16,164 research outputs found
Proof systems for effectively propositional logic
We consider proof systems for effectively propositional logic. First, we show that propositional resolution for effectively propositional logic may have exponentially longer refutations than resolution for this logic. This shows that methods based on ground instantiation may be weaker than non-ground methods. Second, we introduce a generalisation rule for effectively propositional logic and show that resolution for this logic may have exponentially longer proofs than resolution with generalisation. We also discuss some related questions, such as sort assignments for generalisation
Effectively Polynomial Simulations
We introduce a more general notion of efficient simulation between proof systems, which we call effectively-p simulation. We argue that this notion is more natural from a complexity-theoretic point of view, and by revisiting standard concepts in this light we obtain some surprising new results. First, we give several examples where effectively-p simulations are possible between different propositional proof systems, but where p-simulations are impossible (sometimes under complexity assumptions). Secondly, we prove that the rather weak proof system G0 for quantified propositional logic (QBF) can effectively-p simulate any proof system for QBF. Thus our definition sheds new light on the comparative power of proof systems. We also give some evidence that with respect to Frege and Extended Frege systems, an effectively-p simulation may not be possible. Lastly, we prove new relationships between effectively-p simulations, automatizability, and the P versus NP question
Scavenger 0.1: A Theorem Prover Based on Conflict Resolution
This paper introduces Scavenger, the first theorem prover for pure
first-order logic without equality based on the new conflict resolution
calculus. Conflict resolution has a restricted resolution inference rule that
resembles (a first-order generalization of) unit propagation as well as a rule
for assuming decision literals and a rule for deriving new clauses by (a
first-order generalization of) conflict-driven clause learning.Comment: Published at CADE 201
The modal logic of Reverse Mathematics
The implication relationship between subsystems in Reverse Mathematics has an
underlying logic, which can be used to deduce certain new Reverse Mathematics
results from existing ones in a routine way. We use techniques of modal logic
to formalize the logic of Reverse Mathematics into a system that we name
s-logic. We argue that s-logic captures precisely the "logical" content of the
implication and nonimplication relations between subsystems in Reverse
Mathematics. We present a sound, complete, decidable, and compact tableau-style
deductive system for s-logic, and explore in detail two fragments that are
particularly relevant to Reverse Mathematics practice and automated theorem
proving of Reverse Mathematics results
Mass problems and intuitionistic higher-order logic
In this paper we study a model of intuitionistic higher-order logic which we
call \emph{the Muchnik topos}. The Muchnik topos may be defined briefly as the
category of sheaves of sets over the topological space consisting of the Turing
degrees, where the Turing cones form a base for the topology. We note that our
Muchnik topos interpretation of intuitionistic mathematics is an extension of
the well known Kolmogorov/Muchnik interpretation of intuitionistic
propositional calculus via Muchnik degrees, i.e., mass problems under weak
reducibility. We introduce a new sheaf representation of the intuitionistic
real numbers, \emph{the Muchnik reals}, which are different from the Cauchy
reals and the Dedekind reals. Within the Muchnik topos we obtain a \emph{choice
principle} and a \emph{bounding principle} where range over Muchnik
reals, ranges over functions from Muchnik reals to Muchnik reals, and
is a formula not containing or . For the convenience of the
reader, we explain all of the essential background material on intuitionism,
sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems,
Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an
English translation of Muchnik's 1963 paper on Muchnik degrees.Comment: 44 page
The Structure of Differential Invariants and Differential Cut Elimination
The biggest challenge in hybrid systems verification is the handling of
differential equations. Because computable closed-form solutions only exist for
very simple differential equations, proof certificates have been proposed for
more scalable verification. Search procedures for these proof certificates are
still rather ad-hoc, though, because the problem structure is only understood
poorly. We investigate differential invariants, which define an induction
principle for differential equations and which can be checked for invariance
along a differential equation just by using their differential structure,
without having to solve them. We study the structural properties of
differential invariants. To analyze trade-offs for proof search complexity, we
identify more than a dozen relations between several classes of differential
invariants and compare their deductive power. As our main results, we analyze
the deductive power of differential cuts and the deductive power of
differential invariants with auxiliary differential variables. We refute the
differential cut elimination hypothesis and show that, unlike standard cuts,
differential cuts are fundamental proof principles that strictly increase the
deductive power. We also prove that the deductive power increases further when
adding auxiliary differential variables to the dynamics
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