A catalog of Boolean concepts

Our terminology and notation is the same as that of [1], of which this note is a sequel.We wish to show that if we take the natural deduction system (N) described in [1], and delete the rules for the quantifiers, we obtain a complete system for propositional logic. [Of course we now construe “X”, “Y”, “Z” as syntactic variables ranging over formulas of propositional logic, rather than sentences of quantification theory.] Moreover the system serves as a neat decision procedure.

In this paper we describe a teaching tool for proving equivalences between propositional logic formulae, using rewrite rules such as De Morgan’s laws and double negation. This tool is based on an earlier tool for rewriting logical formulae into disjunctive normal form (DNF). Both tools make use of a rewrite strategy, which specifies how an exercise can be solved stepwise. Different types of feedback can be calculated automatically from such a strategy specification. We describe a strategy for constructing expert-like equivalence proofs, and present two techniques for improving the proofs that are generated by the strategy.Keywordspropositional logicequivalencese-learningfeedback

Space Bounds for Resolution

On the hydra number of disconnected graphs

AI systems often need to deal with inconsistent information. For this reason since the early 2000s some AI researchers have developed ways to measure the amount of inconsistency in a knowledge base. By now there is a substantial amount of research about various aspects of inconsistency measuring. The problem is that most of this work applies only to knowledge bases formulated as sets of formulas in propositional logic. Hence this work is not really applicable to the way that information is actually stored. The purpose of this paper is to extend inconsistency measuring to real world information. We first define the concept of general information space which encompasses various types of databases and scenarios in AI systems. Then, we show how to transform any general information space to an inconsistency equivalent propositional knowledge base, and finally apply propositional inconsistency measures to find the inconsistency of the general information space. Our method allows for the direct comparison of the inconsistency of different information spaces, even though the data is presented in different ways. We demonstrate the transformation on three general information spaces: a relational database, a graph database, and a Blocks world scenario, where we apply several inconsistency measures after performing the transformation.

We describe an incremental algorithm for computing interpolants for a pair ϕ A , ϕ B of formulas in propositional logic. In contrast with the common approaches, our method does not require a proof of unsatisfiability of ϕ A ∧ ϕ B , and can be realized using any SAT solver as a black box. We achieve this by combining model enumeration with the ability to easily generate interpolants in the special case that one of the formulas is a cube.

Cut formulas in propositional logic

The article considers non-standard semantics for syllogistic systems. In these systems, the formulas of propositional logic are the meanings of general terms, and the values of categorical statements are determined by the logical relationship between these formulas. If formula A is the meaning of term S, and if formula B is the meaning of term P, then the formula “SaP” is valid if and only if A entails B. Otherwise, the formula of the form “SoP” is valid. The “SeP” formula is valid if and only if A entails ¬B, otherwise the “SiP” formula is valid. This semantics is based on the intensional approach to the construction of syllogistic theories that goes back to Leibniz’s ideas. Firstly, the entailment relation by which categorical statements are interpreted allows, as Voishvillo has shown, an intensional interpretation: A entails B if the logical content of B is a part of the logical content of A. Secondly, propositional formulas can be considered as abbreviations for quantifier-free first-degree formulas with a single free variable, and such formulas, according to Voishvillo, state the contents of concepts. Therefore, in this semantics, the value of a syllogistic formula is determined not by the relationship between the extensions of the subject and the predicate, but by the relationship between their intensions. This article shows how different types of entailment affect the class of universally valid syllogistic formulas. Theories that use the relations of classical entailment and FDE-entailment have already been constructed. The classical entailment can be defined on the set of Carnap-style (consistent and complete) state-descriptions, and the first-degree relevant entailment can be defined on the set of generalized state-descriptions (the latter can be either contradictory or incomplete). In the article, semantics using two other relations: the Hao Wang-style entailment logic (defined on the set of all consistent state-descriptions) and its dual logic entailment (defined on the set of all complete state-descriptions) were built. The deductive capabilities of four syllogistic systems that use different types of logical entailments were compared.

Reaction systems are a qualitative formalism for modeling systems of biochemical reactions characterized by the non-permanency of the elements: molecules disappear if not produced by any enabled reaction. Reaction systems execute in an environment that provides new molecules at each step. Brijder, Ehrenfeucht and Rozemberg introduced the idea of predictors. A predictor of a molecule s, for a given n, is the set of molecules to be observed in the environment to determine whether s is produced or not at step n by the system. We introduced the notion of formula based predictor, that is a propositional logic formula that precisely characterizes environments that lead to the production of s after n steps. In this paper we revise the notion of formula based predictor by defining a specialized version that assumes the environment to provide molecules according to what expressed by a temporal logic formula. As an application, we use specialized formula based predictors to give theoretical grounds to previously obtained results on a model of gene regulation.

There exists a wide range of constraint programming (CP) problems defined on Boolean functions depending on binary variables. One of the approaches to solving CP problems is using specific appropriate solvers, e.g., SAT solvers. An alternative is using the generic solvers for mixed-integer linear programming problems (MILP), but they require transforming expressions with Boolean functions to linear equations or inequalities. Here, we present two methods of such a transformation which applies to any Boolean function defined by explicit rules giving values of the Boolean function for all combinations of its Boolean variables. The first method represents the Boolean function as a linear equation in the original binary variables and, possibly, binary ancillaries, which become additional variables of the MILP problem being composed. The second method represents the Boolean function as a set of linear inequalities in the original binary variables and one additional continuous variable (representing the value of the function). The choice between the first or second method is a trade-off between the number of binary variables and number of linear constraints in the emerging MP problem. The advantage of the proposed approach is that both methods reduce important cryptanalysis problems, such as the preimaging of hash functions or breaking symmetric ciphers as the MILP problems, which are solved by the generic MILP solvers. Furthermore, the first method enables to reduce the binary linear equations to quadratic unconstrained binary optimization (QUBO), by the quantum annealer, e.g., D-Wave.

We explore three applications of geometric sequences in constructing cryptographic Boolean functions. First, we construct 1-resilient functions of n Boolean variables with nonlinearity 2n-1-2(n-1)/2, n odd. The Hadamard transform of these functions is 3-valued, which limits the efficiency of certain stream cipher attacks. From the case for n odd, we construct highly nonlinear 1-resilient functions which disprove a conjecture of Pasalic and Johansson for n even. Our constructions do not have a potential weakness shared by resilient functions which are formed from concatenation of linear functions. Second, we give a new construction for balanced Boolean functions with high nonlinearity, exceeding 2n-1-2(n-1)/2, which is not based on the direct sum construction. Moreover, these functions have high algebraic degree and large linear span. Third, we construct balanced vectorial Boolean functions with nonlinearity 2n-1-2(n-1)/2 and low maximum correlation. They can be used as nonlinear combiners for stream cipher systems with high throughput.

In BDI programs it is quite common to find context conditions of plans which are over-constrained in order to ensure that the most preferred plan is selected for use. This is undesirable for at least two reasons. It makes the plan not available for use at all in situations where it could be of value as a back-up plan, and also it requires incorporation of information that conceptually belongs with the preferred plan. The ability to specify directly in a plan specification, aspects of the situation which would make the plan more or less desirable, enables a dynamically calculated preference ordering which removes the need to over-constrain applicability to obtain the desired plan selection.This paper addresses the issue of dynamically assigning a value to a plan instance, based on the current state and the particulars of the plan instance under consideration. The framework uses specifications based on logical formulae which are evaluated dynamically, using the current state and variable bindings provided via the plan's context condition. These provide a simple mechanism for locally specifying the value of plan instances. This can be regarded as providing a degree of applicability for a plan, rather than simply a boolean value.

We investigate the problem of learning Boolean functions with a short DNF representation using decision trees as a concept description language. Unfortunately, Boolean concepts with a short description may not have a small decision tree representation when the tests at the nodes are limited to the primitive attributes. This representational shortcoming may be overcome by using Boolean features at the decision nodes. We present two new methods that adaptively introduce relevant features while learning a decision tree from examples. We show empirically that these methods outperform a standard decision tree algorithm for learning small random DNF functions when the examples are drawn at random from the uniform distribution.

In the last decade, advances in satisfiability-modulo-theories (SMT) solvers have powered a new generation of software tools for verification and testing. These tools transform various program analysis problems into the problem of satisfiability of formulas in propositional or first-order logic, where they are discharged by SMT solvers, such as Z3 from Microsoft Research. This paper briefly summarizes four initiatives from Microsoft Research that build upon Z3 and move beyond first-order satisfaction: Fixed points—μZ is a scalable, efficient engine for discharging fixed point queries over recursive predicates with logical constraints, integrated in Z3; Interpolants—Interpolating Z3 uses Z3’s proof generation capability to generate Craig interpolants in the first-order theory of uninterpreted functions, arrays and linear arithmetic; Automata—The symbolic automata toolkit lifts classical automata analyses to work modulo symbolic constraints on alphabets; Polynomials—a new decision procedure for the existential theory of the reals allows efficient solving of systems of non-linear arithmetic constraints.KeywordsPoint QueryAbstract DomainSymbolic Model CheckLinear ArithmeticUninterpreted FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Formal verification involves the use of logical and computational methods to establish claims that are expressed in precise mathematical terms. These can include ordinary mathematical theorems, as well as claims that pieces of hardware or software, network protocols, and mechanical and hybrid systems meet their specifications. In practice, there is not a sharp distinction between verifying a piece of mathematics and verifying the correctness of a system: formal verification requires describing hardware and software systems in mathematical terms, at which point establishing claims as to their correctness becomes a form of theorem proving. Conversely, the proof of a mathematical theorem may require a lengthy computation, in which case verifying the truth of the theorem requires verifying that the computation does what it is supposed to do. The gold standard for supporting a mathematical claim is to provide a proof, and twentieth-century developments in logic show most if not all conventional proof methods can be reduced to a small set of axioms and rules in any of a number of foundational systems. With this reduction, there are two ways that a computer can help establish a claim: it can help find a proof in the first place, and it can help verify that a purported proof is correct. Automated theorem proving focuses on the “finding” aspect. Resolution theorem provers, tableau theorem provers, fast satisfiability solvers, and so on provide means of establishing the validity of formulas in propositional and first-order logic. Other systems provide search procedures and decision procedures for specific languages and domains, such as linear or nonlinear expressions over the integers or the real numbers. Architectures like SMT (“satisfiability modulo theories”) combine domain-general search methods with domain-specific procedures. Computer algebra systems and specialized mathematical software packages provide means of carrying out mathematical computations, establishing mathematical bounds, or finding mathematical objects. A calculation can be viewed as a proof as well, and these systems, too, help establish mathematical claims. Automated reasoning systems strive for power and efficiency, often at the expense of guaranteed soundness. Such systems can have bugs, and it can be difficult to ensure that the results they deliver are correct. In contrast, interactive theorem proving focuses on the “verification” aspect of theorem proving, requiring that every claim is supported by a proof in a suitable axiomatic foundation. This sets a very high standard: every rule of inference and every step of a calculation has to be justified by appealing to prior definitions and theorems, all the way down to basic axioms and rules. In fact, most such systems provide fully elaborated “proof objects” that can be communicated to other systems and checked independently. Constructing such proofs typically requires much more input and interaction from users, but it allows us to obtain deeper and more complex proofs. The Lean Theorem Prover aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. The goal is to support both mathematical reasoning and reasoning about complex systems, and to verify claims in both domains.