Syntactic Counterpart Research Articles

The new semantic model in ASIS for Ada 2005

Following the new Ada 2005 standard, the ASIS interface has to be upgraded. In addition to supporting the representation of new features, ASIS has been extended with a semantic subsystem, where queries support the logical views of entities rather than their syntactic counterparts. This paper describes the new concepts introduced by this new subsystem.

Semi-normal forms and functional representation of product fuzzy logic

By McNaughton famous theorem, the class of functions representable by formulas of Łukasiewicz logic is the class of piecewise linear functions with integer coefficients. The first goal of this work to find an analogy of the McNaughton result for product logic. The second goal is to define a conjunctive and disjunctive semi-normal form (CsNF, DsNF) of the formulas of product logic (these forms are a syntactical counterpart of the piecewise monomial functions). These results show us how the functions expressible by the formulas of product logic look like. Furthermore, this is the first step in creating an automated theorem prover for product logic.

Semantical Counting Circuits

Counting functions can be defined syntactically or semantically depending on whether they count the number of witnesses in a non-deterministic or in a deterministic computation on the input. In the Turing-machine-based model these two ways of defining counting were proven to be equivalent for many important complexity classes. In the circuit-based model it was done for #P, but for low-level complexity classes such as #AC 0 and #NC 1 only the syntactical definitions were considered. We give appropriate semantical definitions for these two classes and prove them to be equivalent to the syntactical ones. We also consider semantically defined probabilistic complexity classes corresponding to AC 0 and NC ^{1} and prove that in the case of unbounded error, they are identical to their syntactical counterparts.

GS·Λ Theories: A Syntax for Higher-Order Graphs

Graphs and term graphs have proved strikingly flexible and expressive in modeling and specifying distributed, concurrent systems. However, even if the theory of (term) graph transformation systems is well developed, it has not yet included higher-order features, which are essential, in ordinary programming languages, to guarantee the levels of parametrisation and modularity that are required in practice.Our proposal aims at filling that gap by equipping graph-based formalisms with higher-order constructs. Our starting point has been graph substitution (gs) theories by Corradini and Gadducci, which are proved to be syntactical counterparts for term graphs (in the same way as algebraic theories are for terms). As for cartesian categories, gs categories specify symmetric monoidal categories equipped with two transformations, representing sub-term copying and garbage collection. However, for gs theories the naturality of these transformations does not hold.GS·Λ theories enrich gs theories by requiring the existence of a right adjoint a ⊸ _ to the tensor _⊗ a, for each object a. As such, they can be considered as transformation enriched autonomous categories. The resulting categories are tightly linked to the higher-order sharing theories introduced by Hasegawa: these latter are in fact reflective sub-categories of GS·Λ theories, since they are captured by an equivalent axiomatisation, except for the laws concerning the duplication of a set of values (namely, of λ-abstractions, interpreted as finished computations). Dropping those axioms allow for a neat graphical representation, expanding the wire-and-box notation developed for gs theories in a conservative manner.

Possibilistic Merging and Distance-Based Fusion of Propositional Information

The problem of merging multiple sources information is central in many information processing areas such as databases integrating problems, multiple criteria decision making, expert opinion pooling, etc. Recently, several approaches have been proposed to merge propositional bases, or sets of (non-prioritized) goals. These approaches are in general semantically defined. Like in belief revision, they use implicit priorities, generally based on Dalal's distance, for merging the propositional bases and return a new propositional base as a result. An immediate consequence of the generation of a propositional base is the impossibility of decomposing and iterating the fusion process in a coherent way with respect to priorities since the underlying ordering is lost. This paper presents a general approach for fusing prioritized bases, both semantically and syntactically, when priorities are represented in the possibilistic logic framework. Different classes of merging operators are considered depending on whether the sources are consistent, conflicting, redundant or independent. We show that the approaches which have been recently proposed for merging propositional bases can be embedded in this setting. The result is then a prioritized base, and hence the process can be coherently decomposed and iterated. Moreover, this encoding provides a syntactic counterpart for the fusion of propositional bases.

Peirce's challenge to material implication as a model of 'if'

Suppose there is a sweepstake in which only one thousand people are eligible to participate. Tickets are sold for ?1 each. No participant is permitted to buy more than one ticket. And the winner will take the total of the stakes. Then, intuitively, (3.1) is true while (3.2) is false. Read concludes (p. 6) that classical quantificational logic does not have the resources to express adequately the lack of entailment of the second sentence in (1) or (3) by the first. Moreover, he ascribes this inadequacy to the translation of 'if' by the implication sign of classical quantificational logic (p.10). What I wish to take issue with here is Read's ascription of the source of the observed inadequacy. Consideration of a broader range of data, however, makes clear that the inadequacy noted by Read does not reside in the translation of 'if' as classical material implication. For Peirce's puzzle can be reproduced without any occurrence of the subordinating conjunction 'if', or any occurrence of any of its lexical or syntactic counterparts. Consider the following pair of

Subcategorization and syntax-based theta-role assignment

Classic problems of how to generalize over predicate-argument relations (e.g., buy vs. sell; spray paint vs. spray a wall) have led to postulating semantic representations which are structured differently than deep syntax, such as (linked) theta grids and (lexical) conceptual structures. I argue that such autonomous semantics massively violates parsimony, and that theta-roles are better predicted by using only modestly enhanced, independently justified deep structures. In addition, I claim that several recent generalizations (of Rizzi, Levin and Rappaport, and Randall) are better formulated as deep syntactic properties than in terms of theta-roles. This syntactic approach to predicate-argument relations thus reinitiates a line of research implicit in Chomsky's Aspects but never developed. The first section argues that only this approach faithfully applies the syntactic revolution to lexical (head-complement) semantics. Principles invoked include Chomsky's Full Interpretation and Rule for Agents and Talmy's Figure/Ground separation, along with a new Ground Specification and syntactic counterparts to two formal devices from Jackendoff's Conceptual Structures. The thematic role constellations for many verb classes (mostly but not all from English) are shown to follow from these principles. The conclusion speculates that the theta-roles assigned to a sentence are not its properties at a linguistic level, but rather indicate how that sentence is to modify cognitive representations.

Well-behaved modal logics

Much of the literature on the model theory of modal logics suffers from two weaknesses. Firstly, there is a lack of generality; theorems are proved piecemeal about this or that modal logic, or at best small classes of logics. Much of the literature, e.g. [1], [2], and [3], confines attention to structures with the expanding domain property (i.e., if wRu then Ā(w) ⊆ Ā(u)); the syntactic counterpart of this restriction is assumption of the converse Barcan scheme, a move which offers (in Russell's phrase) “all the advantages of theft over honest toil”. Secondly, I think there has been a failure to hit on the best ways of extending classical model theoretic notions to modal contexts. This weakness makes the literature boring, since a large part of the interest of modal model theory resides in the way in which classical model theoretic notions extend, and in some cases divide, in the modal setting. (The relation between α-recursion theory and classical recursion theory is analogous to that between modal model theory and classical model theory. Much of the work in α-recursion theory involved finding the right definitions (e.g., of recursive-in) and separating concepts which collapse in the classical case (e.g. of finiteness and boundedness).)The notion of a well-behaved modal logic is introduced in §3 to make possible rather general results; of course our attention will not be restricted to structures with the expanding domain property. Rather than prove piecemeal that familiar modal logics are well-behaved, in §4 we shall consider a class of “special” modal logics, which obviously includes many familiar logics and which is included in the class of well-behaved modal logics.

Concrete categories and infinitary languages

By definition, a concrete category is a category of sets which are endowed with an unspecified structure. There have been some attempts to make this structure specific. For example, Blanchard [6] used Bourbaki-type structures and Kurera and Pultr [17] have determined the structure by a functor Set-Set. Our aim is to consider concrete categories as categories of models of first-order theories. However, for these theories to be a syntactic counterpart of concrete categories, they must exceed the usual ones in the following three points: nonlogical symbols are of arbitrary arities, there may be a proper class of them and infinitary logical symbols are admitted. This language might be called ‘an unrestricted f.,,,‘. Its strength is illustrated by the fact that we may imagine any concrete category as a category consisting of models of this language. Of course, in this generality one cannot expect very model-theoretic results. Our approach is more an approach of ‘a working mathematician’, i.e. to see the relation between syntactic properties of theories and semantical properties of their categories of models. This approach has a practical aspect, i.e. to know what may be said about current models from life (e.g. when there exist limits, colimits, free objects or when models form a Cartesian closed category, a symmetric monoidal closed one etc.). There is also a theoretical aspect, i.e. to characterize theories providing a given categorical property or on the contrary to characterize concrete categories of models of theories of a prescribed kind. The pattern of these ‘categorical preservation theorems’ is the Beck-Linton theorem (see MacLane [20, p. 1471) which covers the equational case. A characteristic feature of these preservation theorems is the fact that they work outside any similarity type. The classical preservation theorem corresponding to the above mentioned Beck-Linton one is Birkhoff’s characteriz- ation of varieties of algebras of a given type. The generality of the syntax used means that the categories of models arising need not be legitimate. In Section 2 we establish a smallness property of a theory which ensures the legitimacy of its category of models. Moreover, these theories corres- pond to strongly fibre-small concrete categories in the sense of AdBmek, Herrlich 0022-4049/81/0000-OOOO/SO2.50 0 1981 North-Holland

A second-order relevance logic with modality

In this paper a system, RPF, of second-order relevance logic with S5 necessity is presented which contains a defined, notion of identity for propositions. A complete semantics is provided. It is shown that RPF allows for more than one necessary proposition. RPF contains primitive syntactic counterparts of the following semantic notions: (1) the reflexive, symmetrical, transitive binary alternativeness relation for S5 necessity, (2) the ternary Routley-Meyer alternativeness relation for implication, and (3) the Routley-Meyer notion of a prime intensional theory, as well as defined syntactic counterparts, of the semantic notions of a possible world and the Routley-Meyer * operator.

On the metamathematics of rings and integral domains

Introduction. In this paper we are concerned with the metamathematics of the first order theory of rings Ro and integral domains JDo. The purpose of the paper is to characterize derivability from Ro and JDo respectively by algebraic notions pertaining to the theory of polynomial ideals. The essential tool from logic needed is an improved version of Gentzen's extended Hauptsatz to be derived in ?I. ??II and III contain some remarks and introduce new notations. In ?IV we prove a syntactical counterpart of Hilbert's Nullstellensatz. Although this syntactical result could easily be proved with the aid of Hilbert's Nullstellensatz and the completeness theorem we think that its metamathematical proof has some interest in itself (see Lemma 4*). In ?V we combine the results of ??I and IV in order to prove an algebraic version of Gentzen's extended Hauptsatz for Ro and JDo. Applications of the techniques developed in ??I-IV are presented in ??VI and VII. Lemma 4*, a constructive version of Lemma 4, has been suggested by G. Kreisel. There is an interesting application of Lemma 4* to a problem considered by G. Kreisel. This application lies somewhat outside the scope of this paper, hence we omit it. It will be presented, together with some related topics, in a separate note. NOTATIONS. (1) By I and R we denote the set of integers and the set of rationals respectively. I[xj, .. ., xj] and R[xl, ... , xj] (or briefly I[x], R[x]) are the rings of polynomials in the variables xl, .. ., xn, with coefficients in I and R respectively. Notions such as prime ideal, primary decomposition, basis of an ideal will be used frequently. For details concerning them we refer to [4]. (2) At many places, vectors whose components are terms (from a certain theory) will be used. For particular vectors such as (xi, .. ., xn), (Yi, ..., Ym) we will use sometimes the abbreviations xn, x and Ymi, Y (3) Let gl, .. ., gn be polynomials in R[x]; by B(g1,. ..., gn) we denote the ideal consisting of all polynomials of the form 21 higi with hi E R[x] for i ? n. If g1 ,... gn E I[x] then B*(gl,.. ., gn) denotes the ideal consisting of all polynomials 21 higi with hi E I[x] for i < n. Several notations will be introduced as they will be needed, as, e.g. at the end of ?111. (4) Existential and universal quantifiers will be denoted by ] and V respectively but in order to save space we delete the V in formulas and write universal quantification over x more simply as (x); at some places a sequence of universal quantifiers

Representations of calculi

A previous paper proposed a finitary system of logic which was surmised to be basic in the sense that every finitary system of logic could be defined within it. In the present paper this anticipated result, or rather its syntactical counterpart, is definitely established as Theorem VI. The calculus K is a formalization of the basic logic. Its main features were described in the previous paper, acquaintance with which is presupposed here. Attention is called to the fact that from now on the class Y of atomic U-expressions will be assumed to be finite instead of infinite. This is a relatively minor change and does not affect the previously established results. The account of the nature of calculi, as given in the other paper, is to be made more precise by stating that a calculus is any class of expressions having as a Gödel representation a recursively enumerable class of integers. A system of logic is regarded as being finitary if it is capable of formalization by means of an appropriate calculus.