First-order Predicate Calculus Research Articles

Formal specifications of resource-deadlock prone petri nets

The definition of a class of resource-deadlock prone Petri nets is developed using first-order predicate calculus. From this definition characteristics of resource deadlock can be ascertained. These characteristics form the specifications of a program to be written in Prolog which can detect deadlock in parallel processes represented as Petri nets. An important feature of this class is that it characterizes deadlock in networks that contain nested parallel structures. Using the class defined herein, it should be possible to derive deadlock-free networks as specified in predicate calculus. Furthermore, this methodology, when applied to systems that do not dynamically determine resources to be allocated, allows for the a priori detection and correction of deadlock. An a priori method prevents the need for unnecessary overhead in dynamic deadlock detection as well as the need for the execution of difficult back-out mechanisms for deadlock correction.

An expert system for harmonizing chorales in the style of J.S. Bach

This paper describes an expert system called CHORAL, harmonization of four-part chorales in the style of Johann Sebastian Bach. The system contains about 350 rules, written in a form of first-order predicate calculus. The rules represent musical knowledge from multiple viewpoints of the chorale, such as the chord skeleton, the melodic lines of the individual parts, and Schenkerian voice leading within the descant and bass. The program harmonizes chorale melodies using a generate-and-test method with intelligent backtracking. A substantial number of heuristics are used for biasing the search toward musical solutions. The CHORAL knowledge base provides for style-specific modulations, cadence patterns, and complex encounters of simultaneous inessential notes; it imposes difficult constraints for maintaining melodic interest in the inner voices. Encouraging results have been obtained, and output examples are given. BSL, a new and efficient logic-programming language fundamentally different from PROLOG, was designed to implement the CHORAL system.

Reasoning by model: The case of multiple quantification.

A theory of deductive reasoning is presented fora major class of inferences that has not been investigated by psychologists: inferences that depend on multiply-quantified premises (e.g., of the Princeton letters is in the same place as any of the Dublin letters). It is argued that reasoneis construct mental models based on their knowledge of the meanings of quantifiers (and other terms, including relational expressions). Three experiments corroborate the model theory's prediction that inferences that require the construction of only 1 model will be easier than those that require more than 1 model. The model theory assumes that the logical properties of quantifiers emerge from their meanings and are not mentally represented in rules of inference. How such a semantic process can occur compositionally (i.e., guided by the syntactic analysis of sentences) is described. Deductive reasoning is a process of thought that yields new information from old and aims to establish valid conclusions, that is, conclusions that are necessarily true given the truth of the initial premises or observations. The study of its underlying mental mechanisms is almost as old as experimental psychology, but remains a matter of controversy. There are three main views that have been proposed in both cognitive psychology and artificial intelligence. First, the reasoning mechanism depends on formal rules of inference; second, it depends on content-specific rules of inference; and, third, it depends on semantic procedures that search for interpretations (or models) of the premises that are counterexamples to conclusions. The principal goal of this article is to establish a theory of deductive reasoning for a major class of inferences that has not been investigated before by psychologists: those that depend on multiply-quantified premises. There are theories of relational reasoning and of syllogistic reasoning (i.e., from singly quantified premises), but multiple quantification is more powerful, and its logical analysis calls for the full resources of that branch of logic known as the first-order predicate calculus. In this article we develop a theory based on the manipulation of models and report experimental evidence that confirms this theory. An example of a multiply-quantified assertion is as follows: None of the artists is than any of the beekeepers. Such assertions contain a relational expression—here, a two-place relation, taller than; its arguments are quantified by such expressions as all, some, none, and any. These quantifiers behave in ways that are similar to the quantifiers of the firstorder predicate calculus, but there are other nonstandard

A prolog technology theorem prover: Implementation by an extended prolog compiler

A Prolog technology theorem prover (PTTP) is an extension of Prolog that is complete for the full first-order predicate calculus. It differs from Prolog in its use of unification with the occurs check for soundness, the model-elimination reduction rule that is added to Prolog inferences to make the inference system complete, and depth-first iterative-deepening search instead of unbounded depthfirst search to make the search strategy complete. A Prolog technology theorem prover has been implemented by an extended Prolog-to-LISP compiler that supports these additional features. It is capable of proving theorems in the full first-order predicate calculus at a rate of thousands of inferences per second.

The Formal Verification of Safety-critical Assembly Code

This paper describes the use of the SPADE static analysis and verification tools to model, analyse and formally verify the LUCOL assembly code modules used in the fuel control unit of the Rolls-Royce RB211-524G jet engine. This is probably the first tool-supported formal proof of correctness of safety-critical assembly code in the U.K. The work described is further novel in a number of aspects: the construction of a rigorous model of Z8002 assembly code amenable to analysis and formal verification by SPADE, the rapid development in Prolog of a translator from Z8002 assembly code to FDL (SPADE’s modelling language), the formalisation of the written specifications provided into pre- and post-conditions expressed in first-order predicate calculus, and the use of the SPADE Proof Checker to carry out the proofs of correctness.

A many-valued logic for approximate reasoning

A new system of many-valued logic, the Extended Post system of order p, p ≥ 2, is proposed as a system of logic supporting reasoning with facts and rules which are reliable to a specified extent. In an Extended Post system there are as many operations of logical disjunction and logical conjunction as there are truth values. The truth value associated with a particular operation of disjunction (conjunction) acts as a threshold value controlling the behavior of the operation. The availability of an extended set of logical operations provides improved flexibility in the symbolic translation of sentences from the ordinary word-language. Extended Post systems are equipped with a semantics in which graded rather than crisp sets correspond to predicates. The system exhibits a rich algebraic structure. The p operations of disjunction form a distributivity cycle. To each disjunction there corresponds a dual operation of conjunction, the two operations being distributive to one another. The p conjunctions form a dual distributivity cycle. Both propositional calculus and first-order predicate calculus of EP systems are developed. The application to approximate reasoning is described. It is shown that there exist distinct isomorphic copies of fuzzy logic, each corresponding to a distinct level of approximation and being complete to resolution.

A logical-form and knowledge-base design for natural language generation: Norman K. Sondheimer and Bernhard Nebel Information Sciences Institute, Marina del Rey, November 1986

This paper presents a technique for interpreting output demands by a natural language sentence generator in a formally transparent and efficient way. These demands are stated in a logical language. A network knowledge base organizes the concepts of the application domain into categories known to the generator. The logical expressions are interpreted by the generator using the knowledge base and a restricted, but efficient, hybrid knowledge representation system. This design has been used to allow the NIGEL generator to interpret statements in a first-order predicate calculus using the NIKL and KL-TWO knowledge representation systems. The success of this experiment has led to plans for the inclusion of this design in both the evolving Penman natural language generator and the Janus natural language interface.

Specification statements and refinement

We discuss the development of executable programs from state-based specifications written in the language of first-order predicate calculus. Notable examples of such specifications are those written using the techniques Z and VDM; but our interest will be in the rigorous derivation of the algorithms from which they deliberately abstract. This is of course the role of a development method.

Equality-based binary resolution

A major event in automated reasoning was the introduction by Robinson of resolution as an inference principle that is complete for the first-order predicate calculus. Here the theory of binary resolution, based strictly on unification, is recast to incorporate the axioms of equality. Equality-based binary resolution is complete without making use of paramodulation and leads to refutations that are less than half as long as standard refutations with the equality axioms. A detailed discussion is given of the first major use of a theorem prover based on this new method.

Representation and semiautomatic acquisition of medical knowledge in CADIAG-1 and CADIAG-2

CADIAG-1 and CADIAG-2 (Computer-Assisted DIAGnosis) are medical expert systems especially designed for ill-defined areas such as internal medicine. Both systems are being tested in the setting of a medical information system. With respect to their knowledge representation, CADIAG-1 has obvious advantages in totally ill-defined areas such as syndromes in internal medicine, whereas CADIAG-2 seems more suited for domains with basic laboratory programs, e.g., hepatology or gall bladder and bile duct diseases. The formalization of relationships between medical entities led to first-order predicate calculus formulas in the case of CADIAG-1 and to a model based on fuzzy set theory in the case of CADIAG-2. In both systems two kinds of relationships between medical entities are considered: (1) necessity of occurrence and (2) sufficiency of occurrence. Statistical interpretations using the 2 × 2 table paradigm yield a way to calculate these relationships automatically from samples of patient data. Results obtained by exploiting 3530 patient records from a rheumatological hospital are presented. The described application is a machine-learning program that allows inductive learning from examples under statistical uncertainty.

Truth-Functional Conditionals and Modern vs. Traditional Syllogistic

The failure of nine traditionally valid categorical syllogisms in modern logic,2 as well as the failure of implications, contrariety and sub-contrariety among A, E, I, and 0 propositions with identical subject and predicate terms, is customarily explained by references to the modern doctrine of 'existential import'. But this is a mistake. We wish to show that the differences between what should be counted as laws of logic are due, instead, to the employment of two different conditionals, one truth-functional and one not, in the two logics. By traditional logic we shall mean the standard account given in modern textbooks of the theory of the syllogism as developed from Aristotle's time down through the Port Royale Logic, with 24 valid categorical syllogisms out of 256 possible, and the theory of immediate inference as expressed in the traditional square of opposition, including conversion of E and I, contraposition of A and 0 and obversion of A, E, I, and 0 propositions. By modern logic we mean the logic of Frege, Russell, and Whitehead through the first-order predicate calculus. For convenience we shall use Quine's formulation in giving proofs later in the paper.3

Metalogical control for logic programs

An alternative computational “model” for logic programs is proposed which is based upon control annotations written in first-order predicate calculus. These annotations are used to generate runtime environments within which it is possible to exclude infinite computations. The major advantage of such a model is that it is possible to write conceptually transparent programs which otherwise would not have terminated under an existing naive control mechanism.

An equational approach to theorem proving in first-order predicate calculus

A new approach for proving theorems in first-order predicate calculus is developed based on term rewriting and polynomial simplification methods. A formula is translated into an equivalent set of formulae expressed in terms of 'true', 'false', 'exclusive-or', and 'and' by analyzing the semantics of its top-level operator. In this representation, formulae are polynomials over atomic formulae with 'and' as multiplication and 'exclusive-or' as addition, and they can be manipulated just like polynomials using familiar rules of multiplication and addition. Polynomials representing a formula are converted into rewrite rules which are used to simplify polynomials. New rules are generated by overlapping polynomials using a critical-pair completion procedure closely related to the Knuth- Bendix procedure. This process is repeated until a contradiction is reached or it is no longer possible to generate new rules. It is shown that resolution is subsumed by this method.

Gambit: An Interactive Database Design Tool for Data Structures, Integrity Constraints, and Transactions

The design of a database is a rather complex and dynamic process that requires comprehensive knowledge and experience. There exist many manual design tools and techniques, but the step from a schema to an implementation is still a delicate subject. The interactive database design tool Gambit supports the whole process in an optimal way. It is based on an extended relational-entity relationship model. The designer is assisted in outlining and describing data structures and consistency preserving update transactions. The constraints are formulated using the database programming language Modula/R which is based upon first-order predicate calculus. The update transactions are generated automatically as Modula/R programs and include all defined integrity constraints. They are collected in so-called data modules that represent the only interface to the database apart from read operations. The prototype facility of Gambit allows the designer to test the design of the database. The results can be used as feedback leading to an improvement of the conceptual schema and the transactions.

Refutational theorem proving using term-rewriting systems

In this paper we propose a new approach to theorem proving in first-order logic based on the term-rewriting method. First for propositional calculus, we introduce a canonical term-rewriting system for Boolean algebra. This system enables us to transform the first-order predicate calculus into a form of equational logic, and to develop several complete strategies (both clausal and nonclausal) for first-order theories based on the Knuth-Bendix Completion Procedure. More importantly, our strategies can deal with predicate logic and built-in (equational) theories in a uniform and effective way. We also describe an implementation and comparisons with some other first-order theorem-proving methods.

Probabilities of First-Order Sentences about Unary Functions

Let $f$ be any fixed positive integer and $\sigma$ a sentence in the first-order predicate calculus of $f$ unary functions. For positive integers $n$, an $n$-structure is a model with universe $\{ 0,1, \ldots ,n - 1\}$ and $f$ unary functions, and $\mu (n,\sigma )$ is the ratio of the number of $n$-structures satisfying $\sigma$ to ${n^{nf}}$, the number of $n$-structures. We show that ${\lim _{n \to \infty }}\mu (n,\sigma )$ exists for all such $\sigma$, and its value is given by an expression consisting of integer constants and the operators $+ , - , \cdot ,/$, and ${e^x}$.

Probabilities of first-order sentences about unary functions

Let $f$ be any fixed positive integer and $\sigma$ a sentence in the first-order predicate calculus of $f$ unary functions. For positive integers $n$, an $n$-structure is a model with universe $\{ 0,1, \ldots ,n - 1\}$ and $f$ unary functions, and $\mu (n,\sigma )$ is the ratio of the number of $n$-structures satisfying $\sigma$ to ${n^{nf}}$, the number of $n$-structures. We show that ${\lim _{n \to \infty }}\mu (n,\sigma )$ exists for all such $\sigma$, and its value is given by an expression consisting of integer constants and the operators $+ , - , \cdot ,/$, and ${e^x}$.

Algebraic aspects of deduction theorems

The first known statements of the deduction theorems for the first-order predicate calculus and the classical sentential logic are due to Herbrand [8] and Tarski [14], respectively. The present paper contains an analysis of closure spaces associated with those sentential logics which admit various deduction theorems. For purely algebraic reasons it is convenient to view deduction theorems in a more general form: given a sentential logic C (identified with a structural consequence operation) in a sentential language I, a quite arbitrary set P of formulas of I built up with at most two distinct sentential variables p and q is called a uniform deduction theorem scheme for C if it satisfies the following condition: for every set X of formulas of I and for any formulas α and β, βeC(X∪{{a}}) iff P(α, β) AC(X). [P(α, β) denotes the set of formulas which result by the simultaneous substitution of α for p and β for q in all formulas in P]. The above definition encompasses many particular formulations of theorems considered in the literature to be deduction theorems. Theorem 1.3 gives necessary and sufficient conditions for a logic to have a uniform deduction theorem scheme. Then, given a sentential logic C with a uniform deduction theorem scheme, the lattices of deductive filters on the algebras A similar to the language of C are investigated. It is shown that the join-semilattice of finitely generated (= compact) deductive filters on each algebra A is dually Brouwerian.

The occur-check problem in Prolog

We present a method for preprocessing Prolog programs so that their operational semantics will be given by the first-order predicate calculus. Most Prolog implementations do not use a full unification algorithm, for efficiency reasons. The result is that it is possible to create terms having loops in them, whose semantics is not adequately described by first-order logic. Our method finds places where such loops may be created, and adds tests to detect them. This should not appreciably slow down the execution of most Prolog programs.

Complete problems in the first-order predicate calculus

Some problems concerning the satisfiability of first-order predicate calculus formulae in Schonfinkel-Bernays form provide a natural hierarchy of complete problems for various complexity classes. Also, problems concerning the existence of resolution proofs from sets of clauses not necessarily in Schonfinkel-Bernays form provide another such hierarchy. In this way we obtain problems complete for P, NP, PSPACE, deterministic and nondeterministic exponential, deterministic and nondeterministic double exponential time, and exponential space. The results concerning resolution proofs may have practical implications for the design of resolution theorem proving programs. Also, these results enable us to make precise statements about the relative difficulty of various resolution strategies. Some connections with temporal logic and alternating Turing machines are discussed.