The Ciao clp(FD) library. A modular CLP extension for Prolog

We present a new free library for Constraint Logic Programming over Finite Domains, included with the Ciao Prolog system. The library is entirely written in Prolog, leveraging on Ciao's module system and code transformation capabilities in order to achieve a highly modular design without compromising performance. We describe the interface, implementation, and design rationale of each modular component. The library meets several design goals: a high level of modularity, allowing the individual components to be replaced by different versions; highefficiency, being competitive with other TT> implementations; a glass-box approach, so the user can specify new constraints at different levels; and a Prolog implementation, in order to ease the integration with Ciao's code analysis components. The core is built upon two small libraries which implement integer ranges and closures. On top of that, a finite domain variable datatype is defined, taking care of constraint reexecution depending on range changes. These three libraries form what we call the TT> kernel of the library. This TT> kernel is used in turn to implement several higher-level finite domain constraints, specified using indexicals. Together with a labeling module this layer forms what we name the TT> solver. A final level integrates the CLP (J7©) paradigm with our TT> solver. This is achieved using attributed variables and a compiler from the CLP (J7©) language to the set of constraints provided by the solver. It should be noted that the user of the library is encouraged to work in any of those levels as seen convenient: from writing a new range module to enriching the set of TT> constraints by writing new indexicals

Doctor of Philosophy

dissertationWith the spread of internet and mobile devices, transferring information safely and securely has become more important than ever. Finite fields have widespread applications in such domains, such as in cryptography, error correction codes, among many others. In most finite field applications, the field size - and therefore the bit-width of the operands - can be very large. The high complexity of arithmetic operations over such large fields requires circuits to be (semi-) custom designed. This raises the potential for errors/bugs in the implementation, which can be maliciously exploited and can compromise the security of such systems. Formal verification of finite field arithmetic circuits has therefore become an imperative. This dissertation targets the problem of formal verification of hardware implementations of combinational arithmetic circuits over finite fields of the type F2k . Two specific problems are addressed: i) verifying the correctness of a custom-designed arithmetic circuit implementation against a given word-level polynomial specification over F2k ; and ii) gate-level equivalence checking of two different arithmetic circuit implementations. This dissertation proposes polynomial abstractions over finite fields to model and represent the circuit constraints. Subsequently, decision procedures based on modern computer algebra techniques - notably, Gr¨obner bases-related theory and technology - are engineered to solve the verification problem efficiently. The arithmetic circuit is modeled as a polynomial system in the ring F2k [x1, x2, · · · , xd], and computer algebrabased results (Hilbert's Nullstellensatz) over finite fields are exploited for verification. Using our approach, experiments are performed on a variety of custom-designed finite field arithmetic benchmark circuits. The results are also compared against contemporary methods, based on SAT and SMT solvers, BDDs, and AIG-based methods. Our tools can verify the correctness of, and detect bugs in, up to 163-bit circuits in F2163 , whereas contemporary approaches are infeasible beyond 48-bit circuits

SAT Compilation for Constraints over Structured Finite Domains

A constraint is a formula in first-order logic expressing a relation between values of various domains. In order to solve a constraint, constructing a propositional encoding is a successfully applied technique that benefits from substantial progress made in the development of modern SAT solvers. However, propositional encodings are generally created by developing a problem-specific generator program or by crafting them manually, which often is a time-consuming and error-prone process especially for constraints over complex domains. Therefore, the present thesis introduces the constraint solver CO4 that automatically generates propositional encodings for constraints over structured finite domains written in a syntactical subset of the functional programming language Haskell. This subset of Haskell enables the specification of expressive and concise constraints by supporting user-defined algebraic data types, pattern matching, and polymorphic types, as well as higher-order and recursive functions. The constraint solver CO4 transforms a constraint written in this high-level language into a propositional formula. After an external SAT solver determined a satisfying assignment for the variables in the generated formula, a solution in the domain of discourse is derived. This approach is even applicable for finite restrictions of recursively defined algebraic data types. The present thesis describes all aspects of CO4 in detail: the language used for specifying constraints, the solving process and its correctness, as well as exemplary applications of CO4

High-level Counterexamples for Probabilistic Automata

Providing compact and understandable counterexamples for violated system properties is an essential task in model checking. Existing works on counterexamples for probabilistic systems so far computed either a large set of system runs or a subset of the system's states, both of which are of limited use in manual debugging. Many probabilistic systems are described in a guarded command language like the one used by the popular model checker PRISM. In this paper we describe how a smallest possible subset of the commands can be identified which together make the system erroneous. We additionally show how the selected commands can be further simplified to obtain a well-understandable counterexample