Decidable Theory Research Articles

Natural proofs for data structure manipulation in C using separation logic

The natural proof technique for heap verification developed by Qiu et al. [32] provides a platform for powerful sound reasoning for specifications written in a dialect of separation logic called Dryad. Natural proofs are proof tactics that enable automated reasoning exploiting recursion, mimicking common patterns found in human proofs. However, these proofs are known to work only for a simple toy language [32]. In this work, we develop a framework called VCDryad that extends the Vcc framework [9] to provide an automated deductive framework against separation logic specifications for C programs based on natural proofs. We develop several new techniques to build this framework, including (a) a novel tool architecture that allows encoding natural proofs at a higher level in order to use the existing Vcc framework (including its intricate memory model, the underlying type-checker, and the SMT-based verification infrastructure), and (b) a synthesis of ghost-code annotations that captures natural proof tactics, in essence forcing Vcc to find natural proofs using primarily decidable theories. We evaluate our tool extensively, on more than 150 programs, ranging from code manipulating standard data structures, well-known open source library routines (Glib, OpenBSD), Linux kernel routines, customized OS data structures, etc. We show that all these C programs can be fully automatically verified using natural proofs (given pre/post conditions and loop invariants) without any user-provided proof tactics . VCDryad is perhaps the first deductive verification framework for heap-manipulating programs in a real language that can prove such a wide variety of programs automatically.

Natural proofs for structure, data, and separation

We propose natural proofs for reasoning with programs that manipulate data-structures against specifications that describe the structure of the heap, the data stored within it, and separation and framing of sub-structures. Natural proofs are a subclass of proofs that are amenable to completely automated reasoning, that provide sound but incomplete procedures, and that capture common reasoning tactics in program verification. We develop a dialect of separation logic over heaps, called Dryad, with recursive definitions that avoids explicit quantification. We develop ways to reason with heaplets using classical logic over the theory of sets, and develop natural proofs for reasoning using proof tactics involving disciplined unfoldings and formula abstractions. Natural proofs are encoded into decidable theories of first-order logic so as to be discharged using SMT solvers. We also implement the technique and show that a large class of more than 100 correct programs that manipulate data-structures are amenable to full functional correctness using the proposed natural proof method. These programs are drawn from a variety of sources including standard data-structures, the Schorr-Waite algorithm for garbage collection, a large number of low-level C routines from the Glib library and OpenBSD library, the Linux kernel, and routines from a secure verified OS-browser project. Our work is the first that we know of that can handle such a wide range of full functional verification properties of heaps automatically, given pre/post and loop invariant annotations. We believe that this work paves the way for deductive verification technology to be used by programmers who do not (and need not) understand the internals of the underlying logic solvers, significantly increasing their applicability in building reliable systems.

Fair Synthesis for Asynchronous Distributed Systems

We study the synthesis problem in an asynchronous distributed setting: a finite set of processes interact locally with an uncontrollable environment and communicate with each other by sending signals -- actions controlled by a sender process and that are immediately received by the target process. The fair synthesis problem is to come up with a local strategy for each process such that the resulting fair behaviors of the system meet a given specification. We consider external specifications satisfying some natural closure properties related to the architecture. We present this new setting for studying the fair synthesis problem for distributed systems, and give decidability results for the subclass of networks where communications happen through a strongly connected graph. We claim that this framework for distributed synthesis is natural, convenient and avoids most of the usual sources of undecidability for the synthesis problem. Hence, it may open the way to a decidable theory of distributed synthesis.

A decidable theory of type assignment

This article investigates a theory of type assignment (assigning types to lambda terms) called ETA which is intermediate in strength between the simple theory of type assignment and strong polymorphic theories like Girard's F (Proofs and types. Cambridge University Press, Cambridge, 1989). It is like the simple theory and unlike F in that the typability and type-checking problems are solvable with respect to ETA. This is proved in the article along with three other main results: (1) all primitive recursive functionals of finite type are representable in ETA; (2) every term typable in ETA has a unique normal form; (3) there is a function defined by $${{\varepsilon}_0}$$ -recursion which takes every typable term to a natural number which is an upper bound to the lengths of all β?-reduction sequences starting with that term.

Problem and analysis of stability decidable theory for a class of fractional order nonlinear system

The research on the stability theory of fractional order nonlinear system has an important value for the application of synchronization and the control of fractional order chaotic system. The discussion that the stability discrimination of fractional order nonlinear system is converted into that of corresponding integer order nonlinear system has an important significance. In this paper, through the examples, for time-varying coefficient matrix, we point out the existing mistake of the discrimination theorem that states that if the integer system is stable, then its corresponding fractional system with order less than one is also stable. We also analyze the causes of the mistake.

Producing and verifying extremely large propositional refutations

The importance of producing a certificate of unsatisfiability is increasingly recognized for high performance propositional satisfiability solvers. The leading solvers develop a conflict graph as the basis for deriving (or "learning") new clauses. Extracting a resolution derivation from the conflict graph is theoretically straightforward, but resolution proofs can be extremely long. This paper reports on a tool that has verified proofs more than 1600 gigabytes long. Several other certificate formats have been proposed and studied, but the verifiers for these formats are beyond any hope of automated verification in their own rights. However, some of the alternative formats enjoy the advantages of being easy to produce proofs for, and reasonable in their space requirements. This paper reports progress on developing a practical system for formal verification of a more compact certificate format. Experimental comparisons are presented. A format called RUP (for Reverse Unit Propagation) is introduced and two implementations are evaluated. This method is an extension of conflict-clause proofs introduced by Goldberg and Novikov, and is compatible with conflict-clause minimization. Extracting a resolution derivation from other decidable theories is discussed briefly.

Countably categorical theories

A series of countably categorical theories are constructed based on the Fraisse method. In particular, an example of a decidable countably categorical theory in a finite signature is given for which no decidable model has an infinite computable set of order indiscernibles. Such a theory is used to refute Ershov’s conjecture on the representability of models of c-simple theories over linear orders.

On countable chains having decidable monadic theory

AbstractRationals and countable ordinals are important examples of structures with decidable monadic second-order theories. A chain is an expansion of a linear order by monadic predicates. We show that if the monadic second-order theory of a countable chain C is decidable then C has a non-trivial expansion with decidable monadic second-order theory.

Finite quantifier hierarchies in relational algebras

For a given structure of finite signature, one can construct a hierarchy of classes of relations definable in this structure according to the number of quantifier alternations in the formulas expressing the relations. In ordinary examples, this hierarchy is either infinite (as in the arithmetic of addition and multiplication of natural numbers) or stabilizes very rapidly (in structures with decidable theories, such as the field of real numbers). In the present paper, we construct a series of examples showing that the above-mentioned hierarchy may have an arbitrary finite length. The proof employs a modification of the Ehrenfeucht game.

Sufficient conditions for effective translation of locally generic queries

This paper continues investigations into the database of queries of first-order language theory. It is known that for many decidable theories, the collapse result holds: each locally generic query is equivalent to some restricted query. But, until now, the problem of effective construction of this query remains almost unexplored. We use earlier results of the author on the construction of a method of effective obtaining this query. The method is rather general and applicable, for example, to the Presburger arithmetic and the real number theory.

On constructive models of theories with linear Rudin-Keisler ordering

It is known that the class of Ehrenfeucht theories admits a syntactical characterization and that a finite (Rudin-Keisler) pre-ordering and a function mapping this pre-ordering to naturals play the role of parameters in this characterization. In the article, we construct for any finite linear ordering L, a hereditary decidable Ehrenfeucht theory T possessing L as its Rudin-Keisler pre-ordering. Also, we discuss decidable and computable models of such theories.

Hardware/software partitioning and pipelined scheduling on runtime reconfigurable FPGAs

FPGAs are widely used in today's embedded systems design due to their low cost, high performance, and reconfigurability. Partially RunTime-Reconfigurable (PRTR) FPGAs, such as Virtex-2 Pro and Virtex-4 from Xilinx, allow part of the FPGA area to be reconfigured while the remainder continues to operate without interruption, so that HW tasks can be placed and removed dynamically at runtime. We address two problems related to HW task scheduling on PRTR FPGAs: (1) HW/SW partitioning. Given an application in the form of a task graph with known execution times on the HW (FPGA) and SW (CPU), and known area sizes on the FPGA, find an valid allocation of tasks to either HW or SW and a static schedule with the optimization objective of minimizing the total schedule length (makespan). (2) Pipelined scheduling. Given an input task graph, construct a pipelined schedule on a PRTR FPGA with the goal of maximizing system throughput while meeting a given end-to-end deadline. Both problems are NP-hard. Satisfiability Modulo Theories (SMT) is an extension to SAT by adding the ability to handle arithmetic and other decidable theories. We use the SMT solver Yices with Linear Integer Arithmetic (LIA) theory as the optimization engine for solving the two scheduling problems. In addition, we present an efficient heuristic algorithm based on kernel recognition for the pipelined scheduling problem, a technique borrowed from SW pipelining, to overcome the scalability problem of the SMT-based optimal solution technique.

Decision procedures for algebraic data types with abstractions

We describe a family of decision procedures that extend the decision procedure for quantifier-free constraints on recursive algebraic data types (term algebras) to support recursive abstraction functions. Our abstraction functions are catamorphisms (term algebra homomorphisms) mapping algebraic data type values into values in other decidable theories (e.g. sets, multisets, lists, integers, booleans). Each instance of our decision procedure family is sound; we identify a widely applicable many-to-one condition on abstraction functions that implies the completeness. Complete instances of our decision procedure include the following correctness statements: 1) a functional data structure implementation satisfies a recursively specified invariant, 2) such data structure conforms to a contract given in terms of sets, multisets, lists, sizes, or heights, 3) a transformation of a formula (or lambda term) abstract syntax tree changes the set of free variables in the specified way.

Comments on Interpretability and Decidability in Fuzzy Logic

A particular notion of an interpretation of a theory over a fuzzy predicate logic in another such theory is discussed. For interpretability with the domain defined by a provably crisp formula, which is of course a syntactical notion, a semantic characterization is established. In the last section, we discuss the question of whether the extension of a decidable theory by a single new axiom is decidable and present an erratum to the paper (Hajek, 2007, Fundamenta informaticae, 81, 155–163).

Towards the decidability of the theory of modules over finite commutative rings

On the basis of the Klingler–Levy classification of finitely generated modules over commutative noetherian rings we approach the old problem of classifying finite commutative rings R with a decidable theory of modules. We prove that if R is (finite length) wild, then the theory of all R -modules is undecidable, and verify decidability of this theory for some classes of tame finite commutative rings.

Prime models of computably enumerable degree

AbstractWe examine the computably enumerable (c.e.) degrees of prime models of complete atomic decidable (CAD) theories. A structure has degree d if d is the degree of its elementary diagram. We show that if a CAD theory T has a prime model of c.e. degree c, then T has a prime model of strictly lower c.e. degree b, where, in addition, b is low (b′ = 0′), This extends Csima's result that every CAD theory has a low prime model. We also prove a density result for c.e. degrees of prime models. In particular, if c and d are c.e. degrees with d < c and c not low2 (c″ > 0″), then for any CAD theory T, there exists a c.e. degree b with d < b < c such that T has a prime model of degree b, where b can be chosen so that b′ is any degree c.e. in c with d′ ≤ b′. As a corollary, we show that for any degree c with 0 < c < 0′, every CAD theory has a prime model of low c.e. degree incomparable with c. We show also that every CAD theory has prime models of low c.e. degree that form a minimal pair, extending another result of Csima. We then discuss how these results apply to homogeneous models.

Index sets for some classes of structures

For a class K of structures, closed under isomorphism, the index set is the set I ( K ) of all indices for computable members of K in a universal computable numbering of all computable structures for a fixed computable language. We study the complexity of the index set of class of structures with decidable theories. We first prove the result for the class of all structures in an arbitrary finite nontrivial language. After the complexity is found, we prove similar results for some well-known classes of structures, such as directed graphs, undirected graphs, partial orders and lattices.

Classifying model-theoretic properties

AbstractIn 2004 Csima, Hirschfeldt, Knight, and Soare [1] showed that a set A ≤T 0′ is nonlow2 if and only if A is prime bounding, i.e., for every complete atomic decidable theory T, there is a prime model computable in A. The authors presented nine seemingly unrelated predicates of a set A, and showed that they are equivalent for sets. Some of these predicates, such as prime bounding, and others involving equivalence structures and abelian p-groups come from model theory, while others involving meeting dense sets in trees and escaping a given function come from pure computability theory.As predicates of A, the original nine properties are equivalent for sets; however, they are not equivalent in general. This article examines the (degree-theoretic) relationship between the nine properties. We show that the nine properties fall into three classes, each of which consists of several equivalent properties. We also investigate the relationship between the three classes, by determining whether or not any of the predicates in one class implies a predicate in another class.

The degree spectra of homogeneous models

AbstractMuch previous study has been done on the degree spectra of prime models of a complete atomic decidable theory. Here we study the analogous questions for homogeneous models. We say a countable model has a d-basis if the types realized in are all computable and the Turing degree d can list -indices for all types realized in . We say has a d-decidable copy if there exists a model ≅ such that the elementary diagram of is d-computable. Goncharov, Millar, and Peretyat'kin independently showed there exists a homogeneous with a 0-basis but no decidable copy.We prove that any homogeneous with a 0′-basis has a low decidable copy. This implies Csima's analogous result for prime models. In the case where all types of the theory T are computable and is a homogeneous model with a 0-basis, we show has copies decidable in every nonzero degree. A degree d is 0-homogeneous bounding if any automorphically nontrivial homogeneous with a 0-basis has a d-decidable copy. We show that the nonlow2 degrees are 0-homogeneous bounding.

Elementary equivalence of some rings of definable functions

We characterize elementary equivalences and inclusions between von Neumann regular real closed rings in terms of their boolean algebras of idempotents, and prove that their theories are always decidable. We then show that, under some hypotheses, the map sending an L-structure R to the L-structure of definable functions from Rn to R preserves elementary inclusions and equivalences and gives a structure with a decidable theory whenever R is decidable. We briefly consider structures of definable functions satisfying an extra condition such as continuity.