Deciding combinations of theories

Shostak, Robert E.

doi:10.1007/bfb0000061

Cited by 72 publications

(90 citation statements)

References 8 publications

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“…Nelson and Oppen's method [NO79] combines decision procedures for the individual theories by allowing them to share specific kinds of equality information. Shostak's method [Sho84] extends congruence closure to equational theories that are canonizable and solvable. Nelson and Oppen's method is more generally applicable, but Shostak's method has certain advantages.…”

Section: Shostak's Algorithmmentioning

confidence: 99%

“…We present here the first instance of a verified decision procedure for a combination of theories based on Shostak's ideas. Shostak's algorithm [Sho84] for building decision procedures for the union of canonizable and solvable equational theories has been widely used despite the lack of a convincing correctness proof. Recently, Ruess and Shankar [RS01] showed that this algorithm (even with minor flaws corrected [CLS96]) was both nonterminating and incomplete.…”

Section: Introductionmentioning

confidence: 99%

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Formal Verification of a Combination Decision Procedure

Ford

Shankar

2002

Automated Deduction—CADE-18

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Abstract. Decision procedures for combinations of theories are at the core of many modern theorem provers such as ACL2, Ehdm, PVS, SIM-PLIFY, the Stanford Pascal Verifier, STeP, SVC, and Z/Eves. Shostak, in 1984, published a decision procedure for the combination of canonizable and solvable theories. Recently, Ruess and Shankar showed Shostak's method to be incomplete and nonterminating, and presented a correct version of Shostak's algorithm along with informal proofs of termination, soundness, and completeness. We describe a formalization and mechanical verification of these proofs using the PVS verification system. The formalization itself posed significant challenges and the verification revealed some gaps in the informal argument.

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Section: Shostak's Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Formal Verification of a Combination Decision Procedure

Ford

Shankar

2002

Automated Deduction—CADE-18

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“…Work on theorem proving has focused on decomposition for parallel implementations [8,5,15,43] and has followed decomposition methods guided by lookahead and subgoals, neglecting the types of structural properties we used here. Another related line of work focuses on combining logical systems (e.g., [32,40,3,36,44]). Contrasted with this work, we focus on interactions between theories with overlapping signatures, the efficiency of reasoning, and automatic decomposition.…”

Section: Related Workmentioning

confidence: 99%

Improving the Efficiency of Reasoning Through Structure-Based Reformulation

Amir

McIlraith

2000

Lecture Notes in Computer Science

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Abstract.We investigate the possibility of improving the efficiency of reasoning through structure-based partitioning of logical theories, combined with partitionbased logical reasoning strategies. To this end, we provide algorithms for reasoning with partitions of axioms in first-order and propositional logic. We analyze the computational benefit of our algorithms and detect those parameters of a partitioning that influence the efficiency of computation. These parameters are the number of symbols shared by a pair of partitions, the size of each partition, and the topology of the partitioning. Finally, we provide a greedy algorithm that automatically reformulates a given theory into partitions, exploiting the parameters that influence the efficiency of computation.

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“…SVC is a decision procedure for quantifier-free first-order logic and uses an algorithm similar to the algorithms by Shostak [20,19] and Nelson-Oppen [17]. The input Boolean formula to SVC can contain Boolean operators, uninterpreted functions and interpreted functions, and distinct constants such as the Boolean truth and bit constants.…”

Section: (P! Ai(s)) ^ I(si)mentioning

confidence: 99%

Formally Verifying Data and Control with Weak Reachability Invariants

Dill

Skakkebæk

1998

Formal Methods in Computer-Aided Design

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Existing formal verification methods do not handle systems that combine state machines and data paths very well. Model checking deals with finitestate machines efficiently, but model checking full designs is infeasible because of the large amount of state in the data path. Theorem-proving methods may be effective for verifying data path operations, but verifying the control requires finding and proving inductive invariants that characterize the reachable states of the system. We present a new approach to verification of systems that combine control FSMs and data path operations. Invariants are specified only for a small set of control states, called clean states, where the invariants are especially simple. We avoid the need to specify the invariants for the unclean states by symbolically simulating over all paths to find the possible next clean states. The set of all paths from one clean state to the next is represented by a regular expression, which is extracted from the control FSMs. The number of paths is infinite only if the regular expression contains stars. The method uses a heuristic to generalize the symbolic state to cover all of the paths of the starred expression. We have implemented a prototype tool for guiding an existing symbolic simulator and verification tool and used it successfully to prove properties of the Instruction Fetch Unit of TORCH, a superscalar microprocessor designed at Stanford. With much less effort, we were able to find all the bugs in the unit that were found earlier by manually strengthening the invariants.

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Deciding combinations of theories

Cited by 72 publications

References 8 publications

Formal Verification of a Combination Decision Procedure

Formal Verification of a Combination Decision Procedure

Improving the Efficiency of Reasoning Through Structure-Based Reformulation

Formally Verifying Data and Control with Weak Reachability Invariants

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