Melvin Fitting. First-order logic and automated theorem proving. Texts and monographs in computer science. Springer-Verlag, New York, Berlin, etc., 1990, xv + 242 pp.

1993 ◽  
Vol 58 (2) ◽  
pp. 719-719
Author(s):  
Alfredo Ferro
Keyword(s):  
New York ◽  
Computer Science ◽  
Theorem Proving ◽  
Automated Theorem Proving ◽  
Order Logic ◽  
First Order Logic ◽  
First Order

scholarly journals Different Proofs are Good Proofs

10.29007/kwk9 ◽  
2018 ◽  
Author(s):  
Geoff Sutcliffe ◽  
Cynthia Chang ◽  
Li Ding ◽  
Deborah McGuinness ◽  
Paulo Pinheiro da Silva
Keyword(s):  
Theorem Proving ◽  
Automated Theorem Proving ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Problem Library

In order to compare the quality of proofs, it is necessary to measure artifacts of the proofs, and evaluate the measurements to determine differences between the proofs. This paper discounts the approach of ranking measurements of proof artifacts, and takes the position that different proofs are good proofs. The position is based on proofs in the TSTP solution library, which are generated by Automated Theorem Proving (ATP) systems applied to first-order logic problems in the TPTP problem library.

Proof Procedures for Logic Programming

1998 ◽  
Author(s):  
Donald W. Loveland ◽  
Gopalan Nadathur
Keyword(s):  
Logic Programming ◽  
Theorem Proving ◽  
Automated Theorem Proving ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Proof Procedure ◽  
Sld Resolution ◽  
Work Done ◽  
Resolution Proof

A proof procedure is an algorithm (technically, a semi-decision procedure) which identifies a formula as valid (or unsatisfiable) when appropriate, and may not terminate when the formula is invalid (satisfiable). Since a proof procedure concerns a logic the procedure takes a special form, superimposing a search strategy on an inference calculus. We will consider a certain collection of proof procedures in the light of an inference calculus format that abstracts the concept of logic programming. This formulation allows us to look beyond SLD-resolution, the proof procedure that underlies Prolog, to generalizations and extensions that retain an essence of logic programming structure. The inference structure used in the formulation of the logic programming concept and first realization, Prolog, evolved from the work done in the subdiscipline called automated theorem proving. While many proof procedures have been developed within this subdiscipline, some of which appear in Volume 1 of this handbook, we will present a narrow selection, namely the proof procedures which are clearly ancestors of the first proof procedure associated with logic programming, SLD-resolution. Extensive treatment of proof procedures for automated theorem proving appear in Bibel [Bibel, 1982], Chang and Lee [Chang and Lee, 1973] and Loveland [Loveland, 1978]. Although the consideration of proof procedures for automated theorem proving began about 1958 we begin our overview with the introduction of the resolution proof procedure by Robinson in 1965. We then review the linear resolution procedures, model elimination and SL-resolution procedures. Our exclusion of other proof procedures from consideration here is due to our focus, not because other procedures are less important historically or for general use within automated or semi-automated theorem process. After a review of the general resolution proof procedure, we consider the linear refinement for resolution and then further restrict the procedure format to linear input resolution. Here we are no longer capable of treating full first-order logic, but have forced ourselves to address a smaller domain, in essence the renameable Horn clause formulas. By leaving the resolution format, indeed leaving traditional formula representation, we see there exists a linear input procedure for all of first-order logic.

Some Aspects of Model Theory and Finite Structures

2002 ◽  
Vol 8 (3) ◽  
pp. 380-403 ◽  
Author(s):  
Eric Rosen
Keyword(s):  
Computer Science ◽  
Complexity Theory ◽  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Finite Model ◽  
Finite Model Theory ◽  
First Order ◽  
Single Sentence ◽  
Finite Structures

Model theory is concerned mainly, although not exclusively, with infinite structures. In recent years, finite structures have risen to greater prominence, both within the context of mainstream model theory, e.g., in work of Lachlan, Cherlin, Hrushovski, and others, and with the advent of finite model theory, which incorporates elements of classical model theory, combinatorics, and complexity theory. The purpose of this survey is to provide an overview of what might be called the model theory of finite structures. Some topics in finite model theory have strong connections to theoretical computer science, especially descriptive complexity theory (see [26, 46]). In fact, it has been suggested that finite model theory really is, or should be, logic for computer science. These connections with computer science will, however, not be treated here.It is well-known that many classical results of ‘infinite model theory’ fail over the class of finite structures, including the compactness and completeness theorems, as well as many preservation and interpolation theorems (see [35, 26]). The failure of compactness in the finite, in particular, means that the standard proofs of many theorems are no longer valid in this context. At present, there is no known example of a classical theorem that remains true over finite structures, yet must be proved by substantially different methods. It is generally concluded that first-order logic is ‘badly behaved’ over finite structures.From the perspective of expressive power, first-order logic also behaves badly: it is both too weak and too strong. Too weak because many natural properties, such as the size of a structure being even or a graph being connected, cannot be defined by a single sentence. Too strong, because every class of finite structures with a finite signature can be defined by an infinite set of sentences. Even worse, every finite structure is defined up to isomorphism by a single sentence. In fact, it is perhaps because of this last point more than anything else that model theorists have not been very interested in finite structures. Modern model theory is concerned largely with complete first-order theories, which are completely trivial here.

Sign in / Sign up

Export Citation Format

Share Document