Why horn formulas matter in computer science

General Logics. The connections between logic and computer science are growing rapidly and are becoming deeper. Besides theorem proving, logic programming, and program specification and verification, other areas … Expand. Data Types over Multiple-Values Logics. Highly Influenced.

View 3 excerpts, cites background. Finite Horn Monoids and Near-Semirings. Sequential Composition of Propositional Horn Theories. Boolean Valued Models and Incomplete Specifications. The desirability of Horn clauses in logical deductive systems has long been recognized.

The reasons are at least threefold. Firstly, while inference algorithms for full logics of any reasonable … Expand. View 1 excerpt, cites background. Decision Problems in Predicate Logic. Unit Refutations and Horn Sets. An Evaluation Based Theorem Prover. Horn clause computability. On Closed World Data Bases. Related Papers. By clicking accept or continuing to use the site, you agree to the terms outlined in our Privacy Policy , Terms of Service , and Dataset License. In logic programming Horn formulas are used both as a specification and a programming language because, as R.

Kowalski put it, the allow a proceduralinterpretation cf. Mahr and J. Makowsky [MM83] prove that under certain assumptions for the semantics of algebraic specifications, conditional equations form the largest specification language satisfying these assumptions. L Makowsky and M. Vardi [MV84] characterize various classes of data base dependencies in terms of preservation properties under operations on relations which come from data manipulation.

In logic programming,, it was shown by Tarnlund [Tam77] that Horn logic is enough to program every recursive function, a result, stated in slightly different form in a different context, proven already by S. For an excellent survey see [B].

Additionally, if Tie finite, this set of definable partial functions can be chosen to be finite, too. Without loss of generality can mean two things: Given a specification which is not a set of universal Horn formulas, then either we can find an equivalent set of universal Horn formulas or we have chosen the basic vocabulary set of basic symbols wrongly, and then the theorem tells us that there is a unambiguous way to correct this.

In detail the paper is organized as follows:. In section 3 we characterize first order theories which admit initial term models as the universal Horn theories. This theorem was already proved in [MM83]. Rabin [Ra60], which characterizes first order theories with the intersection property.

In section 5 we state an analogue of Rabin's theorem to obtain our main result. We show that a first order theory admits initial models iff it is a partially functional V3-Horn theory. No proofs are presented in this extended abstract, since the editors of these proceedings put a severe space limit on the papers to be presented.

We hope that the complete paper will appear soon elsewhere. In section 6 we state some conclusions. The reader familiar with the introduction to model theory by G. Kreisel and J. Krivine will realize how much, in spirit, this work is influenced by chapter 6 of [KK66]. I am indebted also to A. Tarlecki for his remarks in our correspondence concerning [MM83], to S. Shelah, who suggested theorem 2. Mahr for the discussions around [MM83].

Initial models and genericity In this section we deal with first order languages with equality. A n, for each sort in t one, together with interpretations for all the function, relation and constant symbols in t. We call t-structures also modets and. Then A is an initial term model for K iff A is A-generic. Then every a A is definable over T by a formula Pa. In other words A I is a pseudo term model.

Characterizing first order theories which admit initial term models In this section we characterize first order theories which admit initial term models. Multiple physical clocks The objective is also to obtain a unique physical time frame within the system so that consistent schedules may be derived from a total chronological ordering of actions occurring in the system.

When several clocks are used it is not enough for the clocks individually to run at approximately the same rate. They must be kept synchronized so that the relative drifting of any two clocks is kept smaller than a predictable constant. In [Lamport78j a solution to accomplish this is given. The system under consideration is modelled after a strongly connected graph ot processes with diameter d.

Every process is provided with a clock and every t, a synchronization sync message is sent over every arc. A sync message contains a physical timestamp T. Upon receiving a sync message, if needed, a process should set forward its local clock to be later than the timestamp value contained in the incoming message.

Let k be the intrinsic accuracy of each clock e. Then, if the upper. Such logical clocks may be implemented by counters. In order to meet condition F, the following rules must be obeyed by producers. Rule i: each producer i increments C i between any two successive actions. Upon receiving a message m, producer j sets C j greater than or equal to its present value and greater than T m. This is why it may be necessary to implement such a system of logical clocks on a system of several physical clocks see previous section.

This may be particularly advantageous in distributed systems Utilization of a circulating privilege Synchronization mechanisms may take advantage of the fact that producers are given unique and permanent names.

This defines a total ordering on the set ot producers. Such an ordering may be used to view producers as being organized as on a chain or as on a loop. Each producer has a unique predecessor and a unique successor. Such a logical structuring does not imply any particular physical topology. Pair-wise shared variables: A synchronization mechanism based on the concept of a logical ring has been presented in [Dijkstra74]. Possession of a control privilege may be inferred by every producer from the observation of a variable shared with one of its two neighbours.

Yn and let k be a natural number. In the next section we want to give a similar characterization for first order theories admitting initial models. Our next goal is to show the existence of initial models for certain theories which have the Intersection Property and which are preserved under products.

For this we need some more definitions Definitions: Let T be a first order theory with the Intersection Property. Then every core model of Tis an 3-term model.

A converse of theorem 4. We fwst want to show that such a theory is equivalent to an v3-horn theory. Next we want to state an analogue of Rabin's theorem theorem 4.

Theorem: Let The a first order theory which admits initial models. Definition: We call a f'lrst order theory which satisfies the conclusion of theorem 5. Then every 3-term model A is a pseudo term model We need another well lmown result from model theory, see e. Theorem: Let Tbe an v3-horn theory. Then Tls preserved under products. Putting everything together we obtain: 5. Theorem: Main theorem Let T be a ten'st order theory.

The following are 6. Conclusions We have given a characterization of universal Horn theories in terms of the existence of initial, or equivalently, A-generic term models theorem 3. The latter essentially says that a first order theory which admits initial models which are not term models does so by oversight: The vocabulary similarity type was badly chosen, such as not to allow that all elements are denoted by some term.

This can be almost remedied: Either by adding definable partial Skolem functions or by allowing pseudo terms, i. The paper also sheds more light into the question why in [ADJ75] Initial structures were proposed as the framework for abstract data types.

We have given in theorem 2. For somebody not familiar with category theory this may be more appealing since it relates directly to or concept of verification by example. However, this characterization has also its technical. Last but not least we have yet added another explanation as to why Horn formulas play such an important role in various branches of computer science. We have shown that universal Horn theories partially functional v3-theories are exactly the framework in which the notion of a generic example can be applied.

This should prevent other researchers from trying to generalize Logic Programming or the semantics abstract data types to larger classes of first order formulas. If it has to be generalized then the direction chosen by R. Burstall and J. Goguen in [GB84] seems to be much more appropriate. Lolli, G. Longo and Am Marcia eds. ACM vol. Logic vol. Mal'cev, North Holland , pp Mahr, B. Longo and A. Marcia eds.

Rabin, M. Universite de Clermont, vol. Longo, G. CompuL vol. The full. Logical Arguments and Formal Proofs 1. Basic Terminology. An axiom is a statement that is given to be true.

A rule of inference is a logical rule that is used to deduce. The axioms which describe the arithmetic of the real numbers. They can be thought as an inference machine with special statements, called provable statements, or sometimes. An Innocent Investigation D. Joyce, Clark University January The beginning. Have you ever wondered why every number is either even or odd?

I don t mean to ask if you ever wondered whether every number. Math Rumbos Spring 1 Handout 1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or. Mathematical Induction Handout March 8, 01 The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,.

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When trying to decide whether a given function is or. Performance Assessment Task Graphs Grade 9 This task challenges a student to use knowledge of graphs and their significant features to identify the linear equations for various lines. A student. Benevides and L. Computer Science Journal of Moldova, vol. Math Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space.

This is understandable. In connection with his investigation of projective planes, M. Hall [2, Theorem 6. Hilbert Polynomials For a monomial ideal, we derive the dimension counting the monomials in the complement, arriving at the notion of the Hilbert polynomial.

The first half of the note is derived from. Simple conventional. Thompson, ISBN Systems of Linear Equations Definition. An n-dimensional vector is a row or a column. Los Angeles, CA. Show that every multiple of a Pythagorean triple. The reader is encouraged to. CS Intro. Scribe notes The purpose of scribe notes is to transcribe our lectures. Although I have formal notes of my. Axioms of Set theory Before presenting the axioms of set theory, we first make a few basic comments about the relevant first order logic.

We will give a somewhat more detailed discussion later, but. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e. The title of Halmos s book was the same as the title of this chapter.

Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example. LMCS, p. Our version of first-order logic will use the following symbols: variables connectives ,,,,.

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Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing.