6,180 research outputs found
Automated Synthesis of a Finite Complexity Ordering for Saturation
We present in this paper a new procedure to saturate a set of clauses with
respect to a well-founded ordering on ground atoms such that A < B implies
Var(A) {\subseteq} Var(B) for every atoms A and B. This condition is satisfied
by any atom ordering compatible with a lexicographic, recursive, or multiset
path ordering on terms. Our saturation procedure is based on a priori ordered
resolution and its main novelty is the on-the-fly construction of a finite
complexity atom ordering. In contrast with the usual redundancy, we give a new
redundancy notion and we prove that during the saturation a non-redundant
inference by a priori ordered resolution is also an inference by a posteriori
ordered resolution. We also prove that if a set S of clauses is saturated with
respect to an atom ordering as described above then the problem of whether a
clause C is entailed from S is decidable
New results on rewrite-based satisfiability procedures
Program analysis and verification require decision procedures to reason on
theories of data structures. Many problems can be reduced to the satisfiability
of sets of ground literals in theory T. If a sound and complete inference
system for first-order logic is guaranteed to terminate on T-satisfiability
problems, any theorem-proving strategy with that system and a fair search plan
is a T-satisfiability procedure. We prove termination of a rewrite-based
first-order engine on the theories of records, integer offsets, integer offsets
modulo and lists. We give a modularity theorem stating sufficient conditions
for termination on a combinations of theories, given termination on each. The
above theories, as well as others, satisfy these conditions. We introduce
several sets of benchmarks on these theories and their combinations, including
both parametric synthetic benchmarks to test scalability, and real-world
problems to test performances on huge sets of literals. We compare the
rewrite-based theorem prover E with the validity checkers CVC and CVC Lite.
Contrary to the folklore that a general-purpose prover cannot compete with
reasoners with built-in theories, the experiments are overall favorable to the
theorem prover, showing that not only the rewriting approach is elegant and
conceptually simple, but has important practical implications.Comment: To appear in the ACM Transactions on Computational Logic, 49 page
Applications of Finite Model Theory: Optimisation Problems, Hybrid Modal Logics and Games.
There exists an interesting relationships between two seemingly distinct fields: logic from the field of Model Theory, which deals with the truth of statements about discrete structures; and Computational Complexity, which deals with the classification of problems by how much of a particular computer resource is required in order to compute a solution. This relationship is known as Descriptive Complexity and it is the primary application of the tools from Model Theory when they are restricted to the finite; this restriction is commonly called Finite Model Theory.
In this thesis, we investigate the extension of the results of Descriptive Complexity from classes of decision problems to classes of optimisation problems. When dealing with decision problems the natural mapping from true and false in logic to yes and no instances of a problem is used but when dealing with optimisation problems, other features of a logic need to be used. We investigate what these features are and provide results in the form of logical frameworks that can be used for describing optimisation problems in particular classes, building on the existing research into this area.
Another application of Finite Model Theory that this thesis investigates is the relative expressiveness of various fragments of an extension of modal logic called hybrid modal logic. This is achieved through taking the Ehrenfeucht-Fraïssé game from Model Theory and modifying it so that it can be applied to hybrid modal logic. Then, by developing winning strategies for the players in the game, results are obtained that show strict hierarchies of expressiveness for fragments of hybrid modal logic that are generated by varying the quantifier depth and the number of proposition and nominal symbols available
Recommended from our members
Efficient recursion termination for function-free horn logic
We present an efficient scheme to terminate infinite recursion in function-free Horn logic. In [BW84], Brough and Walker show that a preorder linear resolution with a goal termination strategy is incomplete, i.e. it must miss some answers. Their theory is true if left-recursion is allowed. The crucial assumption underlying Brough and Walker's theory is that the order of literals in a clause should not be altered. This assumption, however, is not necessary in programs that do not contain any extra-logical features such as the 'cut' symbol of Prolog. This is because the order of literals does not affect the correctness of such programs, only their efficiency. In this paper, we show that left-recursion can always be eliminated. The idea is to transform loops of the input set into safe loops, that are left-recursion free. Consequently, the goal termination strategy guarantees to always terminate properly with all possible answers; thus, it is complete in the domain of safe loops. We further show that all rules in a safe loop can be transformed into rules that begin with a base literal. This permits the implementation of a simple scheme to carry out the goal termination strategy more efficiently. The basic idea of this scheme is to distribute the history containing all executed goals over assertions, rather than maintaining it as a centralized data structure. This reduces the amount of work performed during execution
MaLeS: A Framework for Automatic Tuning of Automated Theorem Provers
MaLeS is an automatic tuning framework for automated theorem provers. It
provides solutions for both the strategy finding as well as the strategy
scheduling problem. This paper describes the tool and the methods used in it,
and evaluates its performance on three automated theorem provers: E, LEO-II and
Satallax. An evaluation on a subset of the TPTP library problems shows that on
average a MaLeS-tuned prover solves 8.67% more problems than the prover with
its default settings
Abstract Canonical Inference
An abstract framework of canonical inference is used to explore how different
proof orderings induce different variants of saturation and completeness.
Notions like completion, paramodulation, saturation, redundancy elimination,
and rewrite-system reduction are connected to proof orderings. Fairness of
deductive mechanisms is defined in terms of proof orderings, distinguishing
between (ordinary) "fairness," which yields completeness, and "uniform
fairness," which yields saturation.Comment: 28 pages, no figures, to appear in ACM Trans. on Computational Logi
- …