Dep. Mathematics, IST TU Lisbon, Portugal 2 Dep. Mathematics, U Aveiro, Portugal 3 SQIG Instituto de Telecomunicações Abstract. Abstract algebraic logic (AAL) is a branch of logic that Abstract algebraic logic (AAL) is a branch of logic that uses universal algebra to study the properties of logical systems by associating them with representative classes of algebras, thus generalizing the Lindenbaum-Tarski process that leads to linking Boolean algebras with classical propositional logic. The theory of AAL then classifies logical systems and their metaproperties, systematically, along bridge theorems that relate them with properties of the associated algebras. Recently, concepts of behavioral algebraic specification have influenced the development of the behavioral approach to AAL. Namely, the notion of behavioral equivalence, imported from computer science, has been used to weaken the traditional...
First order languages: Further syntax and semantics
2011
Third of a series of articles laying down the bases for classical first order model theory. Interpretation of a language in a universe set. Evaluation of a term in a universe. Truth evaluation of an atomic formula. Reassigning the value of a symbol in a given interpretation. Syntax and semantics of a non atomic formula are then defined concurrently (this point is explained in [16], 4.2. 1). As a consequence, the evaluation of any wff string and the relation of logical implication are introduced. Depth of a formula.
Urbana
We propose an abstract formal model of state manipulation in the framework of Girard's linear logic. Two issues motivate this work: how to describe the semantics of higher-order imperative programming languages and how to incorporate state manipulation in functional programming languages. The central idea is that a state is linear and "regenerative", where the latter is the property of a value that generates a new value upon each use. Based on this, we define a type constructor for states and a "modality" type constructor for regenerative values. Just as Girard's "of course" modality allows him to express static values and intuitionistic logic within the framework of linear logic, our regenerative modality allows us to express dynamic values and imperative programs within the same framework. We demonstrate the expressiveness of the model by showing that a higher-order Algol-like language can be embedded in it.
A Linear Logic Model of State ( Extended Abstract )
1993
We propose an abstract formal model of state manipulation in the framework of Girard’s linear logic. Two issues motivate this work: how to describe the semantics of higher-order imperative programming langauges and how to incorporate state manipulation in functional programming languages. The central idea is that a state is linear and “regenerative”, where the latter is the property of a value that generates a new value upon each use. Based on this, we define a type constructor for states and a “modality” type constructor for regenerative values. Just as Girard’s “of course” modality allows him to express static values and intuitionistic logic within the framework of linear logic, our regenerative modality allows us to express dynamic values and imperative programs within the same framework. We demonstrate the expressiveness of the model by showing that a higher-order Algol-like language can be embedded in it.
An algebraic theory of observables
Proceedings of the 1994 International Symposium on Logic Programming, 1994
We give an algebraic formalization of SLD-trees and their abstractions (ob servables). We can state and prove in the framework several useful theorems (AND-compositionality, correctness and full abstraction of the denotation, equivalent top-down and bottom-up constructions) about semantic properties of various observables. Observables are represented by Galois co-insertions and can be used to model abstract interpretation. The constructions and the theorems are inherited by all the observables which can be formalized in the framework. The power of the framework is shown by reconstructing some known examples (answer constraints, call patterns, correct call patterns and ground dependencies call patterns). Corollary 2.7 (correctness and full abstraction) Let p 1 , p 2 be two programs. Then p 1 = id p 2 ⇐⇒ O(p 1) = O(p 2).
Reconciling first-order logic to algebra
Starting from the algebraic method of theorem-proving based on the translation of logic formulas into polynomials over finite fields, and by adapting the case of first-order formulas by employing certain rings equipped with infinitary operations, this paper defines the notion of M-ring, a kind of polynomial ring defined for each first-order structure, by means of generators and relations. The notion of M-ring allows us to operate with some kind of infinitary version of Boolean sums and products, in this way expressing algebraically first-order logic with a new gist. We then show how this polynomial representation of first-order sentences could be seen as a legitimate algebraic semantics for first-order logic, alternative to cylindric and polyadic algebras an with a higher degree of naturalness. We briefly discuss how the method and their generalizations could be successfully lifted to n-valued logics and to other non-classical logics helping to reconcile some lost ties between algebra and logic.
A Linear Logic Model of State ( Preliminary Report )
1992
We propose an abstract formal model of state manipulation in the framework of Girard’s linear logic. This work addresses twin issues: how to incorporate state manipulation in functional programming languages, and how to describe the semantics of higher-order imperative programming languages. In fact, it appears that, given the right model of state, the difference between functional and imperative programming becomes rather thin. Our model of state is based on a new “modality” type constructor for expressing “regenerative values” (values that reproduce themselves each time they are used). Just as Girard’s “of course” modality allows him to express static values and intuitionistic logic within the framework of linear logic, our regenerative modality allows us to express states and state-dependent values within the same framework. We demonstrate the expressiveness of the model by showing that a higher-order Algol-like language can be embedded into it.
Information Sciences, 1989
The existence of full, consistent, and finite systems of inference rules for several languages is discussed. These languages are used for the description of system classes, the system being given by a multiplace relation. The same languages-types of dependencies-are used in relational database theory for describing integrity constraints. The absence of finite axiomatization for different types of dependencies, including join dependencies, is proved.
Fundamental Study Equational programming in A-calculus via SL-systems. Part 2*
1996
A system of equations in the l-calculus is a set of formulas of A (the equations) together with a finite set of variables of ,4 (the unknowns). A system Y is said to be P-solvable (fiq-solvable) iff there exists a simultaneous substitution with closed I-terms for the unknowns that makes the equations of 9' theorems in the theory fi (&). A system 9' can be viewed as a set of specifications (the equations) for a finite set of programs (the unknowns) whereas a solution for Y yields executable codes for such programs. A class G of systems for which the solvability problem is effectively decidable defines an equational programming language and a system solving algorithm for G defines a compiler for such language. This leads us to consider separation-like systems (SL-systems), i.e. systems with equations having form x& = z, where x is an unknown and z is a free variable which is not an unknown. It is known that the /I (/?q)-solvability problem for SL-systems is undecidable. Here we show that there is a class of SL-systems (NP-regular SL-systems) for which the /?-solvability problem is NP-complete. Moreover, we show that any SL-system Y can be transformed into an NP-regular SL-system Y". This transformation consists of adding abstractions to the LHS occurrences of the RHS variables of 9. In this sense NP-regular SL-systems isolate the source of undecidability for SL-systems, namely: a shortage of abstractions on the LHS occurrences of the RHS variables. NP-regular SL-systems yield an equational programming language in which unrestrained self-application is handled, constraints on executable code to be generated by the compiler can be specified by the user and (properties of) data structures can be described in an abstract way. However, existence of executable code satisfying a specification in such language is an *This paper is a revised version of [7, Part 21.