The Experts below are selected from a list of 318 Experts worldwide ranked by ideXlab platform
Jouko Vaananen - One of the best experts on this subject based on the ideXlab platform.
-
Second‐Order Logic and Set Theory
Philosophy Compass, 2015Co-Authors: Jouko VaananenAbstract:Both Second-Order Logic and set theory can be used as a foundation for mathematics, that is, as a formal language in which propositions of mathematics can be expressed and proved. We take it upon ourselves in this paper to compare the two approaches, Second-Order Logic on one hand and set theory on the other hand, evaluating their merits and weaknesses. We argue that we should think of first-order set theory as a very high-order Logic.
-
Categoricity and Consistency in Second-Order Logic
Inquiry, 2015Co-Authors: Jouko VaananenAbstract:We analyse the concept of a Second-Order characterisable structure and divide this concept into two parts—consistency and categoricity—with different strength and nature. We argue that categorical characterisation of mathematical structures in Second-Order Logic is meaningful and possible without assuming that the semantics of Second-Order Logic is defined in set theory. This extends also to the so-called Henkin structures.
-
Boolean-Valued Second-Order Logic
Notre Dame Journal of Formal Logic, 2015Co-Authors: Daisuke Ikegami, Jouko VaananenAbstract:In so-called full Second-Order Logic, the Second-Order variables range over all subsets and relations of the domain in question. In so-called Henkin Second-Order Logic, every model is endowed with a set of subsets and relations which will serve as the range of the Second-Order variables. In our Boolean-valued Second-Order Logic, the Second-Order variables range over all Boolean-valued subsets and relations on the domain. We show that under large cardinal assumptions Boolean-valued Second-Order Logic is more robust than full Second-Order Logic. Its validity is absolute under forcing, and its Hanf and Lowenheim numbers are smaller than those of full Second-Order Logic.
-
On Second Order Logic
Philosophical Inquiry, 2015Co-Authors: Jouko VaananenAbstract:I defend here the view that (full) second order Logic, if considered as a foundation of mathematics, is best understood as a fragment of set theory. This view is very common among set theorists, and therefore would perhaps not require defending. However, there are many non-set theorists who hold quite a different view, namely that (full) second order Logic is a better foundation for mathematics because it is capable to characterise—unlike the first order Logic underlying set theory—important mathematical structures up to isomorphism. Let us first discuss what do I mean when I say that I consider second order Logic as a foundation of mathematics. I have in mind the following: Whatever mathematical structures mathematicians have need of can be characterised in second order Logic. Then truths about these structures can be identified with Logical consequences of those characterisations. If a mathematician doubts the truth of a statement, he or she needs only to figure out what is the structure that the claim is concerned with and then check whether the characterisation of the structure Logically implies the claim. There are several points in this scenario which need clarification. First of all, in mathematical practice one sometimes needs to appeal to third order Logic, that is, second order is not enough; but I consider this an irrelevant point. Secondly, is it really true that whatever mathematical structures mathematicians have need of can indeed be characterised in second order Logic up to isomorphism? After all, there are only countably many possible
-
Epistemology versus Ontology - Second Order Logic, Set Theory and Foundations of Mathematics
Epistemology versus Ontology, 2012Co-Authors: Jouko VaananenAbstract:The question, whether second order Logic is a better foundation for mathematics than set theory, is addressed. The main difference between second order Logic and set theory is that set theory builds up a transfinite cumulative hierarchy while second order Logic stays within one application of the power sets. It is argued that in many ways this difference is illusory. More importantly, it is argued that the often stated difference, that second order Logic has categorical characterizations of relevant mathematical structures, while set theory has non-standard models, amounts to no difference at all. Second order Logic and set theory permit quite similar categoricity results on one hand, and similar non-standard models on the other hand.
Peter Fletcher - One of the best experts on this subject based on the ideXlab platform.
-
From Second-Order Calculus of Proof Functions to Second-Order Logic of Partial Terms
Truth Proof and Infinity, 1998Co-Authors: Peter FletcherAbstract:In this chapter, the syntax of formulae in the Logic of Partial Terms (LPT) is extended to include the atomic formula x = α and the Second-Order universal quantifier ∀2, interpreted as a combination of ∀’ and ∀. Axioms and rules for these new constructs are added to LPT, giving Second-Order Logic of Partial Terms (2LPT). In this chapter I shall extend the interpretation of LPT in CPF given in Chapter 29 to an interpretation of 2LPT in 2CPF.
-
Second-Order Logic
Truth Proof and Infinity, 1998Co-Authors: Peter FletcherAbstract:Second-Order Logic (2L) consists of first-order Logic (L, Chapter 26), with the metavariables that denote variables restricted to range only over object variables, plus the following definitions and theorems. In this chapter, metavariables that denote variables are interpreted as follows.
-
Second-Order Logic of Partial Terms
Truth Proof and Infinity, 1998Co-Authors: Peter FletcherAbstract:Second-Order Logic of Partial Terms (2LPT) is obtained from Logic of Partial Terms (LPT, Chapter 30) as follows. In this chapter, metavariables that denote variables are interpreted as follows.
-
From Second-Order Logic to Second-Order Calculus of Proof Functions
Truth Proof and Infinity, 1998Co-Authors: Peter FletcherAbstract:Second-Order Calculus of Proof Functions (2CPF) is obtained from Calculus of Proof Functions (CPF) by adding axioms and rules for the Logical constants introduced in Second-Order Logic. In this chapter I shall extend the arguments of Chapter 27 (‘From Logic to Calculus of Proof Functions’) to incorporate these new axioms and rules.
-
From the Second-Order Coding of Trees to Second-Order Logic
Truth Proof and Infinity, 1998Co-Authors: Peter FletcherAbstract:Second-Order Logic (2L) is obtained from first-order Logic (L) by adding type predicates and the Logical constants apply, val’, predify’ and ‘∀’. In this chapter I shall extend the arguments of Chapter 25 (‘From the Coding of Trees to Logic’) to include these new Logical constants and their properties. Some notions defined in Chapter 25 need to be modified slightly to take account of the difference between object and type variables.
Stephan Kreutzer - One of the best experts on this subject based on the ideXlab platform.
-
LICS - Quantitative Monadic Second-Order Logic
2013 28th Annual ACM IEEE Symposium on Logic in Computer Science, 2013Co-Authors: Stephan Kreutzer, Cristian RiverosAbstract:While monadic Second-Order Logic is a prominent Logic for specifying languages of finite words, it lacks the power to compute quantitative properties, e.g. to count. An automata model capable of computing such properties are weighted automata, but Logics equivalent to these automata have only recently emerged. We propose a new framework for adding quantitative properties to Logics specifying Boolean properties of words. We use this to define Quantitative Monadic Second-Order Logic (QMSO). In this way we obtain a simple Logic which is equally expressive to weighted automata. We analyse its evaluation complexity, both data and combined complexity, and show completeness results for combined complexity. We further refine the analysis of this Logic and obtain fragments that characterise exactly subclasses of weighted automata defined by the level of ambiguity allowed in the automata. In this way, we define a quantitative Logic which has good decidability properties while being resonably expressive and enjoying a simple syntactical definition.
-
Lower Bounds for the Complexity of Monadic Second-Order Logic
arXiv: Logic in Computer Science, 2010Co-Authors: Stephan Kreutzer, Siamak TazariAbstract:Courcelle's famous theorem from 1990 states that any property of graphs definable in monadic Second-Order Logic (MSO) can be decided in linear time on any class of graphs of bounded treewidth, or in other words, MSO is fixed-parameter tractable in linear time on any such class of graphs. From a Logical perspective, Courcelle's theorem establishes a sufficient condition, or an upper bound, for tractability of MSO-model checking. Whereas such upper bounds on the complexity of Logics have received significant attention in the literature, almost nothing is known about corresponding lower bounds. In this paper we establish a strong lower bound for the complexity of monadic Second-Order Logic. In particular, we show that if C is any class of graphs which is closed under taking subgraphs and whose treewidth is not bounded by a polylogarithmic function (in fact, $\log^c n$ for some small c suffices) then MSO-model checking is intractable on C (under a suitable assumption from complexity theory).
-
LICS - Lower Bounds for the Complexity of Monadic Second-Order Logic
2010 25th Annual IEEE Symposium on Logic in Computer Science, 2010Co-Authors: Stephan Kreutzer, Siamak TazariAbstract:Courcelle's famous theorem from 1990 states that any property of graphs definable in monadic Second-Order Logic (MSO_2) can be decided in linear time on any class of graphs of bounded tree-width, or in other words, MSO_2 is fixed-parameter tractable in linear time on any such class of graphs. From a Logical perspective, Courcelle's theorem establishes a sufficient condition, or an upper bound, for tractability of MSO_2-model checking. Whereas such upper bounds on the complexity of Logics have received significant attention in the literature, almost nothing is known about corresponding lower bounds. In this paper we establish a strong lower bound for the complexity of monadic Second-Order Logic. In particular, we show that if C is any class of graphs which is closed under taking sub-graphs and whose tree-width is not bounded by a poly-logarithmic function (in fact, log^c n for some small c suffices) then MSO_2-model checking is intractable on C (under a suitable assumption from complexity theory).
Siamak Tazari - One of the best experts on this subject based on the ideXlab platform.
-
Lower Bounds for the Complexity of Monadic Second-Order Logic
arXiv: Logic in Computer Science, 2010Co-Authors: Stephan Kreutzer, Siamak TazariAbstract:Courcelle's famous theorem from 1990 states that any property of graphs definable in monadic Second-Order Logic (MSO) can be decided in linear time on any class of graphs of bounded treewidth, or in other words, MSO is fixed-parameter tractable in linear time on any such class of graphs. From a Logical perspective, Courcelle's theorem establishes a sufficient condition, or an upper bound, for tractability of MSO-model checking. Whereas such upper bounds on the complexity of Logics have received significant attention in the literature, almost nothing is known about corresponding lower bounds. In this paper we establish a strong lower bound for the complexity of monadic Second-Order Logic. In particular, we show that if C is any class of graphs which is closed under taking subgraphs and whose treewidth is not bounded by a polylogarithmic function (in fact, $\log^c n$ for some small c suffices) then MSO-model checking is intractable on C (under a suitable assumption from complexity theory).
-
LICS - Lower Bounds for the Complexity of Monadic Second-Order Logic
2010 25th Annual IEEE Symposium on Logic in Computer Science, 2010Co-Authors: Stephan Kreutzer, Siamak TazariAbstract:Courcelle's famous theorem from 1990 states that any property of graphs definable in monadic Second-Order Logic (MSO_2) can be decided in linear time on any class of graphs of bounded tree-width, or in other words, MSO_2 is fixed-parameter tractable in linear time on any such class of graphs. From a Logical perspective, Courcelle's theorem establishes a sufficient condition, or an upper bound, for tractability of MSO_2-model checking. Whereas such upper bounds on the complexity of Logics have received significant attention in the literature, almost nothing is known about corresponding lower bounds. In this paper we establish a strong lower bound for the complexity of monadic Second-Order Logic. In particular, we show that if C is any class of graphs which is closed under taking sub-graphs and whose tree-width is not bounded by a poly-logarithmic function (in fact, log^c n for some small c suffices) then MSO_2-model checking is intractable on C (under a suitable assumption from complexity theory).
Marijke H. L. Bodlaender - One of the best experts on this subject based on the ideXlab platform.
-
IFIP TCS - Probabilistic inference and monadic second order Logic
Lecture Notes in Computer Science, 2012Co-Authors: Marijke H. L. BodlaenderAbstract:This paper combines two classic results from two different fields: the result by Lauritzen and Spiegelhalter [21] that the probabilistic inference problem on probabilistic networks can be solved in linear time on networks with a moralization of bounded treewidth, and the result by Courcelle [10] that problems that can be formulated in counting monadic second order Logic can be solved in linear time on graphs of bounded treewidth. It is shown that, given a probabilistic network whose moralization has bounded treewidth and a property P of the network and the values of the variables that can be formulated in counting monadic second order Logic, one can determine in linear time the probability that P holds.
