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Implementing Mathematics with The Nuprl Proof Development System

This article presents EXE, an effective bug-finding tool that automatically generates inputs that crash real code. Instead of running code on manually or randomly constructed input, EXE runs it on symbolic input initially allowed to be anything. As checked code runs, EXE tracks the constraints on each symbolic (i.e., input-derived) memory location. If a statement uses a symbolic value, EXE does not run it, but instead adds it as an input-constraint; all other statements run as usual. If code conditionally checks a symbolic expression, EXE forks execution, constraining the expression to be true on the true branch and false on the other. Because EXE reasons about all possible values on a path, it has much more power than a traditional runtime tool: (1) it can force execution down any feasible program path and (2) at dangerous operations (e.g., a pointer dereference), it detects if the current path constraints allow any value that causes a bug. When a path terminates or hits a bug, EXE automatically generates a test case by solving the current path constraints to find concrete values using its own co-designed constraint solver, STP. Because EXE’s constraints have no approximations, feeding this concrete input to an uninstrumented version of the checked code will cause it to follow the same path and hit the same bug (assuming deterministic code).EXE works well on real code, finding bugs along with inputs that trigger them in: the BSD and Linux packet filter implementations, the dhcpd DHCP server, the pcre regular expression library, and three Linux file systems.

EXE: Automatically Generating Inputs of Death

Satisfiability Modulo Theories (SMT) refers to the problem of determining whether a first-order formula is satisfiable with respect to some logical theory. Solvers based on SMT are used as back-end engines in model-checking applications such as bounded, interpolation-based, and predicate-abstraction-based model checking. After a brief illustration of these uses, we survey the predominant techniques for solving SMT problems with an emphasis on the lazy approach, in which a propositional satisfiability (SAT) solver is combined with one or more theory solvers. We discuss the architecture of a lazy SMT solver, give examples of theory solvers, show how to combine such solvers modularly, and mention several extensions of the lazy approach. We also briefly describe the eager approach in which the SMT problem is reduced to a SAT problem. Finally, we discuss how the basic framework for determining satisfiability can be extended with additional functionality such as producing models, proofs, unsatisfiable cores, and interpolants.

Satisfiability Modulo Theories

Knowledge representation is at the very core of a radical idea for understanding intelligence. Instead of trying to understand or build brains from the bottom up, its goal is to understand and build intelligent behavior from the top down, putting the focus on what an agent needs to know in order to behave intelligently, how this knowledge can be represented symbolically, and how automated reasoning procedures can make this knowledge available as needed. 

This landmark text takes the central concepts of knowledge representation developed over the last 50 years and illustrates them in a lucid and compelling way. Each of the various styles of representation is presented in a simple and intuitive form, and the basics of reasoning with that representation are explained in detail. This approach gives readers a solid foundation for understanding the more advanced work found in the research literature. The presentation is clear enough to be accessible to a broad audience, including researchers and practitioners in database management, information retrieval, and object-oriented systems as well as artificial intelligence. This book provides the foundation in knowledge representation and reasoning that every AI practitioner needs.

*Authors are well-recognized experts in the field who have applied the techniques to real-world problems 
* Presents the core ideas of KR&R in a simple straight forward approach, independent of the quirks of research systems 
*Offers the first true synthesis of the field in over a decade

Table of Contents

1 Introduction * 2 The Language of First-Order Logic *3 Expressing Knowledge * 4 Resolution * 5 Horn Logic * 6 Procedural Control of Reasoning * 7 Rules in Production Systems * 8 Object-Oriented Representation * 9 Structured Descriptions * 10 Inheritance * 11 Numerical Uncertainty *12 Defaults *13 Abductive Reasoning *14 Actions * 15 Planning *16 A Knowledge Representation Tradeoff * Bibliography * Index

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Knowledge Representation and Reasoning

In learning their native language children develop a remarkable set of capabilities They acquire knowledge and skills that enable them to pro duce and comprehend an inde nite number of novel utterances and to make quite subtle judgments about certain of their properties The ma jor goal of psycholinguistic research is to devise an explanatory account of the mental operations that underlie these linguistic abilities In pursuing this goal we have adopted what we call the Competence Hypothesis as a methodological principle We assume that an explana tory model of human language performance will incorporate a theoreti cally justi ed representation of the native speaker s linguistic knowledge a grammar as a component separate both from the computational mech anisms that operate on it a processor and from other nongrammatical processing parameters that might in uence the processor s behavior To a certain extent the various components that we postulate can be studied independently guided where appropriate by the well established methods and evaluation standards of linguistics computer science and experimen tal psychology However the requirement that the various components ultimately must t together in a consistent and coherent model imposes even stronger constraints on their structure and operation

Lexical Functional Grammar A Formal System for Grammatical Representation

The decision algorithms rely on an algorithm for computing the congruence closure of a relation on a graph. Let G = (V,E) be a directed graph and R a (binary) relation on V. Define the congruence closure ~ of R on G to be the unique minimal extension of R, such that (1) ~ is an equivalence relation, and (2) for any two vertices u and" in V, if all corresponding successors of u and v are equivalent under~, then so are u and v. We give a simple algorithm for computing congruence closures in O(mn) time, where n is the number of vertices and m the number of edges in G.

Fast decision algorithms based on union and find

A method for combining decision procedures for several theories into a single decision procedure for their combination is described, and a simplifier based on this method is discussed. The simplifier finds a normal form for any expression formed from individual variables, the usual Boolean connectives, the equality predicate =, the conditional function if-then-else, the integers, the arithmetic functions and predicates +, -, and ≤, the Lisp functions and predicates car, cdr, cons, and atom, the functions store and select for storing into and selecting from arrays, and uninterpreted function symbols. If the expression is a theorem it is simplified to the constant true, so the simplifier can be used as a decision procedure for the quantifier-free theory containing these functions and predicates. The simplifier is currently used in the Stanford Pascal Verifier.

Simplification by Cooperating Decision Procedures

The notion of the congruence closure of a relation on a graph is defined and several algorithms for computing it are surveyed. A simple proof is given that the congruence closure algorithm provides a decision procedure for the quantifier-free theory of equality. A decision procedure is then given for the quantifier-free theory of LISP list structure based on the congruence closure algorithm. Both decision procedures determine the satisfiability of a conjunction of literals of length n in average time O(n log n) using the fastest known congruence closure algorithm. It is also shown that if the axiomatization of the theory of list structure is changed slightly, the problem of determining the satisfiability of a conjunction of literals becomes NP-complete. The decision procedures have been implemented in the authors' simplifier for the Stanford Pascal Verifier.

Fast Decision Procedures Based on Congruence Closure

We describe a simplifier for use in program manipulation and verification. The simplifier finds a normal form for any expression over the language consisting of individual variables, the usual boolean connectives, the conditional function cond (denoting if-then-else), the integers (numerals), the arithmetic functions and predicates +, - and ≤, the LISP constants, functions and predicates nil, car, cdr, cons and atom, the functions store and select for storing into and selecting from arrays, and uninterpreted function symbols. Individual variables range over the union of the rationals, the set of arrays, the LISP s-expressions and the booleans true and false. The constant, function and predicate symbols take their natural interpretations.The simplifier is complete; that is, it simplifies every valid formula to true. Thus it is also a decision procedure for the quantifier-free theory of rationals, arrays and s-expressions under the above functions and predicates.The organization of the simplifier is based on a method for combining decision algorithms for several theories into a single decision algorithm for a larger theory containing the original theories. More precisely, given a set S of functions and predicates over a fixed domain, a satisfiability program for S is a program which determines the satisfiability of conjunctions of literals (signed atomic formulas) whose predicates and function signs are in S. We give a general procedure for combining satisfiability programs for sets S and T into a single satisfiability program for S ∪ T, given certain conditions on S and T. We show how a satisfiability program for a set S can be used to write a complete simplifier for expressions containing functions and predicates of S as well as uninterpreted function symbols.The simplifier described in this paper is currently used in the Stanford Pascal Verifier.

A simplifier based on efficient decision algorithms

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Fast decision algorithms based on union and find

Simplification by Cooperating Decision Procedures

Fast Decision Procedures Based on Congruence Closure

A simplifier based on efficient decision algorithms