1,456 research outputs found
Introduction to clarithmetic I
"Clarithmetic" is a generic name for formal number theories similar to Peano
arithmetic, but based on computability logic (see
http://www.cis.upenn.edu/~giorgi/cl.html) instead of the more traditional
classical or intuitionistic logics. Formulas of clarithmetical theories
represent interactive computational problems, and their "truth" is understood
as existence of an algorithmic solution. Imposing various complexity
constraints on such solutions yields various versions of clarithmetic. The
present paper introduces a system of clarithmetic for polynomial time
computability, which is shown to be sound and complete. Sound in the sense that
every theorem T of the system represents an interactive number-theoretic
computational problem with a polynomial time solution and, furthermore, such a
solution can be efficiently extracted from a proof of T. And complete in the
sense that every interactive number-theoretic problem with a polynomial time
solution is represented by some theorem T of the system. The paper is written
in a semitutorial style and targets readers with no prior familiarity with
computability logic
Build your own clarithmetic I: Setup and completeness
Clarithmetics are number theories based on computability logic (see
http://www.csc.villanova.edu/~japaridz/CL/ ). Formulas of these theories
represent interactive computational problems, and their "truth" is understood
as existence of an algorithmic solution. Various complexity constraints on such
solutions induce various versions of clarithmetic. The present paper introduces
a parameterized/schematic version CLA11(P1,P2,P3,P4). By tuning the three
parameters P1,P2,P3 in an essentially mechanical manner, one automatically
obtains sound and complete theories with respect to a wide range of target
tricomplexity classes, i.e. combinations of time (set by P3), space (set by P2)
and so called amplitude (set by P1) complexities. Sound in the sense that every
theorem T of the system represents an interactive number-theoretic
computational problem with a solution from the given tricomplexity class and,
furthermore, such a solution can be automatically extracted from a proof of T.
And complete in the sense that every interactive number-theoretic problem with
a solution from the given tricomplexity class is represented by some theorem of
the system. Furthermore, through tuning the 4th parameter P4, at the cost of
sacrificing recursive axiomatizability but not simplicity or elegance, the
above extensional completeness can be strengthened to intensional completeness,
according to which every formula representing a problem with a solution from
the given tricomplexity class is a theorem of the system. This article is
published in two parts. The present Part I introduces the system and proves its
completeness, while Part II is devoted to proving soundness
Logics for Unranked Trees: An Overview
Labeled unranked trees are used as a model of XML documents, and logical
languages for them have been studied actively over the past several years. Such
logics have different purposes: some are better suited for extracting data,
some for expressing navigational properties, and some make it easy to relate
complex properties of trees to the existence of tree automata for those
properties. Furthermore, logics differ significantly in their model-checking
properties, their automata models, and their behavior on ordered and unordered
trees. In this paper we present a survey of logics for unranked trees
On the Scope of the Universal-Algebraic Approach to Constraint Satisfaction
The universal-algebraic approach has proved a powerful tool in the study of
the complexity of CSPs. This approach has previously been applied to the study
of CSPs with finite or (infinite) omega-categorical templates, and relies on
two facts. The first is that in finite or omega-categorical structures A, a
relation is primitive positive definable if and only if it is preserved by the
polymorphisms of A. The second is that every finite or omega-categorical
structure is homomorphically equivalent to a core structure. In this paper, we
present generalizations of these facts to infinite structures that are not
necessarily omega-categorical. (This abstract has been severely curtailed by
the space constraints of arXiv -- please read the full abstract in the
article.) Finally, we present applications of our general results to the
description and analysis of the complexity of CSPs. In particular, we give
general hardness criteria based on the absence of polymorphisms that depend on
more than one argument, and we present a polymorphism-based description of
those CSPs that are first-order definable (and therefore can be solved in
polynomial time).Comment: Extended abstract appeared at 25th Symposium on Logic in Computer
Science (LICS 2010). This version will appear in the LMCS special issue
associated with LICS 201
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
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Imdlawn Tashlhiyt Berber Syllabification is Quantifier-Free
Imdlawn Tashlhiyt Berber (ITB) is unusual due to its tolerance of non-vocalic syllabic nuclei. Rule-based and constraint-based accounts of ITB syllabification do not directly address the question of how complex the process is. Model theory and formal logic allow for comparison of complexity across different theories of phonology by identifying the computational power (or expressivity) of linguistic formalisms in a grammar-independent way. With these tools, I develop a mathematical formalism for representing ITB syllabification using Quantifier-Free (QF) logic, one of the least powerful logics known. This result indicates that ITB syllabification is relatively simple from a computational standpoint and that grammatical formalisms could succeed with even less powerful mechanisms than are currently accepted
Ptarithmetic
The present article introduces ptarithmetic (short for "polynomial time
arithmetic") -- a formal number theory similar to the well known Peano
arithmetic, but based on the recently born computability logic (see
http://www.cis.upenn.edu/~giorgi/cl.html) instead of classical logic. The
formulas of ptarithmetic represent interactive computational problems rather
than just true/false statements, and their "truth" is understood as existence
of a polynomial time solution. The system of ptarithmetic elaborated in this
article is shown to be sound and complete. Sound in the sense that every
theorem T of the system represents an interactive number-theoretic
computational problem with a polynomial time solution and, furthermore, such a
solution can be effectively extracted from a proof of T. And complete in the
sense that every interactive number-theoretic problem with a polynomial time
solution is represented by some theorem T of the system.
The paper is self-contained, and can be read without any previous familiarity
with computability logic.Comment: Substantially better versions are on their way. Hence the present
article probably will not be publishe
A Uniform Method for Proving Lower Bounds of the Computational Complexity of Logical Theories
https://deepblue.lib.umich.edu/bitstream/2027.42/154178/1/39015100081655.pd
Unary Pushdown Automata and Straight-Line Programs
We consider decision problems for deterministic pushdown automata over a
unary alphabet (udpda, for short). Udpda are a simple computation model that
accept exactly the unary regular languages, but can be exponentially more
succinct than finite-state automata. We complete the complexity landscape for
udpda by showing that emptiness (and thus universality) is P-hard, equivalence
and compressed membership problems are P-complete, and inclusion is
coNP-complete. Our upper bounds are based on a translation theorem between
udpda and straight-line programs over the binary alphabet (SLPs). We show that
the characteristic sequence of any udpda can be represented as a pair of
SLPs---one for the prefix, one for the lasso---that have size linear in the
size of the udpda and can be computed in polynomial time. Hence, decision
problems on udpda are reduced to decision problems on SLPs. Conversely, any SLP
can be converted in logarithmic space into a udpda, and this forms the basis
for our lower bound proofs. We show coNP-hardness of the ordered matching
problem for SLPs, from which we derive coNP-hardness for inclusion. In
addition, we complete the complexity landscape for unary nondeterministic
pushdown automata by showing that the universality problem is -hard, using a new class of integer expressions. Our techniques have
applications beyond udpda. We show that our results imply -completeness for a natural fragment of Presburger arithmetic and coNP lower
bounds for compressed matching problems with one-character wildcards
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