We prove that the problem of determining the minimum propositional proof length is NP- hard to approximate within a factor of 2 log 1 - o(1) n . These results are very robust in that they hold for almost all natural proof systems, including: Frege systems, extended Frege systems, resolution, Horn resolution, the polynomial calculus, the sequent calculus, the cut-free sequent calculus, as well as the polynomial calculus. Our hardness of approximation results usually apply to proof length measured either by (...) number of symbols or by number of inferences, for tree-like or dag-like proofs. We introduce the Monotone Minimum (Circuit) Satisfying Assignment problem and reduce it to the problems of approximation of the length of proofs. (shrink)

From proofs in any classical first-order theory that proves the existence of at least two elements, one can eliminate definitions in polynomial time. From proofs in any classical first-order theory strong enough to code finite functions, including sequential theories, one can also eliminate Skolem functions in polynomial time.

In this note we show that the classical modal technology of Sahlqvist formulas gives quick proofs of the completeness theorems in [8] (D. Gregory, Completeness and decidability results for some propositional modal logics containing actually operators, Journal of Philosophical Logic 30(1): 57–78, 2001) and vastly generalizes them. Moreover, as a corollary, interpolation theorems for the logics considered in [8] are obtained. We then compare Gregory's modal language enriched with an actually operator with the work of Arthur Prior now known under (...) the name of hybrid logic. This analysis relates the actually axioms to standard hybrid axioms, yields the decidability results in [8], and provides a number of complexity results. Finally, we use a bisimulation argument to show that the hybrid language is strictly more expressive than Gregory's language. (shrink)

Many of the formalisms used in Attribute Value grammar are notational variants of languages of propositional modal logic, and testing whether two Attribute Value Structures unify amounts to testing for modal satisfiability. In this paper we put this observation to work. We study the complexity of the satisfiability problem for nine modal languages which mirror different aspects of AVS description formalisms, including the ability to express re-entrancy, the ability to express generalisations, and the ability to express recursive constraints. Two main (...) techniques are used: either Kripke models with desirable properties are constructed, or modalities are used to simulate fragments of Propositional Dynamic Logic. Further possibilities for the application of modal logic in computational linguistics are noted. (shrink)

There is an exponential speed-up in the number of lines of the quantified propositional sequent calculus over Substitution Frege Systems, if one considers proofs as trees. Whether this is true also for the number of symbols, is still an open problem.

We give resource bounded versions of the Completeness Theorem for propositional and predicate logic. For example, it is well known that every computable consistent propositional theory has a computable complete consistent extension. We show that, when length is measured relative to the binary representation of natural numbers and formulas, every polynomial time decidable propositional theory has an exponential time (EXPTIME) complete consistent extension whereas there is a nondeterministic polynomial time (NP) decidable theory which has no polynomial time complete consistent extension (...) when length is measured relative to the binary representation of natural numbers and formulas. It is well known that a propositional theory is axiomatizable (respectively decidable) if and only if it may be represented as the set of infinite paths through a computable tree (respectively a computable tree with no dead ends). We show that any polynomial time decidable theory may be represented as the set of paths through a polynomial time decidable tree. On the other hand, the statement that every polynomial time decidable, relative to the tally representation of natural numbers and formulas, is equivalent to P = NP. For predicate logic, we develop a complexity theoretic version of the Henkin construction to prove a complexity theoretic version of the Completeness Theorem. Our results imply that that any polynomial space decidable theory △ possesses a polynomial space computable model which is exponential space decidable and thus △ has an exponential space complete consistent extension. Similar results are obtained for other notions of complexity. (shrink)

We show the completeness of a Hilbert-style system LK defined by M. Valiev involving the knowledge operator K dedicated to the reasoning with incomplete information. The completeness proof uses a variant of Makinson's canonical model construction. Furthermore we prove that the theoremhood problem for LK is co-NP-complete, using techniques similar to those used to prove that the satisfiability problem for propositional S5 is NP-complete.

This book asks not only how the study of white-collar crime can enrich our understanding of crime and justice more generally, but also how criminological ...

The main powerful method for establishing termination of term rewriting systems was discovered by Nachum Dershowitz through the introduction of certain natural well founded orderings (lexicographic path orderings). This leads to natural decision problems which may be of the highest computational complexity of any decidable problems appearing in a natural established computer science context.

While the theory of belief change has attracted a lot of interest from researchers, work on implementing belief change and actually putting it to use in real-world problems is still scarce. In this paper, we present an implementation of propositional belief change using Binary Decision Diagrams. Upper complexity bounds for the algorithm are presented and discussed. The approach is presented both in the general case, as well as on specific belief change operators from the literature. In an effort to gain (...) a better understanding of the empirical efficiency of the algorithms involved, a fault diagnosis problem on combinational circuits is presented, implemented and evaluated. (shrink)

We identify the computational complexity of the satisfiability problem for FO 2 , the fragment of first-order logic consisting of all relational first-order sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity of its decision problem has not been pinpointed so far. In 1975 Mortimer proved that FO 2 has the finite-model property, which means that if an FO 2 -sentence is satisfiable, then it has a finite (...) model. Moreover, Mortimer showed that every satisfiable FO 2 -sentence has a model whose size is at most doubly exponential in the size of the sentence. In this paper, we improve Mortimer's bound by one exponential and show that every satisfiable FO 2 -sentence has a model whose size is at most exponential in the size of the sentence. As a consequence, we establish that the satisfiability problem for FO 2 is NEXPTIME-complete. (shrink)

We propose a new interpretation of objective deterministic chances in statistical physics based on physical computational complexity. This notion applies to a single physical system (be it an experimental set--up in the lab, or a subsystem of the universe), and quantifies (1) the difficulty to realize a physical state given another, (2) the 'distance' (in terms of physical resources) from a physical state to another, and (3) the size of the set of time--complexity functions that are compatible with the physical (...) resources required to reach a physical state from another. (shrink)

Automated theorem proving amounts to solving search problems in usually tremendous search spaces. A lot of research therefore focuses on search space reductions. Our approach reduces the search space which arises when using so-called connection tableau calculi for first-order automated theorem proving. It uses disjunctive constraints over first-order equations to compress certain parts of this search space. We present the basics of our constrained-connection-tableau calculi, a constraint extension of connection tableau calculi, and deal with the efficient handling of constraints during (...) the search process. The new techniques are integrated into the automated connection tableau prover Setheo. (shrink)

Aleksandar Ignjatović. Delineating Classes of Computational Complexity via Second Order Theories with Weak Set Existence Principles (I).

We study definability of second-order generalized quantifiers. We show that the question whether a second-order generalized quantifier $\sQ_1$ is definable in terms of another quantifier $\sQ_2$, the base logic being monadic second-order logic, reduces to the question if a quantifier $\sQ^{\star}_1$ is definable in $\FO(\sQ^{\star}_2,<,+,\times)$ for certain first-order quantifiers $\sQ^{\star}_1$ and $\sQ^{\star}_2$. We use our characterization to show new definability and non-definability results for second-order generalized quantifiers. In particular, we show that the monadic second-order majority quantifier $\most^1$ is not definable (...) in second-order logic. (shrink)

We consider collective quantification in natural language. For many years the common strategy in formalizing collective quantification has been to define the meanings of collective determiners, quantifying over collections, using certain type-shifting operations. These type-shifting operations, i.e., lifts, define the collective interpretations of determiners systematically from the standard meanings of quantifiers. All the lifts considered in the literature turn out to be definable in second-order logic. We argue that second-order definable quantifiers are probably not expressive enough to formalize all collective (...) quantification in natural language. (shrink)

We consider the problem about the length of proofs of the sentences $\operatorname{Con}_S(\underline{n})$ saying that there is no proof of contradiction in S whose length is ≤ n. We show the relation of this problem to some problems about propositional proof systems.

Some problems rarely discussed in traditional philosophy of science are mentioned: The empirical sciences using mathematico-quantitative theoretical models are frequently confronted with several types of computational problems posing primarily methodological limitations on explanatory and prognostic matters. Such limitations may arise from the appearances of deterministic chaos and (too) high computational complexity in general. In many cases, however, scientists circumvent such limitations by utilizing reductional approximations or complexity reductions for intractable problem formulations, thus constructing new models which are computationally tractable. (...) Such activities are compared with reduction types (more) established in philosophy of science. (shrink)

Given any simply consistent formal theory F of the state complexity L(S) of finite binary sequences S as computed by 3-tape-symbol Turing machines, there exists a natural number L(F ) such that L(S) > n is provable in F only if n < L(F ). On the other hand, almost all finite binary sequences S satisfy L(S) > L(F ). The proof resembles Berry’s..

I argue that considerations about computational complexity show that all finite agents need characteristics like those that have been called epistemic virtues. The necessity of these virtues follows in part from the nonexistence of shortcuts, or efficient ways of finding shortcuts, to cognitively expensive routines. It follows that agents must possess the capacities – metavirtues –of developing in advance the cognitive virtues they will need when time and memory are at a premium.

I argue that considerations about computational complexity show that all finite agents need characteristics like those that have been called epistemic virtues. The necessity of these virtues follows in part from the nonexistence of shortcuts, or efficient ways of finding shortcuts, to cognitively expensive routines. It follows that agents must possess the capacities – metavirtues –of developing in advance the cognitive virtues they will need when time and memory are at a premium.

Hilbert's ε-calculus is based on an extension of the language of predicate logic by a term-forming operator ex. Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand's Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential (...) theorems obtained by this elimination procedure. (shrink)

We consider the notion of everyday language. We claim that everyday language is semantically bounded by the properties expressible in the existential fragment of second–order logic. Two arguments for this thesis are formulated. Firstly, we show that so–called Barwise's test of negation normality works properly only when assuming our main thesis. Secondly, we discuss the argument from practical computability for finite universes. Everyday language sentences are directly or indirectly verifiable. We show that in both cases they are bounded by second–order (...) existential properties. Moreover, there are known examples of everyday language sentences which are the most difficult in this class (NPTIME–complete). (shrink)

The problem of computational complexity of semantics for some natural language constructions – considered in [M. Mostowski, D. Wojtyniak 2004] – motivates an interest in complexity of Ramsey quantifiers in finite models. In general a sentence with a Ramsey quantifier R of the following form Rx, yH(x, y) is interpreted as ∃A(A is big relatively to the universe ∧A2 ⊆ H). In the paper cited the problem of the complexity of the Hintikka sentence is reduced to the problem of computational (...) complexity of the Ramsey quantifier for which the phrase “A is big relatively to the universe” is interpreted as containing at least one representative of each equivalence class, for some given equvalence relation. In this work we consider quantifiers Rf, for which “A is big relatively to the universe” means “card(A) > f (n), where n is the size of the universe”. Following [Blass, Gurevich 1986] we call R mighty if Rx, yH(x, y) defines N P – complete class of finite models. Similarly we say that Rf is N P –hard if the corresponding class is N P –hard. We prove the following theorems. (shrink)

Currently, the widely used notion of activity is increasingly present in computer science. However, because this notion is used in specific contexts, it becomes vague. Here, the notion of activity is scrutinized in various contexts and, accordingly, put in perspective. It is discussed through four scientific disciplines: computer science, biology, economics, and epistemology. The definition of activity usually used in simulation is extended to new qualitative and quantitative definitions. In computer science, biology and economics disciplines, the new simulation activity definition (...) is first applied critically. Then, activity is discussed generally. In epistemology, activity is discussed, in a prospective way, as a possible framework in models of human beliefs and knowledge. (shrink)

In this paper we give some formal examples of ideas developed by Penco in two papers on the tension inside Frege's notion of sense (see Penco 2003). The paper attempts to compose the tension between semantic and cognitive aspects of sense, through the idea of sense as proof or procedure – not as an alternative to the idea of sense as truth condition, but as complementary to it (as it happens sometimes in the old tradition of procedural semantics).

The numerically definite syllogistic is the fragment of English obtained by extending the language of the classical syllogism with numerical quantifiers. The numerically definite relational syllogistic is the fragment of English obtained by extending the numerically definite syllogistic with predicates involving transitive verbs. This paper investigates the computational complexity of the satisfiability problem for these fragments. We show that the satisfiability problem (= finite satisfiability problem) for the numerically definite syllogistic is strongly NP-complete, and that the satisfiability problem (= finite (...) satisfiability problem) for the numerically definite relational syllogistic is NEXPTIME-complete, but perhaps not strongly so. We discuss the related problem of probabilistic (propositional) satisfiability, and thereby demonstrate the incompleteness of some proof-systems that have been proposed for the numerically definite syllogistic. (shrink)

This introduction to certain mathematical topics central to theoretical computer science treats computability and recursive functions, formal languages and automata, computational complexity, and cruptography. The presentation is essentially self-contained with detailed proofs of all statements provided. Although it begins with the basics, it proceeds to some of the most important recent developments in theoretical computer science.

Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.

We study the computational complexity of polyadic quantifiers in natural language. This type of quantification is widely used in formal semantics to model the meaning of multi-quantifier sentences. First, we show that the standard constructions that turn simple determiners into complex quantifiers, namely Boolean operations, iteration, cumulation, and resumption, are tractable. Then, we provide an insight into branching operation yielding intractable natural language multi-quantifier expressions. Next, we focus on a linguistic case study. We use computational complexity results to investigate semantic (...) distinctions between quantified reciprocal sentences. We show a computational dichotomy<br>between different readings of reciprocity. Finally, we go more into philosophical speculation on meaning, ambiguity and computational complexity. In particular, we investigate a possibility to<br>revise the Strong Meaning Hypothesis with complexity aspects to better account for meaning shifts in the domain of multi-quantifier sentences. The paper not only contributes to the field of the formal<br>semantics but also illustrates how the tools of computational complexity theory might be successfully used in linguistics and philosophy with an eye towards cognitive science. (shrink)

In the dissertation we study the complexity of generalized quantifiers in natural language. Our perspective is interdisciplinary: we combine philosophical insights with theoretical computer science, experimental cognitive science and linguistic theories. -/- In Chapter 1 we argue for identifying a part of meaning, the so-called referential meaning (model-checking), with algorithms. Moreover, we discuss the influence of computational complexity theory on cognitive tasks. We give some arguments to treat as cognitively tractable only those problems which can be computed in polynomial time. (...) Additionally, we suggest that plausible semantic theories of the everyday fragment of natural language can be formulated in the existential fragment of second-order logic. -/- In Chapter 2 we give an overview of the basic notions of generalized quantifier theory, computability theory, and descriptive complexity theory. -/- In Chapter 3 we prove that PTIME quantifiers are closed under iteration, cumulation and resumption. Next, we discuss the NP-completeness of branching quantifiers. Finally, we show that some Ramsey quantifiers define NP-complete classes of finite models while others stay in PTIME. We also give a sufficient condition for a Ramsey quantifier to be computable in polynomial time. -/- In Chapter 4 we investigate the computational complexity of polyadic lifts expressing various readings of reciprocal sentences with quantified antecedents. We show a dichotomy between these readings: the strong reciprocal reading can create NP-complete constructions, while the weak and the intermediate reciprocal readings do not. Additionally, we argue that this difference should be acknowledged in the Strong Meaning hypothesis. -/- In Chapter 5 we study the definability and complexity of the type-shifting approach to collective quantification in natural language. We show that under reasonable complexity assumptions it is not general enough to cover the semantics of all collective quantifiers in natural language. The type-shifting approach cannot lead outside second-order logic and arguably some collective quantifiers are not expressible in second-order logic. As a result, we argue that algebraic (many-sorted) formalisms dealing with collectivity are more plausible than the type-shifting approach. Moreover, we suggest that some collective quantifiers might not be realized in everyday language due to their high computational complexity. Additionally, we introduce the so-called second-order generalized quantifiers to the study of collective semantics. -/- In Chapter 6 we study the statement known as Hintikka's thesis: that the semantics of sentences like ``Most boys and most girls hate each other'' is not expressible by linear formulae and one needs to use branching quantification. We discuss possible readings of such sentences and come to the conclusion that they are expressible by linear formulae, as opposed to what Hintikka states. Next, we propose empirical evidence confirming our theoretical predictions that these sentences are sometimes interpreted by people as having the conjunctional reading. -/- In Chapter 7 we discuss a computational semantics for monadic quantifiers in natural language. We recall that it can be expressed in terms of finite-state and push-down automata. Then we present and criticize the neurological research building on this model. The discussion leads to a new experimental set-up which provides empirical evidence confirming the complexity predictions of the computational model. We show that the differences in reaction time needed for comprehension of sentences with monadic quantifiers are consistent with the complexity differences predicted by the model. -/- In Chapter 8 we discuss some general open questions and possible directions for future research, e.g., using different measures of complexity, involving game-theory and so on. -/- In general, our research explores, from different perspectives, the advantages of identifying meaning with algorithms and applying computational complexity analysis to semantic issues. It shows the fruitfulness of such an abstract computational approach for linguistics and cognitive science. (shrink)

We study the computational complexity of reciprocal sentences with quantified antecedents. We observe a computational dichotomy between different interpretations of reciprocity, and shed some light on the status of the so-called Strong Meaning Hypothesis.

We discuss McMillan et al. (2005) paper devoted to study brain activity during comprehension of sentences with generalized quantifiers. According to the authors their results verify a particular computational model of natural language quantifier comprehension posited by several linguists and logicians (e. g. see van Benthem, 1986). We challenge this statement by invoking the computational difference between first-order quantifiers and divisibility quantifiers (e. g. see Mostowski, 1998). Moreover, we suggest other studies on quantifier comprehension, which can throw more light on (...) the role of working memory in processing quantifiers. (shrink)

The paper presents an experimental evidence on differences in the sentence-picture verification under additional memory load between parity and proportional quantifiers. We asked subjects to memorize strings of 4 or 6 digits, then to decide whether a quantifier sentence is true at a given picture, and finally to recall the initially given string of numbers. The results show that: (a) proportional quantifiers are more difficult than parity quantifiers with respect to reaction time and accuracy; (b) maintaining either 4 or 6 (...) elements in the working memory has the same effect on the processing of parity quantifiers; (c) however, in the case of proportional quantifiers subjects performed better in the verification tasks under the 6-digit load condition, and (d) even though the strings of 4 numbers were better recalled by subjects after judging parity there is no difference between quantifiers in the case of the 6-element condition. We briefly outline two alternative explanations for the observed phenomena rooted in the computational model of quantifier verification and the different theories of working memory. (shrink)

We compare time needed for understanding different types of quantifiers. We show that the computational distinction between quantifiers recognized by finite-automata and pushdown automata is psychologically relevant. Our research improves upon hypothesis and explanatory power of recent neuroimaging studies as well as provides evidence for the claim that human linguistic abilities are constrained by computational complexity.

We examine the verification of simple quantifiers in natural language from a computational model perspective. We refer to previous neuropsychological investigations of the same problem and suggest extending their experimental setting. Moreover, we give some direct empirical evidence linking computational complexity predictions with cognitive reality.
In the empirical study we compare time needed for understanding different types of quantifiers. We show that the computational distinction between quantifiers recognized by finite-automata and push-down automata is psychologically relevant. Our research improves upon hypothesis and (...) explanatory power of recent neuroimaging studies as well as provides
evidence. (shrink)

Szymanik (2007) suggested that the distinction between first-order and higher-order quantifiers does not coincide with the computational resources required to compute the meaning of quantifiers. Cognitive difficulty of quantifier processing might be better assessed on the basis of complexity of the minimal corresponding automata. For example, both logical and numerical quantifiers are first-order. However, computational devices recognizing logical quantifiers have a fixed number of states while the number of states in automata corresponding to numerical quantifiers grows with the rank of (...) the quantifier. This observation partially explains the differences in processing between those two types of quantifiers (Troiani et al. 2009) and links them to the computational model. Taking this perspective, below, we suggest the experimental setting extending the one by McMillan et al. (2005) and Troiani et al. (2009). (shrink)

Human intentional communication is marked by its flexibility and context sensitivity. Hypothesized brain mechanisms can provide convincing and complete explanations of the human capacity for intentional communication only insofar as they can match the computational power required for displaying that capacity. It is thus of importance for cognitive neuroscience to know how computationally complex intentional communication actually is. Though the subject of considerable debate, the computational complexity of communication remains so far unknown. In this paper we defend the position that (...) the computational complexity of communication is not a constant, as some views of communication seem to hold, but rather a function of situational factors. We present a methodology for studying and characterizing the computational complexity of communication under different situational constraints. We illustrate our methodology for a model of the problems solved by receivers and senders during a communicative exchange. This approach opens the way to a principled identification of putative model parameters that control cognitive processes supporting intentional communication. (shrink)

As brightly shown by Mainzer [24], the science of complexity has many distinct origins in many disciplines. Those various origins has led to “an interdisciplinary methodology to explain the emergence of certain macroscopic phenomena via the nonlinear interactions of microscopic elements” (ibid.). This paper suggests that the parallel and strong expansion of modeling and simulation - especially after the Second World War and the subsequent development of computers - is a rationale which also can be counted as an explanation of (...) this emergence. With the benefit of hindsight, one can find three periods in the methodologies of modeling in the empirical sciences: 1st the simple modeling of the simple, 2nd the simple modeling of the complex, 3rd the complex modeling and simulation of the complex. Our main thesis is that the current spreading (since the 90’s) of complex computer simulations of systems of models (where a simulation is no more a step by step calculus of a unique logico-mathematical model) is another promising dimension of the science of complexity. Following this claim, we propose to distinguish three different types of computer simulations in the context of complex systems’ modeling. Finally, we show that these types of simulations lead to three different types of weak emergence, too. (shrink)

We compared the processing of natural language quantifiers in a group of patients with schizophrenia and a healthy control group. In both groups, the difficulty of the quantifiers was consistent with computational predictions, and patients with schizophrenia took more time to solve the problems. However, they were significantly less accurate only with proportional quantifiers, like more than half. This can be explained by noting that, according to the complexity perspective, only proportional quantifiers require working memory engagement.