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In this paper by using a model-theoretic approach, we prove Craig interpolation property for Formal Propositional Logic, FPL, Basic propositional logic, BPL and the uniform left-interpolation property for FPL. We also show that there are countably infinite extensions of FPL with the uniform interpolation property.

Machine learning methods are growing in relevance for biometrics and personal information processing in domains such as forensics, e-health, recruitment, and e-learning. In these domains, white-box (human-readable) explanations of systems built on machine learning methods become crucial. Inductive logic programming (ILP) is a subfield of symbolic AI aimed to automatically learn declarative theories about the processing of data. Learning from interpretation transition (LFIT) is an ILP technique that can learn a propositional logic theory equivalent to a given black-box system (under certain conditions). The present work takes a first step to a general methodology to incorporate accurate declarative explanations to classic machine learning by checking the viability of LFIT in a specific AI application scenario: fair recruitment based on an automatic tool generated with machine learning methods for ranking Curricula Vitae that incorporates soft biometric information (gender and ethnicity). We show the expressiveness of LFIT for this specific problem and propose a scheme that can be applicable to other domains. In order to check the ability to cope with other domains no matter the machine learning paradigm used, we have done a preliminary test of the expressiveness of LFIT, feeding it with a real dataset about adult incomes taken from the US census, in which we consider the income level as a function of the rest of attributes to verify if LFIT can provide logical theory to support and explain to what extent higher incomes are biased by gender and ethnicity.

Proof-theoretic methods are developed for subsystems of Johansson's logic obtained by extending the positive fragment of intuitionistic logic with weak negations. These methods are exploited to establish properties of the logical systems. In particular, cut-free complete sequent calculi are introduced and used to provide a proof of the fact that the systems satisfy the Craig interpolation property. Alternative versions of the calculi are later obtained by means of an appropriate loop-checking history mechanism. Termination of the new calculi is proved, and used to conclude that the considered logical systems are PSPACE-complete.

We introduce a new formal logic, called equality propositional logic. It has two basic connectives, $\\boldsymbol{\\wedge}$ (conjunction) and $\\equiv$ (equivalence). Moreover, the $\\Rightarrow$ (implication) connective can be derived as $A\\Rightarrow B:=(A\\boldsymbol{\\wedge}B)\\equiv A$. We formulate the equality propositional logic and demonstrate that the resulting logic has reasonable properties such as Modus Ponens(MP) rule, Hypothetical Syllogism(HS) rule and completeness, etc. Especially, we provide two ways to prove the completeness of this logic system. We also introduce two extensions of equality propositional logic. The first one is involutive equality propositional logic, which is equality propositional logic with double negation. The second one adds prelinearity which is rich enough to enjoy the strong completeness property. Finally, we introduce additional connective $\\Delta$(delta) in equality propositional logic and demonstrate that the resulting logic holds soundness and completeness.

We present sound and complete semantics and a sequent calculus for the Lambek calculus extended with intuitionistic propositional logic.

We show that the relational semantics of the Lambek calculus, both nonassociative and associative, is also sound and complete for its extension with classical propositional logic. Then, using nitrations, we obtain the finite model property for the nonassociative Lambek calculus extended with classical propositional logic.

Relying simply on bitwise operators, the recently introduced Tsetlin machine (TM) has provided competitive pattern classification accuracy in several benchmarks, including text understanding. In this paper, we introduce the regression Tsetlin machine (RTM), a new class of TMs designed for continuous input and output, targeting nonlinear regression problems. In all brevity, we convert continuous input into a binary representation based on thresholding, and transform the propositional formula formed by the TM into an aggregated continuous output. Our empirical comparison of the RTM with state-of-the-art regression techniques reveals either superior or on par performance on five datasets. This article is part of the theme issue ‘Harmonizing energy-autonomous computing and intelligence’.

We give a new proof of the following result (originally due to Linial and Post): it is undecidable whether a given calculus, that is a finite set of propositional formulas together with the rules of modus ponens and substitution, axiomatizes the classical logic. Moreover, we prove the same for every superintuitionistic calculus. As a corollary, it is undecidable whether a given calculus is consistent, whether it is superintuitionistic, whether two given calculi have the same theorems, whether a given formula is derivable in a given calculus. The proof is by reduction from the undecidable halting problem for the so-called tag systems introduced by Post. We also give a historical survey of related results.

Research on intelligent tutoring systems has been exploring data-driven methods to deliver effective adaptive assistance. While much work has been done to provide adaptive assistance when students seek help, they may not seek help optimally. This had led to the growing interest in proactive adaptive assistance, where the tutor provides unsolicited assistance upon predictions of struggle or unproductivity. Determining when and whether to provide personalized support is a well-known challenge called the assistance dilemma. Addressing this dilemma is particularly challenging in open-ended domains, where there can be several ways to solve problems. Researchers have explored methods to determine when to proactively help students, but few of these methods have taken prior hint usage into account. In this paper, we present a novel data-driven approach to incorporate students’ hint usage in predicting their need for help. We explore its impact in an intelligent tutor that deals with the open-ended and well-structured domain of logic proofs. We present a controlled study to investigate the impact of an adaptive hint policy based on predictions of HelpNeed that incorporate students’ hint usage. We show empirical evidence to support that such a policy can save students a significant amount of time in training and lead to improved posttest results, when compared to a control without proactive interventions. We also show that incorporating students’ hint usage significantly improves the adaptive hint policy’s efficacy in predicting students’ HelpNeed, thereby reducing training unproductivity, reducing possible help avoidance, and increasing possible help appropriateness (a higher chance of receiving help when it was likely to be needed). We conclude with suggestions on the domains that can benefit from this approach as well as the requirements for adoption.

In the decision making logic it is often necessary solving of logical equations for which, due to the features of disjunction and conjunction, no admissible solutions exist. An approach is suggested in which by introducing of Imaginary Logical Variables (ILV) the classical propositional logic is extended to a complex one. This provides a possibility to solve a large class of logical equations. The real and imaginary variables each satisfy the axioms of the Boolean algebra and of the lattice. It is shown that the Complex Logical Variables (CLV) observe the requirements of the Boolean algebra and the lattice axioms. Suitable definitions are found for these variables for the operations disjunction, conjunction, and negation. A series of results are obtained, included also the truth tables of the operations disjunction, conjunction, negation, implication, and equivalence for complex variables. Inference rules are deduced for them analogous to Modus Ponens and Modus Tollens in the classical propositional logic. Values of the complex variables are obtained, corresponding to TRUE (T) and FALSE (F) in the classic propositional logic. A conclusion may be made from the initial assumptions and the results attained, that the imaginary logical variable i introduced hereby is “truer” than the condition “T” of the classic propositional logic and ¬i—“falser” than the condition “F”, respectively. Possibilities for further investigations of this class of complex logical structures are pointed out.