Proof Generation from Delta-Decisions

The ability of a theorem prover to generate explicit derivations for the theorems it proves has major benefits for the testing and maintenance of the prover. It also eliminates the need to trust the correctness of the prover at the expense of trusting a much simpler proof checker. However, it is not always obvious how to generate explicit proofs in a theorem prover that uses decision procedures whose operation does not directly model the axiomatization of the underlying theories. In this paper we describe the modifications that are necessary to support proof generation in a congruence-closure decision procedure for equality and in a Simplex-based decision procedure for linear arithmetic. Both of these decision procedures have been integrated using a modified Nelson-Oppen cooperation mechanism in the Touchstone theorem prover, which we use to produce proof-carrying code. Our experience with designing and implementing Touchstone is that proof generation has a relatively low cost in terms of design complexity and proving time and we conclude that the software-engineering benefits of proof generation clearly outweighs these costs.

We propose a practical path-based framework for deriving and simplifying source-tracking information for term unification in the empty theory. Such a framework is useful for debugging unification-based systems, including the diagnosis of ill-typed programs and the generation of success and failure proofs in logic programming.The objects of source-tracking are deductions in the logic of unification. The semantics of deductions are paths over a unification graph whose labels form the language of suffixes of a semi-Dyck set. Based on this framework, two algorithms for generating proofs are presented: the first uses context-free shortest-path algorithms to generate optimal (shortest) proofs in time O(n 3), where n is the number of vertices of the unification graph. The second algorithm integrates easily with standard unification algorithms, entailing an overhead of only a constant factor, but generates non-optimal proofs. These non-optimal proofs may be further simplified by group rewrite rules.

We address the problem of determining the satisfiability of a Boolean combination of convex constraints over the real numbers, which is common in the context of hybrid system verification and control. We first show that a special type of logic formulas, termed monotone Satisfiability Modulo Convex (SMC) formulas, is the most general class of formulas over Boolean and nonlinear real predicates that reduce to convex programs for any satisfying assignment of the Boolean variables. For this class of formulas, we develop a new satisfiability modulo convex optimization procedure that uses a lazy combination of SAT solving and convex programming to provide a satisfying assignment or determine that the formula is unsatisfiable. Our approach can then leverage the efficiency and the formal guarantees of state-of-the-art algorithms in both the Boolean and convex analysis domains. A key step in lazy satisfiability solving is the generation of succinct infeasibility proofs that can support conflict-driven learning and decrease the number of iterations between the SAT and the theory solver. For this purpose, we propose a suite of algorithms that can trade complexity with the minimality of the generated infeasibility certificates. Remarkably, we show that a minimal infeasibility certificate can be generated by simply solving one convex program for a sub-class of SMC formulas, namely ordered positive unate SMC formulas, that have additional monotonicity properties. Perhaps surprisingly, ordered positive unate formulas appear themselves very frequently in a variety of practical applications. By exploiting the properties of monotone SMC formulas, we can then build and demonstrate effective and scalable decision procedures for problems in hybrid system verification and control, including secure state estimation and robotic motion planning.

This paper describes a tool for Knuth-Bendix completion. In its interactive mode the user only has to select the orientation of equations into rewrite rules; all other computations (including necessary termination checks) are performed internally. Apart from the interactive mode, the tool also provides a fully automatic mode. Moreover, the generation of (dis)proofs in equational logic is supported. Finally, the tool outputs proofs in a certifiable format.

Real or natural number interpretation and their effect on complexity

As part of a project on automatic generation of proofs involving both logic and computation, we have automated the production of some proofs involving epsilon-delta arguments. These proofs involve two or three quantifiers on the logical side, and on the computational side, they involve algebra, trigonometry, and some calculus. At the border of logic and computation, they involve several types of arguments involving inequalities, including transitivity chaining and several types of bounding arguments, in which bounds are sought that do not depend on certain variables. Control mechanisms have been developed for intermixing logical deduction steps with computational steps and with inequality reasoning. Problems discussed here as examples involve the continuity and uniform continuity of various specific functions.

Logic of proofs

Over the last decade, first-order constraints have been efficiently used in the constraint programming world to model many kinds of complex problems such as: scheduling, resource allocation, computer graphics and bio-informatics. Recently, a new property called decomposability has been introduce and many first-order theories have been proved to be decomposable: finite or infinite trees, rational and real numbers, linear dense order, ... etc. A decision procedure in the form of 5 rewriting rules has also been developed but this later can only decide if a formula without free variables is true or not in any decomposable theory T. Unfortunately, this decision procedure is not enough when we want to find the values of the free variables of any first-order constraint φ so that φ is true in any decomposable theory T. These kind of problems are generally known as first-order constraint satisfaction problems. We present in this paper not only a decision procedure but a full first-order constraint solver for any decomposable theory T. Our solver is given in the form of six rewriting rules which transform any first-order constraint (which can possibly contain free variables) into an equivalent solved formula which is either the formula true, or the formula false or a formula having at least one free variable and written in a very simple and explicit solved form. We also show the efficiency of our solver by solving complex first-order constraints containing a huge number of imbricated quantifiers and negations. This is the first full first-order constraint solver over any decomposable theory.

Over the last decade, first-order constraints have been efficiently used in the artificial intelligence world to model many kinds of complex problems such as: scheduling, resource allocation, computer graphics and bio-informatics. Recently, a new property called decomposability has been introduced and many first-order theories have been proved to be decomposable: finite or infinite trees, rational and real numbers, linear dense order,...etc. A decision procedure in the form of 5 rewriting rules has also been developed. This latter can decide if a first-order formula without free variables is true or not in any decomposable theory. Unfortunately, this decision procedure is not enough when we want to express the solutions of a first-order constraint having free variables. These kind of problems are generally known as first-order constraint satisfaction problems. We present in this paper, not only a decision procedure but a full first-order constraint solver for decomposable theories. Our solver is given in the form of nine rewriting rules which transform any first-order constraint ϕ (which can possibly contain free variables) into an equivalent formula φ which is either the formula true, or the formula false or a simple solved formula having at least one free variable and being equivalent neither to true nor to false. We show the efficiency of our solver by solving complex first-order constraints over finite or infinite trees containing a huge number of imbricated quantifiers and negations and compare the performances with those obtained using the most recent and efficient dedicated solver for finite or infinite trees. This is the first full first-order constraint solver for any decomposable theory.

Over the last decade, first-order constraints have been efficiently used in the artificial intelligence world to model many kinds of complex problems such as: scheduling, resource allocation, computer graphics and bio-informatics. Recently, a new property called decomposability has been introduced and many first-order theories have been proved to be decomposable: finite or infinite trees, rational and real numbers, linear dense order,...etc. A decision procedure in the form of five rewriting rules has also been developed. This latter can decide if a first-order formula without free variables is true or not in any decomposable theory. Unfortunately, this decision procedure is not enough when we want to express the solutions of a first-order constraint having free variables. These kind of problems are generally known as first-order constraint satisfaction problems. We present in this paper, not only a decision procedure but a full first-order constraint solver for decomposable theories. Our solver is given in the form of nine rewriting rules which transform any first-order constraint ? (which can possibly contain free variables) into an equivalent formula ? which is either the formula true, or the formula false or a simple solved formula having at least one free variable and being equivalent neither to true nor to false. We show the efficiency of our solver by solving complex first-order constraints over finite or infinite trees containing a huge number of imbricated quantifiers and negations and compare the performances with those obtained using the most recent and efficient dedicated solver for finite or infinite trees. This is the first full first-order constraint solver for any decomposable theory.

Over the last decade, first-order constraints have been efficiently used in the artificial intelligence world to model many kinds of complex problems such as: scheduling, resource allocation, computer graphics and bio-informatics. Recently, a new property called decomposability has been introduced and many first-order theories have been proved to be decomposable such as finite or infinite trees, rational and real numbers, linear dense order,...etc. A decision procedure in the form of five rewriting rules has also been developed. It decides if a first-order formula without free variables (proposition) is true or not in any decomposable theory. Unfortunately, this later needs to normalize the initial proposition before starting the solving process. This transformation generates many nested negations and quantifications which greatly slow down the performances of this decision procedure. We present in this paper an efficient decision procedure for functional decomposable theories, i.e. theories whose set of relation is reduced to { = , ? }. This new decision procedure does not need to normalize the formulas and transforms any first-order proposition with any logical symbols into a boolean combination of basic formulas which are either equivalent to true or to false. We show the efficiency of our algorithm (in time and space) and compare its performances with those of the classical decision procedure for decomposable theories. Our algorithm is able to solve first-order propositions involving many nested alternated quantifiers of the form $\exists\bar{x}\forall\bar{y}$ over different functional decomposable theories.

From decomposable to residual theories

Deciding polynomial-transcendental problems

This paper presents a decision procedure for a certain class of sentences of first order logic involving integral polynomials and the exponential function in which the variables range over the real numbers. The inputs to the decision procedure are prenex sentences in which only the outermost quantified variable can occur in the exponential function. The decision procedure has been implemented in the computer logic system REDLOG. Closely related work is reported in [2, 7, 16, 20, 24].

Ranking fuzzy numbers play as a key tool in many applied models in the world and in particular decision-making procedures. We are going to present a new method based on the ranking the fuzzy number and real number. The problem of ranking the fuzzy number and real number is proposed with ranking function and then this approach to extend the ranking of two fuzzy numbers with ranking function. The proposed method is illustrated by some numerical examples and in particular the results of ranking by the proposed method and some common and existing methods for ranking fuzzy sets is compared to verify the advantage of the new approach. We will see that against of most existing ranking approaches where for two fuzzy sets are the exact ranking, the above men sioned method can give a ranking fuzzy numbers with acceptance rate smaller as fuzzy.