Prenex Normal Form Research Articles

Feferman–Vaught Decompositions for Prefix Classes of First Order Logic

The Feferman–Vaught theorem provides a way of evaluating a first order sentence \(\varphi \) on a disjoint union of structures by producing a decomposition of \(\varphi \) into sentences which can be evaluated on the individual structures and the results of these evaluations combined using a propositional formula. This decomposition can in general be non-elementarily larger than \(\varphi \). We introduce a “tree” generalization of the prenex normal form (PNF) for first order sentences, and show that for an input sentence in this form having a fixed number of quantifier alternations, a Feferman–Vaught decomposition can be obtained in time elementary in the size of the sentence. The sentences in the decomposition are also in tree PNF, and further have the same number of quantifier alternations and the same quantifier rank as the input sentence. We extend this result by considering binary operations other than disjoint union, in particular sum-like operations such as join, ordered sum and NLC-sum, that are definable using quantifier-free translation schemes.

PRENEX NORMAL FORM THEOREMS IN SEMI-CLASSICAL ARITHMETIC

AbstractAkama et al. [1] systematically studied an arithmetical hierarchy of the law of excluded middle and related principles in the context of first-order arithmetic. In that paper, they first provide a prenex normal form theorem as a justification of their semi-classical principles restricted to prenex formulas. However, there are some errors in their proof. In this paper, we provide a simple counterexample of their prenex normal form theorem [1, Theorem 2.7], then modify it in an appropriate way which still serves to largely justify the arithmetical hierarchy. In addition, we characterize a variety of prenex normal form theorems by logical principles in the arithmetical hierarchy. The characterization results reveal that our prenex normal form theorems are optimal. For the characterization results, we establish a new conservation theorem on semi-classical arithmetic. The theorem generalizes a well-known fact that classical arithmetic is $\Pi _2$ -conservative over intuitionistic arithmetic.

The Skolemization of prenex formulas in intermediate logics

The skolem class of a logic consists of the formulas for which the derivability of the formula is equivalent to the derivability of its Skolemization. In contrast to classical logic, the skolem classes of many intermediate logics do not contain all formulas. In this paper it is proven for certain classes of propositional formulas that any instance of them by (independent) predicate sentences in prenex normal form belongs to the skolem class of any intermediate logic complete with respect to a class of well-founded trees. In particular, all prenex sentences belong to the skolem class of these logics, and this result extends to the constant domain versions of these logics.

О базисах тождеств многообразий группоидов отношений

Множество бинарных отношений, замкнутое относительно некоторой совокупности операций над ними, образует алгебру, называемую алгеброй отношений. Всякую такую алгебру можно рассматривать как упорядоченную отношением теоретико-множественного включения. Для заданного множества Ω операций над бинарными отношениями обозначим через V ar{Ω} [V ar{Ω,⊂}] многообразие, порождённое алгебрами [соответственно упорядоченными алгебрами] отношений с операциями из Ω. Операции над отношениями, как правило, задаются формулами исчисления предикатов первого порядка. Такие операции называются логическими. Важным классом логических операция является класс диофантовых операций. Операция называется диофантовой, если она может быть задана с помощью формулы, которая в своей предваренной нормальной форме содержит лишь операции конъюнкции и кванторы существования. В работе изучаются алгебры отношений с одной бинарной диофантовой операцией, то есть группоиды отношений. В качестве рассматриваемой операции выступает диофантова операция *, определяемая следующим образом: ρ*σ = {( x,у) ∈ X × X : (∃z)( x, z) ∈ ρ∧( x, z) ∈ σ}. Отношение ρ*σ представляет собой результат цилиндрификации пересечения ρ∩σ бинарных отношений ρ и σ. В работе находятся конечные базисы тождеств для многообразий V ar{*} иVar{*,⊂}. Группоид (A, ) принадлежит многообразиюV ar{*} тогда и только тогда, когда он удовлетворяет тождествам: xy = уx (1), (xу)2= xу (2), (xу)у = xу (3), x2у2= x2у (4), (x2у2)z = x2(у2z) (5). Упорядоченный группоид (A,·,≤) принадлежит многообразию V ar{*,⊂} тогда и только тогда, когда он удовлетворяет тождествам (1)–(5) и тождествам: х ≤ x2(6), xу ≤ x2(7). В качестве следствия также получен конечный базис тождеств многообразияVar{*,∪}.

Decidability of the AE-theory of the lattice of $${\varPi }_1^0$$ Π 1 0 classes

An AE-sentence is a sentence in prenex normal form with all universal quantifiers preceding all existential quantifiers, and the AE-theory of a structure is the set of all AE-sentences true in the structure. We show that the AE-theory of \((\mathscr {L}({\varPi }_1^0), \cap , \cup , 0, 1)\) is decidable by giving a procedure which, for any AE-sentence in the language, determines the truth or falsity of the sentence in our structure.

Solving quantified linear arithmetic by counterexample-guided instantiation

This paper presents a framework to derive instantiation-based decision procedures for satisfiability of quantified formulas in first-order theories, including its correctness, implementation, and evaluation. Using this framework we derive decision procedures for linear real arithmetic and linear integer arithmetic formulas with one quantifier alternation. We discuss extensions of these techniques for handling mixed real and integer arithmetic, and to formulas with arbitrary quantifier alternations. For the latter, we use a novel strategy that handles quantified formulas that are not in prenex normal form, which has advantages with respect to existing approaches. All of these techniques can be integrated within the solving architecture used by typical SMT solvers. Experimental results on standardized benchmarks from model checking, static analysis, and synthesis show that our implementation in the SMT solver cvc4 outperforms existing tools for quantified linear arithmetic.

A note on the size of prenex normal forms

The textbook method for converting a first-order logic formula to prenex normal form potentially leads to an exponential growth of the formula size, if the formula is allowed to use all of the classical logical connectives ∧, ∨, →, ↔, ¬. This note presents a short proof which shows that the conversion is possible with polynomial growth of the formula size.

One quantifier alternation in first-order logic with modular predicates

© 2015 EDP Sciences. Adding modular predicates yields a generalization of first-order logic FO over words. The expressive power of FO[ < , MOD] with order comparison x < y and predicates for x ≡ i mod n has been investigated by Barrington et al. The study of FO[ < , MOD]-fragments was initiated by Chaubard et al. More recently, Dartois and Paperman showed that definability in the two-variable fragment FO 2 [ < , MOD] is decidable. In this paper we continue this line of work. We give an effective algebraic characterization of the word languages in Σ 2 [ < , MOD]. The fragment Σ 2 consists of first-order formulas in prenex normal form with two blocks of quantifiers starting with an existential block. In addition we show that Δ 2 [ < , MOD], the largest subclass of Σ 2 [ < , MOD] which is closed under negation, has the same expressive power as two-variable logic FO 2 [ < , MOD]. This generalizes the result FO 2 [ < ] = Δ 2 [ < ] of Therien and Wilke to modular predicates. As a byproduct, we obtain another decidable characterization of FO 2 [ < , MOD].

Reasoning About Strategies

In open systems verification, to formally check for reliability, one needs an appropriate formalism to model the interaction between agents and express the correctness of the system no matter how the environment behaves. An important contribution in this context is given by modal logics for strategic ability, in the setting of multiagent games, such as A tl , A tl *, and the like. Recently, Chatterjee, Henzinger, and Piterman introduced Strategy Logic , which we denote here by CHP-S l , with the aim of getting a powerful framework for reasoning explicitly about strategies. CHP-S l is obtained by using first-order quantifications over strategies and has been investigated in the very specific setting of two-agents turned-based games, where a nonelementary model-checking algorithm has been provided. While CHP-S l is a very expressive logic, we claim that it does not fully capture the strategic aspects of multiagent systems. In this article, we introduce and study a more general strategy logic, denoted S l , for reasoning about strategies in multiagent concurrent games. As a key aspect, strategies in S l are not intrinsically glued to a specific agent, but an explicit binding operator allows an agent to bind to a strategy variable. This allows agents to share strategies or reuse one previously adopted. We prove that S l strictly includes CHP-S l , while maintaining a decidable model-checking problem. In particular, the algorithm we propose is computationally not harder than the best one known for CHP-S l . Moreover, we prove that such a problem for S l is N on E lementary . This negative result has spurred us to investigate syntactic fragments of S l , strictly subsuming A tl *, with the hope of obtaining an elementary model-checking problem. Among others, we introduce and study the sublogics S l [ ng ], S l [ bg ], and S l [1 g ]. They encompass formulas in a special prenex normal form having, respectively, nested temporal goals, Boolean combinations of goals, and, a single goal at a time. Intuitively, for a goal, we mean a sequence of bindings, one for each agent, followed by an L tl formula. We prove that the model-checking problem for S l [1 g ] is 2E xp T ime - complete , thus not harder than the one for A tl *. In contrast, S l [ ng ] turns out to be N on E lementary -hard, strengthening the corresponding result for S l . Regarding S l [ bg ], we show that it includes CHP-S l and its model-checking is decidable with a 2E xp T ime lower-bound. It is worth enlightening that to achieve the positive results about S l [1 g ], we introduce a fundamental property of the semantics of this logic, called behavioral , which allows to strongly simplify the reasoning about strategies. Indeed, in a nonbehavioral logic such as S l [ bg ] and the subsuming ones, to satisfy a formula, one has to take into account that a move of an agent, at a given moment of a play, may depend on the moves taken by any agent in another counterfactual play.

Realizability interpretation of PA by iterated limiting PCA

For any partial combinatory algebra (PCA for short) $\mathcal{A}$, the class of $\mathcal{A}$-representable partial functions from $\mathbb{N}$ to $\mathcal{A}$ quotiented by the filter of cofinite sets of $\mathbb{N}$ is a PCA such that the representable partial functions are exactly the limiting partial functions of $\mathcal{A}$-representable partial functions (Akama 2004). The n-times iteration of this construction results in a PCA that represents any n-iterated limiting partial recursive function, and the inductive limit of the PCAs over all n is a PCA that represents any arithmetical partial function. Kleene's realizability interpretation over the former PCA interprets the logical principles of double negation elimination for Σ0n-formulae, and over the latter PCA, it interprets Peano's arithmetic (PA for short). A hierarchy of logical systems between Heyting's arithmetic (HA for short) and PA is used to discuss the prenex normal form theorem, relativised independence-of-premise schemes, and the statement ‘PA is an unbounded extension of HA’.

Prenex normal form in linguistic quantifiers modeled by Sugeno integrals

In this paper, Ying's prenex normal form theorem for linguistic quantifiers is strengthened so that it behaves like the one in the classical logic. This result further shows elegant logical properties of Ying's framework.

Linguistic quantifiers based on Choquet integrals

The introduction of linguistic quantifiers has provided an important tool to model a large number of issues in intelligent systems. Ying [M.S. Ying, Linguistic quantifiers modeled by Sugeno integrals, Artificial Intelligence 170 (2006) 581–606] recently introduced a new framework for modeling quantifiers in natural languages in which each linguistic quantifier is represented by a family of fuzzy measures, and the truth value of a quantified proposition is evaluated by using Sugeno’s integral. Representing linguistic quantifiers by fuzzy measures, this paper evaluates linguistic quantified propositions by the Choquet integral. Some elegant logical properties of linguistic quantifiers are derived within this approach, including a prenex normal form theorem stronger than that in Ying’s model. In addition, our Choquet integral approach to the evaluation of quantified statements is compared with others, in particular with Ying’s Sugeno integral approach.

On Minimal Models

We investigate some logics which use the concept of minimal models in their definition. Minimal objects are widely used in Logic and Computer Science. They are applied in the context of Inductive Definitions, Logic Programming and Artificial Intelligence. An example of logic which uses this concept is the MIN(FO) logic due to van Benthem [20]. He shows that MIN(FO) is equivalent to the Least Fixed Point logic (LFP) in expressive power. In [6], we extended MIN(FO) to the MIN Logic and proved it is equivalent to second-order logic in expressive power. Here, we exhibit a fragment of MIN, the MIN Δ logic, which is more expressive than LFP, less expressive than MIN and closed under boolean connectives and first-order quantification. In order to do this, in the Section 2, we prove that the Downward Lowenheim-Skolem Theorem holds for arbitrary countable sets of LFP-formulas by showing that every infinite structure has a countable LFP-substructure. The method may be used to generalize this theorem to any set of LFP-formulas. We also analyse the expressive power of the Nested Abnormality Theories (NATs) of Lifschitz, another formalism based on minimal models used in Artificial Intelligence, and we demonstrate that for each second-order theory Γ there is a NAT which is a conservative extension of Γ. We give a translation from second-order sentences into such NATs which is linear in the size of the sentence in prenex normal form. Finally, we establish a hierarchy of expressiveness of these logics that deal with the concept of minimal models.

On the First-Order Prefix Hierarchy

We investigate the expressive power of fragments of first-order logic that are defined in terms of prefixes. The main result establishes a strict hierarchy among these fragments over the signature consisting of a single binary relation. It implies that for each prefix p, there is a sentence $\phi_p$ in prenex normal form with prefix p, over a single binary relation, such that for all sentences θ in prenex normal form, if θ is equivalent to $\phi_p$, then p can be embedded in the prefix of θ. This strengthens a theorem of Walkoe.

Existential second-order logic over graphs

Fagin's theorem, the first important result of descriptive complexity, asserts that a property of graphs is in NP if and only if it is definable by an existential second-order formula. In this article, we study the complexity of evaluating existential second-order formulas that belong to prefix classses of existential second-order logic, where a prefix class is the collection of all existential second-order formulas in prenex normal form such that the second-order and the first-order quantifiers obey a certain quantifier pattern. We completely characterize the computational complexity of prefix classes of existential second-order logic in three different contexts: (1) over directed graphs, (2) over undirected graphs with self-loops and (3) over undirected graphs without self-loops. Our main result is that in each of these three contexts a dichotomy holds, that is to say, each prefix class of existential second-order logic either contains sentences that can express NP-complete problems, or each of its sentences expresses a polynomial-time solvable problem. Although the boundary of the dichotomy coincides for the first two cases, it changes, as one moves to undirected graphs without self-loops. The key difference is that a certain prefix class, based on the well-known Ackermann class of first-order logic, contains sentences that can express NP-complete problems over graphs of the first two types, but becomes tractable over undirected graphs without self-loops. Moreover, establishing the dichotomy over undirected graphs without self-loops turns out to be a technically challenging problem that requires the use of sophisticated machinery from graph theory and combinatorics, including results about graphs of bounded tree-width and Ramsey's theorem.

Polynomial Computability of Certain Rudimentary Predicates

The class of rudimentary predicates is defined as the smallest class of numerical predicates that contains the equality and concatenation predicates and is closed under the operations of propositional logic, explicit transformations, and bounded quantification. Two classes of rudimentary predicates are considered. The first of them consists of the predicates whose prenex normal form of a special type has the quantifier prefix of the form $$\exists \forall $$ . Predicates of the second class can have an arbitrary quantifier prefix, but restrictions are imposed on the Skolem deciding functions. It is proved that any predicate from each of these classes can be computed by a suitable deterministic algorithm in polynomial time.

Sequences of n-diagrams

We consider only computable languages, and countable structures, with universe a subset of ω, which we think of as a set of constants. We identify sentences with their Gödel numbers. Thus, for a structure , the complete (elementary) diagram, Dc(), and the atomic diagram, D(), are subsets of ω. We classify formulas as usual. A formula is both Σ0 and Π0 if it is open. For n &gt; 0, a formula, in prenex normal form, is Σn, or Πn, if it has n blocks of like quantifiers, beginning with ∃, or ∀. For a formula θ, in prenex normal form, we let neg(θ) denote the dual formula that is logically equivalent to ¬θ—if θ is Σn, then neg(θ) is Πn, and vice versa.

Detection of infeasible paths using Presburger arithmetic

Detecting infeasible paths (IFPs) allows accurate computation of various kinds of program slices, and accurate detection of semantic errors that may occur when two variants of a program are merged. We propose a method of efficiently determining the truth of a prenex normal form Presburger sentence (P-sentence) bounded only by existential quantifiers, which is suitable for detecting IFPs. In this method, a coefficients matrix is converted into a triangular matrix based on the method proposed by Cooper (1972). If the rank of the matrix is lower than the degree of the matrix, the matrix is triangulated by using a method for solving one linear equation with three or more unknowns, so that the matrix can be back-substituted. This paper shows that an implementation of our method provides a slower increase in computation time than the previous method and reduces computation time by up to 3,000,000 times.

Cut normal forms and proof complexity

Statman and Orevkov independently proved that cut-elimination is of nonelementary complexity. Although their worst-case sequences are mathematically different the syntax of the corresponding cut formulas is of striking similarity. This leads to the main question of this paper: to what extent is it possible to restrict the syntax of formulas and — at the same time— keep their power as cut formulas in a proof? We give a detailed analysis of this problem for negation normal form (NNF), prenex normal form (PNF) and monotone formulas. We show that proof reduction to NNF is quadratic, while PNF behaves differently: although there exists a quadratic transformation into a proof in PNF too, additional cuts are needed; the elimination of these cuts requires nonelementary expense in general. By restricting the logical operators to {⋀,⋁,∀,∃}, we obtain the type of monotone formulas. We prove that the elimination of monotone cuts can be of nonelementary complexity (here generalized disjunctions in the antecedents of sequents play a central role). On the other hand, we define a large class of problems (including all Horn theories) where cut-elimination of monotone cuts is only exponential and show that this bound is tight. This implies that the elimination of monotone cuts in equational theories is at most exponential. Particularly, there are no short proofs of Statman's sequence with monotone cuts.

Evaluation of recursive queries with extended rules in deductive databases

In order to extend the expressive power of deductive databases, a formula that can have existential quantifiers in prenex normal form in a restricted way is defined as an extended rule. With the extended rule, we can easily define a virtual view that requires a division operation of relational algebra to evaluate. The paper addresses a recursive query evaluation where at least one formula in a recursive rule set is of an extended rule. We investigate transformable recursions as well as four cases of non-transformable recursions of transitive-closure-like and linear type. The work reveals that occurrence of an existentially quantified variable in the extended recursive body predicate might dramatically limit the level of recursive search. In particular, the number of iterations to answer extended queries can be determined, independently of database contents.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>