Introducing Quantified Cuts in Logic with Equality

Cut-introduction is a technique for structuring and compressing formal proofs. In this paper we generalize our cut-introduction method for the introduction of quantified lemmas of the form $\forall x.A$ (for quantifier-free $A$) to a method generating lemmas of the form $\forall x_1\ldots\forall x_n.A$. Moreover, we extend the original method to predicate logic with equality. The new method was implemented and applied to the TSTP proof database. It is shown that the extension of the method to handle equality and quantifier-blocks leads to a substantial improvement of the old algorithm

Elimination of Cuts in First-order Finite-valued Logics

A uniform construction for sequent calculi for finite-valued first-order logics with distribution quantifiers is exhibited. Completeness, cut-elimination and midsequent theorems are established. As an application, an analog of Herbrand’s theorem for the four-valued knowledge-representation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information

Bounded Quantifier Instantiation for Checking Inductive Invariants

We consider the problem of checking whether a proposed invariant $\varphi$ expressed in first-order logic with quantifier alternation is inductive, i.e. preserved by a piece of code. While the problem is undecidable, modern SMT solvers can sometimes solve it automatically. However, they employ powerful quantifier instantiation methods that may diverge, especially when $\varphi$ is not preserved. A notable difficulty arises due to counterexamples of infinite size. This paper studies Bounded-Horizon instantiation, a natural method for guaranteeing the termination of SMT solvers. The method bounds the depth of terms used in the quantifier instantiation process. We show that this method is surprisingly powerful for checking quantified invariants in uninterpreted domains. Furthermore, by producing partial models it can help the user diagnose the case when $\varphi$ is not inductive, especially when the underlying reason is the existence of infinite counterexamples. Our main technical result is that Bounded-Horizon is at least as powerful as instrumentation, which is a manual method to guarantee convergence of the solver by modifying the program so that it admits a purely universal invariant. We show that with a bound of 1 we can simulate a natural class of instrumentations, without the need to modify the code and in a fully automatic way. We also report on a prototype implementation on top of Z3, which we used to verify several examples by Bounded-Horizon of bound 1

Formulas as Programs

We provide here a computational interpretation of first-order logic based on a constructive interpretation of satisfiability w.r.t. a fixed but arbitrary interpretation. In this approach the formulas themselves are programs. This contrasts with the so-called formulas as types approach in which the proofs of the formulas are typed terms that can be taken as programs. This view of computing is inspired by logic programming and constraint logic programming but differs from them in a number of crucial aspects. Formulas as programs is argued to yield a realistic approach to programming that has been realized in the implemented programming language ALMA-0 (Apt et al.) that combines the advantages of imperative and logic programming. The work here reported can also be used to reason about the correctness of non-recursive ALMA-0 programs that do not include destructive assignment.Comment: 34 pages, appears in: The Logic Programming Paradigm: a 25 Years Perspective, K.R. Apt, V. Marek, M. Truszczynski and D.S. Warren (eds), Springer-Verlag, Artificial Intelligence Serie

Set-based design of mechanical systems with design robustness integrated

This paper presents a method for parameter design of mechanical products based on a set-based approach. Set-based concurrent engineering emphasises on designing in a multi-stakeholder environment with concurrent involvement of the stakeholders in the design process. It also encourages flexibility in design through communication in terms of ranges instead of fixed point values and subsequent alternative solutions resulting from intersection of these ranges. These alternative solutions can then be refined and selected according to the designers’ preferences and clients’ needs. This paper presents a model and tools for integrated flexible design that take into account the manufacturing variations as well as the design objectives for finding inherently robust solutions using QCSP transformation through interval analysis. In order to demonstrate the approach, an example of design of rigid flange coupling with a variable number of bolts and a choice of bolts from ISO M standard has been resolved and demonstrated

Decomposable Theories

We present in this paper a general algorithm for solving first-order formulas in particular theories called "decomposable theories". First of all, using special quantifiers, we give a formal characterization of decomposable theories and show some of their properties. Then, we present a general algorithm for solving first-order formulas in any decomposable theory "T". The algorithm is given in the form of five rewriting rules. It transforms a first-order formula "P", which can possibly contain free variables, into a conjunction "Q" of solved formulas easily transformable into a Boolean combination of existentially quantified conjunctions of atomic formulas. In particular, if "P" has no free variables then "Q" is either the formula "true" or "false". The correctness of our algorithm proves the completeness of the decomposable theories. Finally, we show that the theory "Tr" of finite or infinite trees is a decomposable theory and give some benchmarks realized by an implementation of our algorithm, solving formulas on two-partner games in "Tr" with more than 160 nested alternated quantifiers