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18
Theorem Proving with the Real Numbers
, 1996
"... This thesis discusses the use of the real numbers in theorem proving. Typically, theorem provers only support a few `discrete' datatypes such as the natural numbers. However the availability of the real numbers opens up many interesting and important application areas, such as the verification ..."
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Cited by 117 (14 self)
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This thesis discusses the use of the real numbers in theorem proving. Typically, theorem provers only support a few `discrete' datatypes such as the natural numbers. However the availability of the real numbers opens up many interesting and important application areas, such as the verification of floating point hardware and hybrid systems. It also allows the formalization of many more branches of classical mathematics, which is particularly relevant for attempts to inject more rigour into computer algebra systems. Our work is conducted in a version of the HOL theorem prover. We describe the rigorous definitional construction of the real numbers, using a new version of Cantor's method, and the formalization of a significant portion of real analysis. We also describe an advanced derived decision procedure for the `Tarski subset' of real algebra as well as some more modest but practically useful tools for automating explicit calculations and routine linear arithmetic reasoning. Finally,...
Limitations of SelfAssembly at Temperature 1
, 2008
"... We prove that if a set X ⊆ Z 2 weakly selfassembles at temperature 1 in a deterministic (Winfree) tile assembly system satisfying a natural condition known as pumpability, then X is a finite union of doubly periodic sets. This shows that only the most simple of infinite shapes and patterns can be c ..."
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Cited by 23 (13 self)
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We prove that if a set X ⊆ Z 2 weakly selfassembles at temperature 1 in a deterministic (Winfree) tile assembly system satisfying a natural condition known as pumpability, then X is a finite union of doubly periodic sets. This shows that only the most simple of infinite shapes and patterns can be constructed using pumpable temperature 1 tile assembly systems, and gives strong evidence for the thesis that temperature 2 or higher is required to carry out generalpurpose computation in a tile assembly system. We employ this result to show that, unlike the case of temperature 2 selfassembly, no discrete selfsimilar fractal weakly selfassembles at temperature 1 in a pumpable tile assembly system. Finally, we show that generalpurpose computation is possible at temperature 1 if negative glue strengths are allowed in the tile assembly model.
On the Automata Size for Presburger Arithmetic
 In Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LICS 2004
, 2004
"... Automata provide an effective mechanization of decision procedures for Presburger arithmetic. However, only crude lower and upper bounds are known on the sizes of the automata produced by this approach. In this paper, we prove that the number of states of the minimal deterministic automaton for a Pr ..."
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Cited by 11 (1 self)
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Automata provide an effective mechanization of decision procedures for Presburger arithmetic. However, only crude lower and upper bounds are known on the sizes of the automata produced by this approach. In this paper, we prove that the number of states of the minimal deterministic automaton for a Presburger arithmetic formula is triple exponentially bounded in the length of the formula. This upper bound is established by comparing the automata for Presburger arithmetic formulas with the formulas produced by a quantifier elimination method. We also show that this triple exponential bound is tight (even for nondeterministic automata). Moreover, we provide optimal automata constructions for linear equations and inequations.
Automating elementary numbertheoretic proofs using Gröbner bases
"... Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain numbertheoretic predicates ..."
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Cited by 6 (1 self)
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Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain numbertheoretic predicates such as ‘divisible by’, ‘congruent ’ and ‘coprime’; one notable example in this class is the Chinese Remainder Theorem (for a specific number of moduli). The method is based on a reduction to ideal membership assertions that are then solved using Gröbner bases. As well as illustrating the usefulness of the procedure on examples, and considering some extensions, we prove a limited form of completeness for properties that hold in all rings. 1
A framework for the flexible integration of a class of decision procedures into theorem provers
 FEDRA, K., GIS AND ENVIRONMENTAL MODELING
, 1999
"... The role of decision procedures is often essential in theorem proving. Decision procedures can reduce the search space of heuristic components of a prover and increase its abilities. However, in some applications only a small number of conjectures fall within the scope of the available decision proc ..."
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Cited by 6 (2 self)
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The role of decision procedures is often essential in theorem proving. Decision procedures can reduce the search space of heuristic components of a prover and increase its abilities. However, in some applications only a small number of conjectures fall within the scope of the available decision procedures. Some of these conjectures could in an informal sense fall ‘just outside’ that scope. In these situations a problem arises because lemmas have to be invoked or the decision procedure has to communicate with the heuristic component of a theorem prover. This problem is also related to the general problem of how to exibly integrate decision procedures into heuristic theorem provers. In this paper we address such problems and describe a framework for the exible integration of decision procedures into other proof methods. The proposed framework can be used in different theorem provers, for different theories and for different decision procedures. New decision procedures can be simply ‘pluggedin’ to the system. As an illustration, we describe an instantiation of this framework within the Clam proofplanning system, to which it is well suited. We report on some results using this implementation.
Verification using Satisfiability Checking, Predicate Abstraction, and Craig Interpolation
, 2008
"... not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the sponsoring institutions, the U.S. Government or any other entity. Keywords: Formal methods, model checking, abstraction, refinement, bounded model checking, Boolean satisfiabilit ..."
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Cited by 4 (1 self)
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not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the sponsoring institutions, the U.S. Government or any other entity. Keywords: Formal methods, model checking, abstraction, refinement, bounded model checking, Boolean satisfiability, nonclausal SAT solvers, DPLL, general matings, unsatisfiable core, craig interpolation, proofs of unsatisfiability, linear diophantine equations, linear modular equations (linear congruences), linear diophantine Automatic verification of hardware and software implementations is crucial for building reliable computer systems. Most verification tools rely on decision procedures to check the satisfiability of various formulas that are generated during the verification process. This thesis develops new techniques for building efficient decision procedures and adds new capabilities to the existing decision procedures for certain logics. Boolean satisfiability (SAT) solvers are used heavily in verification tools as decision procedures for propositional logic. Most stateoftheart SAT solvers are
Strict General Setting for Building Decision Procedures into Theorem Provers
 THE 1ST INTERNATIONAL JOINT CONFERENCE ON AUTOMATED REASONING (IJCAR2001) — SHORT PAPERS
, 2001
"... The efficient and flexible incorporating of decision procedures into theorem provers is very important for their successful use. There are several approaches for combining and augmenting of decision procedures; some of them support handling uninterpreted functions, congruence closure, lemma invoking ..."
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Cited by 2 (1 self)
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The efficient and flexible incorporating of decision procedures into theorem provers is very important for their successful use. There are several approaches for combining and augmenting of decision procedures; some of them support handling uninterpreted functions, congruence closure, lemma invoking etc. In this paper we present a variant of one general setting for building decision procedures into theorem provers (gs framework [18]). That setting is based on macro inference rules motivated by techniques used in different approaches. The general setting enables a simple describing of different combination/augmentation schemes. In this paper, we further develop and extend this setting by an imposed ordering on the macro inference rules. That ordering leads to a ”strict setting”. It makes implementing and using variants of wellknown or new schemes within this framework a very easy task even for a nonexpert user. Also, this setting enables easy comparison of different combination/augmentation schemes and combination of their ideas.
On Lattices of Regular Sets of Natural Integers Closed under Decrementation, Submitted, 2013. Preprint version on arXiv
, 2013
"... We consider lattices of regular sets of non negative integers, i.e. of sets definable in Presbuger arithmetic. We prove that if such a lattice is closed under decrement then it is also closed under many other functions: quotients by an integer, roots, etc. Keywords. Lattices, lattices of subsets of ..."
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Cited by 2 (1 self)
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We consider lattices of regular sets of non negative integers, i.e. of sets definable in Presbuger arithmetic. We prove that if such a lattice is closed under decrement then it is also closed under many other functions: quotients by an integer, roots, etc. Keywords. Lattices, lattices of subsets of N, regular subsets of N, closure properties.