Resolution-based Theorem Proving for Many-valued Logics

There are three main families of inference algorithms in first-order logic: direct inference and its application to deductive databases and production systems; backward inference procedures and logic programming systems; theorem proving systems based on the resolution method. When solving specific problems, the most effective algorithms are those that allow you to cover all the facts and axioms and must be taken into account in the process of inference. An example is considered in which it is necessary to prove the guilt of a person in murder. On the basis of statements, a knowledge base is formed from expressions, with the help of which an expression of first-order logic is compiled and proved using direct logical inference. The proof of the reasoning obtained in direct inference using the proof tree is given. However, direct inference provides for the implementation of all admissible stages of logical inference based on all known facts. The article also considers a method based on the resolution when implementing the reverse inference, taking into account the expression obtained in the direct inference. This expression is converted into a conjunctive normal formula using the laws of Boolean algebra and is proved by the elimination of events using the conjunction operation.

Automated theorem provers for first-order logic usually operate on sets of first-order clauses. It is well-known that the translation of a formula in full first-order logic to a clausal normal form (CNF) can crucially affect performance of a theorem prover. In our recent work we introduced a modification of first-order logic extended by the first class boolean sort and syntactical constructs that mirror features of programming languages. We called this logic FOOL. Formulas in FOOL can be translated to ordinary first-order formulas and checked by first-order theorem provers. While this translation is straightforward, it does not result in a CNF that can be efficiently handled by state-of-the-art theorem provers which use superposition calculus. In this paper we present a new CNF translation algorithm for FOOL that is friendly and efficient for superposition-based first-order provers. We implemented the algorithm in the Vampire theorem prover and evaluated it on a large number of problems coming from formalisation of mathematics and program analysis. Our experimental results show an increase of performance of the prover with our CNF translation compared to the naive translation.

The universe of propositional approximations

A logic family is a bunch of logics that belong together in some way. First-order logic is one of the examples. Logics organized into a structure occur in abstract model theory, institution theory and in algebraic logic. Logic families play a role in adopting methods for investigating sentential logics to first-order like logics. We thoroughly discuss the notion of logic families as defined in the recent Universal Algebraic Logic book.

Given a finite set of vectors over a finite totally ordered domain, we study the problem of computing a constraint in conjunctive normal form such that the set of solutions for the produced constraint is identical to the original set. We develop an efficient polynomial-time algorithm for the general case, followed by specific polynomial-time algorithms producing Horn, dual Horn, and bijunctive formulas for sets of vectors closed under the operations of conjunction, disjunction, and median, respectively. Our results generalize the work of Dechter and Pearl on relational data, as well as the papers by Hébrard and Zanuttini. They complement the results of Hähnle et al. on multivalued logics and Jeavons et al. on the algebraic approach to constraints.

Chapter 20 - Automated Deduction for Many-Valued Logics

Dynamic logic is a powerful framework for reasoning about imperative programs. An extension with a concurrent operator, called concurrent propositional dynamic logic (CPDL) [20], was introduced to formalise programs running in parallel. In a different direction, other authors proposed a systematic method for generating multi-valued propositional dynamic logics to reason about weighted programs [15]. This paper presents the first step of combining these two frameworks to introduce uncertainty in concurrent computations. In the proposed framework, a weight is assigned to each branch of the parallel execution, resulting in a (possible) asymmetric parallelism, inherent to the fuzzy programming paradigm [2, 23]. By adopting such an approach, a family of logics is obtained, called multi-valued concurrent propositional dynamic logics (\(\mathcal {GCDL}(\mathbf {A})\)), parametric on an action lattice \(\mathbf {A}\) specifying a notion of “weight” assigned to program execution. Additionally, the validity of some axioms of CPDL is discussed in the new family of generated logics.

One aspect that has been poorly studied in multiple-valued logics, and in particular in Lukasiewicz logics, is the generation of instances of varying difficulty for evaluating, comparing and improving satisfiability solvers. In this paper we present a new class of clausal forms, called Lukasiewicz (L-)clausal forms, motivate their usefulness, study their complexity, and report on an empirical investigation that shows an easy-hard-easy pattern and a phase transition phenomenon when testing the satisfiability of L-clausal forms.

This paper examines the questions of functional completeness and canonical completeness in many-valued logics, offering proofs for several theorems on these topics.A skeletal description of the domain for these theorems is as follows. We are concerned with a proper logic L, containing a denumerably infinite class of propositional symbols, P, Q, R, …, a finite set of unary operations, U1, U2,…, Ub, and a finite set of binary operations, B1, B2, …, Bc. Well-formed formulas in L are recursively defined by the conventional set of rules. With L there is associated an integer, M ≧ 2, and the integers m, where (1 ≦m≦M), are the truth values of L.

The language of signed formulas offers a first-order classical logic framework for automated reasoning in multiple-valued logics. It is sufficiently general to include both annotated logics and fuzzy operator logics. Signed resolution unifies the two inference rules of annotated logics, thus enabling the development of an SLD-style proof procedure for annotated logic programs. Signed resolution also captures fuzzy resolution. The logic of signed formulas offers a means of adapting most classical inference techniques to multiple-valued logics.

We outline a method for proving theorems in first-order logic with equality using some equational logics and their associated multiple-valued propositional logics, and describe an application of the method that makes use of the automated theorem-prover Otter to prove a range of theorems from the TPTP library of problems in first-order logic with equality.

There are a number of board games such as Checkers [2], Go [5], and Gobang [8], which are known to be PSPACE-hard. This means that the problem to determine the player having a winning strategy in a given situation on an nA—n board of one of these games is as hard to solve as everything computable in polynomial space. PSPACE-completeness has been previously proven for some combinatorial games played on graphs or by logical formulas [1, 9]. In this paper we will show that the same holds for the game of Hex. The crucial point of the proof is to establish PSPACE-hardness for a generalization of Hex played on planar graphs. This will be done by showing that the problem, whether a given quantified Boolean formula in conjunctive normal form is true, is polynomial time-reducible to the decision problem for generalized Hex. In order to do this we will use methods, which were also used to prove PSPACE-completeness of planar Geography in [5]. Therefore our proof is quite different from the proof provided by Even and Tarjan [1], who showed PSPACE-completeness of generalized Hex played on arbitrary graphs. Since it is easy to see that the decision problem for Hex is in PS PACE, the decision problem for Hex is PSPACE-complete.

In this paper, we present a system that translates first order logic (FOL) formulas into natural language (NL) sentences. The motivation comes from an intelligent tutoring system teaching logic as a knowledge representation language, where it is used as a means for feedback to the users. FOL to NL conversion is achieved by using a rule-based approach, where we exploit the pattern matching capabilities of rules. So, the system consists of a rule-based component and a lexicon. The rule-based unit implements the conversion process, which is based on a linguistic analysis of a FOL sentence, and the lexicon provides lexical and grammatical information that helps in producing the NL sentences. The whole system is implemented in Jess, a java-based expert system shell. The conversion process currently covers a restricted set of FOL formulas.

Many-valued logics in general and 3-valued logic in particular is an old subject which had its beginning in the work of Łukasiewicz [Łuk]. Recently there is a revived interest in this topic, both for its own sake (see, for example, [Ho]), and also because of its potential applications in several areas of computer science, such as proving correctness of programs [Jo], knowledge bases [CP] and artificial intelligence [Tu]. There are, however, a huge number of 3-valued systems which logicians have studied throughout the years. The motivation behind them and their properties are not always clear, and their proof theory is frequently not well developed. This state of affairs makes both the use of 3-valued logics and doing fruitful research on them rather difficult.Our first goal in this work is, accordingly, to identify and characterize a class of 3-valued logics which might be called natural. For this we use the general framework for characterizing and investigating logics which we have developed in [Av1]. Not many 3-valued logics appear as natural within this framework, but it turns out that those that do include some of the best known ones. These include the 3-valued logics of Łukasiewicz, Kleene and Sobociński, the logic LPF used in the VDM project, the logic RM3 from the relevance family and the paraconsistent 3-valued logic of [dCA]. Our presentation provides justifications for the introduction of certain connectives in these logics which are often regarded as ad hoc. It also shows that they are all closely related to each other. It is shown, for example, that Łukasiewicz 3-valued logic and RM3 (the strongest logic in the family of relevance logics) are in a strong sense dual to each other, and that both are derivable by the same general construction from, respectively, Kleene 3-valued logic and the 3-valued paraconsistent logic.

Many-valued logic based interval reasoning plays an increasingly important role in fuzzy and other many-valued extensions of two-valued (crisp) logic. It is often overlooked that many-valued logic (MVL) reasoning has a richer inference rule base than the crisp (i.e. classical two-valued) logic. Indeed some rules that are not possible in the crisp logic (which is point-based) do come to existence when we accept intervals as the basic elements of the semantic space ([0, 1] or a more general lattice) into which the logic expressions are valuated [11, 8, 37].