8,101 research outputs found
Transforming Equality Logic to Propositional Logic
We investigate and compare various ways of transforming equality formulas to propositional formulas, in order to be able to solve satisfiability in equality logic by means of satisfiability in propositional logic. We propose equality substitution as a new approach combining desirable properties of earlier methods, we prove its correctness and show its applicability by experiments
Two solutions to incorporate zero, successor and equality in binary decision diagrams
In this article we extend BDDs (binary decision diagrams) for plain propositional logic to the fragment of first order logic, consisting of quantifier free logic with equality, zero and successor. We insert equations with zero and successor in BDDs, and call these objects (0,S,=)-BDDs. We extend the notion of {em Ordered} BDDs in the presence of equality, zero and successor. (0,S,=)-BDDs can be transformed to equivalent Ordered (0,S,=)-BDD s by applying a number of rewrite rules. All paths in these extended OBDDs are satisfiable. The major advantage of transforming a formula to an equivalent Ordered (0,S,=)-BDD is that on the latter it can be observed in constant time whether the formula is a tautology, a contradiction, or just satisfiable
Virtual Evidence: A Constructive Semantics for Classical Logics
This article presents a computational semantics for classical logic using
constructive type theory. Such semantics seems impossible because classical
logic allows the Law of Excluded Middle (LEM), not accepted in constructive
logic since it does not have computational meaning. However, the apparently
oracular powers expressed in the LEM, that for any proposition P either it or
its negation, not P, is true can also be explained in terms of constructive
evidence that does not refer to "oracles for truth." Types with virtual
evidence and the constructive impossibility of negative evidence provide
sufficient semantic grounds for classical truth and have a simple computational
meaning. This idea is formalized using refinement types, a concept of
constructive type theory used since 1984 and explained here. A new axiom
creating virtual evidence fully retains the constructive meaning of the logical
operators in classical contexts.
Key Words: classical logic, constructive logic, intuitionistic logic,
propositions-as-types, constructive type theory, refinement types, double
negation translation, computational content, virtual evidenc
Processor Verification Using Efficient Reductions of the Logic of Uninterpreted Functions to Propositional Logic
The logic of equality with uninterpreted functions (EUF) provides a means of
abstracting the manipulation of data by a processor when verifying the
correctness of its control logic. By reducing formulas in this logic to
propositional formulas, we can apply Boolean methods such as Ordered Binary
Decision Diagrams (BDDs) and Boolean satisfiability checkers to perform the
verification.
We can exploit characteristics of the formulas describing the verification
conditions to greatly simplify the propositional formulas generated. In
particular, we exploit the property that many equations appear only in positive
form. We can therefore reduce the set of interpretations of the function
symbols that must be considered to prove that a formula is universally valid to
those that are ``maximally diverse.''
We present experimental results demonstrating the efficiency of this approach
when verifying pipelined processors using the method proposed by Burch and
Dill.Comment: 46 page
Proving Craig and Lyndon Interpolation Using Labelled Sequent Calculi
We have recently presented a general method of proving the fundamental
logical properties of Craig and Lyndon Interpolation (IPs) by induction on
derivations in a wide class of internal sequent calculi, including sequents,
hypersequents, and nested sequents. Here we adapt the method to a more general
external formalism of labelled sequents and provide sufficient criteria on the
Kripke-frame characterization of a logic that guarantee the IPs. In particular,
we show that classes of frames definable by quantifier-free Horn formulas
correspond to logics with the IPs. These criteria capture the modal cube and
the infinite family of transitive Geach logics
Introducing Quantified Cuts in Logic with Equality
Cut-introduction is a technique for structuring and compressing formal
proofs. In this paper we generalize our cut-introduction method for the
introduction of quantified lemmas of the form (for
quantifier-free ) to a method generating lemmas of the form . Moreover, we extend the original method to predicate
logic with equality. The new method was implemented and applied to the TSTP
proof database. It is shown that the extension of the method to handle equality
and quantifier-blocks leads to a substantial improvement of the old algorithm
Do Hard SAT-Related Reasoning Tasks Become Easier in the Krom Fragment?
Many reasoning problems are based on the problem of satisfiability (SAT).
While SAT itself becomes easy when restricting the structure of the formulas in
a certain way, the situation is more opaque for more involved decision
problems. We consider here the CardMinSat problem which asks, given a
propositional formula and an atom , whether is true in some
cardinality-minimal model of . This problem is easy for the Horn
fragment, but, as we will show in this paper, remains -complete (and
thus -hard) for the Krom fragment (which is given by formulas in
CNF where clauses have at most two literals). We will make use of this fact to
study the complexity of reasoning tasks in belief revision and logic-based
abduction and show that, while in some cases the restriction to Krom formulas
leads to a decrease of complexity, in others it does not. We thus also consider
the CardMinSat problem with respect to additional restrictions to Krom formulas
towards a better understanding of the tractability frontier of such problems
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