## Hypothetical Reasoning with Intuitionistic Logic (1994)

Venue: | Non-Standard Queries and Answers, Studies on Logic and Computation, chapter 8 |

Citations: | 5 - 4 self |

### BibTeX

@INPROCEEDINGS{Bonner94hypotheticalreasoning,

author = {Anthony J. Bonner},

title = {Hypothetical Reasoning with Intuitionistic Logic},

booktitle = {Non-Standard Queries and Answers, Studies on Logic and Computation, chapter 8},

year = {1994},

pages = {187--219},

publisher = {Oxford University Press}

}

### OpenURL

### Abstract

This paper addresses a limitation of most deductive database systems: They cannot reason hypothetically. Although they reason effectively about the world as it is, they are poor at tasks such as planning and design, where one must explore the consequences of hypothetical actions and possibilities. To address this limitation, this paper presents a logic-programming language in which a user can create hypotheses and draw inferences from them. Two types of hypothetical operations are considered: the insertion of tuples into a database, and the creation of new constant symbols. These two operations are interesting, not only because they extend the capabilities of database systems, but also because they fit neatly into a well-established logical framework, namely intuitionistic logic. This paper presents the proof theory for the logic, outlines its intuitionistic model theory, and summarizes results on its complexity and on its ability to express database queries. Our results establish a st...

### Citations

1492 | The stable model semantics for logic programming
- Gelfond, Lifschitz
- 1988
(Show Context)
Citation Context ...o define semantics for arbitrary Horn programs with negation. There have been numerous attempts here too. Perhaps the best known are the well-founded semantics [28, 52] and the stable model semantics =-=[29]-=-. Although, these semantics offer different interpretations of recursion through negation, they are equivalent when rulebases are stratified or locally stratified. Several researchers have extended th... |

844 | The well-founded semantics for general logic programs
- Gelder, Ross, et al.
- 1991
(Show Context)
Citation Context ...proach to negation-as-failure attempts to define semantics for arbitrary Horn programs with negation. There have been numerous attempts here too. Perhaps the best known are the well-founded semantics =-=[28, 52]-=- and the stable model semantics [29]. Although, these semantics offer different interpretations of recursion through negation, they are equivalent when rulebases are stratified or locally stratified. ... |

634 |
Modal Logic, An Introduction
- Chellas
- 1980
(Show Context)
Citation Context ...e propositional case [8], and then for the full predicate case [15]. The proof is not based on the fixpoint constructions of logic programming, but on the canonical-model constructions of Modal logic =-=[21]-=-. The rest of this section outlines intuitionistic model-theory, specializing the presentation for the special case of embedded implications. Note that the syntax of the logic is firstorder, but its m... |

610 |
Towards a theory of declarative knowledge
- Apt, Blair, et al.
- 1988
(Show Context)
Citation Context ...do not arise. Perhaps the best known of these classes is the stratified rulebases. These rulebases are layered, and within each layer, a negated premise refers only to rules found in the layers below =-=[4]-=-. In this way, recursion never occurs through a negated premise, so the semantics of negation is 4 This example, taken from [14], is an improvement by McCarty on an example originally due to Bonner [1... |

530 |
The complexity of relational query languages
- Vardi
- 1982
(Show Context)
Citation Context ... form R+ DB ` /. Informally, the data complexity of this system is the complexity of inference when the rulebase is fixed and the database varies (acts as input). More formal definitions are given in =-=[18, 53, 11]-=-. We first show that inference with embedded implications is semi-decidable, and more specifically, that its data complexity is complete for re. We then show that the data complexity can be reduced---... |

271 | Relational queries computable in polynomial time
- Immerman
- 1986
(Show Context)
Citation Context ...imb one level in the polynomial-time hierarchy. As a corollary, we conclude that stratified linear rulebases express exactly the second-order definable queries. Unlike other results in the literature =-=[32, 53, 19]-=-, the results in this section do not assume that the data domain is linearly ordered. The assumption of ordered domains is a technical device that is often used to achieve expressibility results, but ... |

254 |
On the declarative semantics of deductive databases and logic programs
- Przymusinski
- 1988
(Show Context)
Citation Context ...iginally due to Bonner [10]. always well-defined. Another approach has been to allow limited recursion through negation. This approach generalizes the notion of stratification to local stratification =-=[45]-=-. Intuitively, a rulebase is locally stratified if its ground instantiation is stratified (with possibly infinitely many strata). Locally stratified rulebases are perhaps the largest class of logic pr... |

244 | Structure and complexity of relational queries
- Chandra, Harel
- 1982
(Show Context)
Citation Context ... form R+ DB ` /. Informally, the data complexity of this system is the complexity of inference when the rulebase is fixed and the database varies (acts as input). More formal definitions are given in =-=[18, 53, 11]-=-. We first show that inference with embedded implications is semi-decidable, and more specifically, that its data complexity is complete for re. We then show that the data complexity can be reduced---... |

230 | Languages that capture complexity classes
- Immerman
- 1987
(Show Context)
Citation Context ...ively complete, and can therefore express any computable generic query. In database theory, the ability to express generic queries is the normal measure of expressiveness for database query languages =-=[2, 3, 17, 18, 32, 33]-=-. Genericity captures the idea that constant symbols are uninterpreted, i.e., that they have no innate meaning. To make this idea more precise, consider the effect on a query of renaming the constant ... |

221 |
Unfounded Sets and Well-founded Semantics for General Logic
- Gelder, Ross, et al.
- 1989
(Show Context)
Citation Context ...proach to negation-as-failure attempts to define semantics for arbitrary Horn programs with negation. There have been numerous attempts here too. Perhaps the best known are the well-founded semantics =-=[28, 52]-=- and the stable model semantics [29]. Although, these semantics offer different interpretations of recursion through negation, they are equivalent when rulebases are stratified or locally stratified. ... |

196 |
An amateur’s introduction to recursive query processing strategies
- Bancilhon, Ramakrishnan
- 1986
(Show Context)
Citation Context ...clause logic, "linear rules play an important role because, (i) there is a belief that most `real life' recursive rules are linear, and (ii) algorithms have been developed to handle them efficien=-=tly" [6]-=-. A precise definition of linear recursion for embedded implications is given in [11, 9]. The following example illustrates the idea. Example 5.1 A Linear Rulebase A Non-Linear Rulebase A / (B / D 1 )... |

184 |
Contributions to the theory of logic programming
- Apt, Emden
- 1982
(Show Context)
Citation Context ... C) is interpreted as the failure to prove B / C. Thus, A is inferred if B / C cannot be inferred. This semantics is similar to the semantics of negation as commonly defined in Horn logic programming =-=[5]-=-. Example 6.1 The rules below are part of a rulebase that defines a student's eligibility for financial aid. 4 Intuitively, a student s is eligible for a stipend if he is a near-graduate but not a gra... |

152 |
Datalog extensions for database queries and updates
- Abiteboul, Vianu
- 1991
(Show Context)
Citation Context ...ively complete, and can therefore express any computable generic query. In database theory, the ability to express generic queries is the normal measure of expressiveness for database query languages =-=[2, 3, 17, 18, 32, 33]-=-. Genericity captures the idea that constant symbols are uninterpreted, i.e., that they have no innate meaning. To make this idea more precise, consider the effect on a query of renaming the constant ... |

147 |
A logical analysis of modules in logic programming
- Miller
- 1989
(Show Context)
Citation Context ... as well. Vieille, et al, for instance, have developed a deductive database along these lines [54], and Warren and Manchanda have used hypothetical rules to reason about database updates [55, 36]. In =-=[41]-=-, Miller shows that hypothetical insertions can structure the runtime environment of logic programs, resulting in programs that are more elegant, more efficient, and easier to maintain. In [42], he de... |

101 |
Semantical analysis of intuitionistic logic I, Formal Systems and Recursive
- Kripke
- 1963
(Show Context)
Citation Context ...mplications. Note that the syntax of the logic is firstorder, but its model theory is modal, i.e., is based on a set of possible worlds. A complete development of intuitionistic logic can be found in =-=[23, 35]. Defini-=-tion 4.1 (Structures) An intuitionistic structure is a triple M = hS; ; ��i, where ffl S is a non-empty set, fflsis a transitive, reflexive relation on S, ffl �� is a mapping from elements of ... |

84 | Computable queries for relational databases
- Chandra, Harel
- 1980
(Show Context)
Citation Context ... Complete definitions are given in [10, 11, 15]. Here, we summarize the main ideas. Formally, a database query is a mapping that takes a database as input and returns a set of ground tuples as output =-=[17]-=-. We can use embedded implications with free variables to define such mappings. Some care is required though, since the inference system of Definition 3.2 is defined only for embedded implications tha... |

84 |
Intuitionistic Logic Model Theory and Forcing
- Fitting
- 1969
(Show Context)
Citation Context ...programming community. Most of this work focuses on the hypothetical insertion of atoms into a database. These updates, it turns out, fit neatly into a well-known logical system, intuitionistic logic =-=[23]-=-. Most of this work has been semantic, first showing that hypothetical insertion is indeed intuitionistic, and then casting the semantics in terms of a least fixpoint theory, in the logic-programming ... |

82 |
N-prolog: an extension of prolog with hypothetical implications. i
- Gabbay, Reyle
- 1984
(Show Context)
Citation Context ...ion is indeed intuitionistic, and then casting the semantics in terms of a least fixpoint theory, in the logic-programming tradition. Gabbay first showed that hypothetical insertion is intuitionistic =-=[25]-=-. Working independently, McCarty and Miller extended this result to operations that create new constant symbols during inference, and they developed fixpoint semantics based on intuitionistic logic [3... |

65 | Lexical scoping as universal quantification - Miller - 1989 |

59 |
Intuitionistic propositional logic is polynomial-space complete
- Statman
- 1979
(Show Context)
Citation Context ... symbols during inference, and they developed fixpoint semantics based on intuitionistic logic [38, 41]. Earlier, Statman showed that intuitionistic logic is PSPACE-complete in the propositional case =-=[48]-=-. Several researchers have also investigated the the semantics of negation-as-failure for hypothetical insertions [25, 14, 31, 43, 30, 22]. We discuss this work in Section 6. This paper adds to the pi... |

57 |
A language for legal discourse - i. basic features
- McCarty
- 1989
(Show Context)
Citation Context ... motivated by legal applications, has developed a wide class of hypothetical rules for computer-based consultation systems, especially systems for reasoning about corporate tax law and estate tax law =-=[38, 40, 46]-=-. Theoretical work on hypothetical inference has also been carried out, largely by the logic-programming community. Most of this work focuses on the hypothetical insertion of atoms into a database. Th... |

55 |
Theory of database queries
- Chandra
- 1988
(Show Context)
Citation Context ...ink between an important complexity class, polynomial space, and a well-known logic, intuitionistic logic. Other database query languages have been developed that are expressively complete for PSPACE =-=[16, 18, 2, 3]-=-, but the language of stratified embedded implications is the only one based on a model theoretic semantics, and the only one to have an Prolog-style proof procedure, one based on resolution and unifi... |

50 |
A logic-based language for database updates
- Manchandra, Warren
- 1988
(Show Context)
Citation Context ...cal databases as well. Vieille, et al, for instance, have developed a deductive database along these lines [54], and Warren and Manchanda have used hypothetical rules to reason about database updates =-=[55, 36]-=-. In [41], Miller shows that hypothetical insertions can structure the runtime environment of logic programs, resulting in programs that are more elegant, more efficient, and easier to maintain. In [4... |

41 |
The polynomial time hierarchy
- Stockmeyer
- 1977
(Show Context)
Citation Context ...ded implications. The polynomial-time hierarchy is a sequence of complexity classes between P and PSPACE. It is based on the idea of an oracle Turing-machine and can be defined recursively as follows =-=[49]-=-: ffl \Sigma P 0 = \Delta P 0 = P. ffl \Delta P k+1 = P \Sigma P k = those languages accepted in deterministic polynomial time by an oracle machine whose oracle is a language in \Sigma P k . ffl \Sigm... |

37 | Fixpoint logics, relational machines, and computational conlplexity
- Abiteboul, Vardi, et al.
- 1992
(Show Context)
Citation Context ...that the data domain is linearly ordered. The assumption of ordered domains is a technical device that is often used to achieve expressibility results, but it is not an intrinsic feature of databases =-=[1]-=-. Embedded implications do not need this artificial assumption, since they can generate a linear order on the domain hypothetically. Thus, the results of this section are for arbitrary databases, orde... |

36 | Hypothetical datalog: Complexity and expressibility
- Bonner
- 1990
(Show Context)
Citation Context ...aper adds to the picture by summarizing some of our own results on hypothetical inference. Our work has considered a variety of hypothetical operations, including insertion, deletion and bulk updates =-=[11, 10, 15, 12, 13, 14, 9, 8]. This paper, howeve-=-r, focuses on two specific operations: (i) adding formulas to a database hypothetically ("insertion"), and (ii) creating new constant symbols hypothetically ("creation"). The abili... |

35 |
Clausal intuitionistic logic II. tableau proof procedure
- McCarty
- 1988
(Show Context)
Citation Context ...essary to actually check n k ground inferences, since unification-based proof procedures exist for embedded implications with free variables. One of the first such procedures was developed by McCarty =-=[39]-=- for the negation-free case. Having defined what it means to express queries with embedded implications, the rest of this section characterizes the exact sets of queries that can be expressed. As show... |

33 |
H.; Decision support systems: putting theory into practice , Prentice-Hall
- Sprague
- 1993
(Show Context)
Citation Context ...eed for computer systems that reason hypothetically. Decision support systems (DSS) are a good example, especially in domains like financial planning where many "what if" scenarios must be c=-=onsidered [34, 44]-=-. A typical example is an analyst who must predict a company's deficit for the upcoming year assuming that employee salaries are increased by a given percentage. Or he might want a table of deficit pr... |

25 |
EKS-V1, a short overview
- Vieille, Bayer, et al.
- 1990
(Show Context)
Citation Context ...pany's deficit for the upcoming year assuming that employee salaries are increased by a given percentage. Or he might want a table of deficit predictions for a number of hypothetical salary increases =-=[54]-=-. Similar problems occur in computer aided design (CAD). Here, one must evaluate the effect on the overall design of local design alternatives and of various external factors [24, 47]. For example, an... |

24 |
Clausal intuitionistic logic I - fixed-point semantics
- McCarty
- 1988
(Show Context)
Citation Context ... motivated by legal applications, has developed a wide class of hypothetical rules for computer-based consultation systems, especially systems for reasoning about corporate tax law and estate tax law =-=[38, 40, 46]-=-. Theoretical work on hypothetical inference has also been carried out, largely by the logic-programming community. Most of this work focuses on the hypothetical insertion of atoms into a database. Th... |

20 |
A transcation language complete for database update and specification
- Abiteboul, Vianu
- 1987
(Show Context)
Citation Context ...ively complete, and can therefore express any computable generic query. In database theory, the ability to express generic queries is the normal measure of expressiveness for database query languages =-=[2, 3, 17, 18, 32, 33]-=-. Genericity captures the idea that constant symbols are uninterpreted, i.e., that they have no innate meaning. To make this idea more precise, consider the effect on a query of renaming the constant ... |

17 |
Database Updates in Pure Prolog
- Warren
- 1984
(Show Context)
Citation Context ...cal databases as well. Vieille, et al, for instance, have developed a deductive database along these lines [54], and Warren and Manchanda have used hypothetical rules to reason about database updates =-=[55, 36]-=-. In [41], Miller shows that hypothetical insertions can structure the runtime environment of logic programs, resulting in programs that are more elegant, more efficient, and easier to maintain. In [4... |

15 | A logic for hypothetical reasoning
- Bonner
- 1988
(Show Context)
Citation Context ...aper adds to the picture by summarizing some of our own results on hypothetical inference. Our work has considered a variety of hypothetical operations, including insertion, deletion and bulk updates =-=[11, 10, 15, 12, 13, 14, 9, 8]. This paper, howeve-=-r, focuses on two specific operations: (i) adding formulas to a database hypothetically ("insertion"), and (ii) creating new constant symbols hypothetically ("creation"). The abili... |

11 | Adding negation-as-failure to intuitionistic logic programming, ill
- Bonner, McCarty
- 1990
(Show Context)
Citation Context ... instance, it is possible that a rulebase may entail 8 x A(x), and yet it may not entail A(b) for any constant, b. Such issues have received little attention in the literature, and we address them in =-=[7]-=-. 7 Expressibility This section studies the impact of negation-as-failure on the ability of embedded implications to express database queries. We first show that with negation, embedded implications a... |

11 | Hypothetical datalog negation and linear recursion
- BONNER
- 1989
(Show Context)
Citation Context ...aper adds to the picture by summarizing some of our own results on hypothetical inference. Our work has considered a variety of hypothetical operations, including insertion, deletion and bulk updates =-=[11, 10, 15, 12, 13, 14, 9, 8]. This paper, howeve-=-r, focuses on two specific operations: (i) adding formulas to a database hypothetically ("insertion"), and (ii) creating new constant symbols hypothetically ("creation"). The abili... |

10 |
Hypothetical Reasoning in Deductive Databases
- Bonner
- 1991
(Show Context)
Citation Context ...ecursion through negation, they are equivalent when rulebases are stratified or locally stratified. Several researchers have extended these ideas to embedded implications with negation. For instance, =-=[14, 10, 11]-=- develop a semantics of stratified rulebases, including a proof theory and model theory. Examples 6.1, 7.1 and 7.2 show stratified rulebases and describe their semantics informally. In the special cas... |

9 |
Expressing database queries with intuitionistic logic
- Bonner, McCarty, et al.
- 1989
(Show Context)
Citation Context ...is not expressible in many other relational query languages. Before continuing, we must define what it means to express a database query using embedded implications. Complete definitions are given in =-=[10, 11, 15]-=-. Here, we summarize the main ideas. Formally, a database query is a mapping that takes a database as input and returns a set of ground tuples as output [17]. We can use embedded implications with fre... |

8 |
Declarative Semantics of Hypothetical Logic Programming with Negation as Failure
- Dung
- 1992
(Show Context)
Citation Context ...man showed that intuitionistic logic is PSPACE-complete in the propositional case [48]. Several researchers have also investigated the the semantics of negation-as-failure for hypothetical insertions =-=[25, 14, 31, 43, 30, 22]-=-. We discuss this work in Section 6. This paper adds to the picture by summarizing some of our own results on hypothetical inference. Our work has considered a variety of hypothetical operations, incl... |

7 |
Negation as failure in intuitionistic logic programming
- Giordano, Olivetti
- 1992
(Show Context)
Citation Context ...man showed that intuitionistic logic is PSPACE-complete in the propositional case [48]. Several researchers have also investigated the the semantics of negation-as-failure for hypothetical insertions =-=[25, 14, 31, 43, 30, 22]-=-. We discuss this work in Section 6. This paper adds to the picture by summarizing some of our own results on hypothetical inference. Our work has considered a variety of hypothetical operations, incl... |

7 | A Kripke-like Model for Negation as Failure
- Harland
- 1989
(Show Context)
Citation Context ...man showed that intuitionistic logic is PSPACE-complete in the propositional case [48]. Several researchers have also investigated the the semantics of negation-as-failure for hypothetical insertions =-=[25, 14, 31, 43, 30, 22]-=-. We discuss this work in Section 6. This paper adds to the picture by summarizing some of our own results on hypothetical inference. Our work has considered a variety of hypothetical operations, incl... |

7 |
A Dynamic Logic Programming Language for Relational Updates
- Manchanda
- 1987
(Show Context)
Citation Context ...lications are first-order formulas that can hypothetically insert formulas into a database, and can hypothetically create new constant symbols. Such systems have been developed by several researchers =-=[26, 38, 41, 10, 15, 37]-=-. This section defines a simplified version of these systems, one that retains many of the essential properties of the more elaborate systems while admitting a clean theoretical analysis. One notable ... |

7 |
Embedding expert knowledge and hypothetical databases into a data base system
- Stonebraker, Keller
- 1980
(Show Context)
Citation Context ...ssumptions, or updates. Early work in this area was done by Stonebraker, who showed that hypothetical databases can be efficiently implemented by slight extensions to conventional database mechanisms =-=[51, 50]. He -=-pointed out that hypothetical databases are useful for debugging purposes, for generating test data, and for carrying out a variety of simulations. He also argued that "there are advantages to ma... |

6 |
EPS-II: Estate planning with prototypes
- Schlobohm, McCarty
- 1989
(Show Context)
Citation Context ... motivated by legal applications, has developed a wide class of hypothetical rules for computer-based consultation systems, especially systems for reasoning about corporate tax law and estate tax law =-=[38, 40, 46]-=-. Theoretical work on hypothetical inference has also been carried out, largely by the logic-programming community. Most of this work focuses on the hypothetical insertion of atoms into a database. Th... |

4 | The reuse and Modification of Rulebases by predicate substitution
- Bonner, Imielinski
- 1990
(Show Context)
Citation Context |

4 |
Hypothetical databases as views
- Stonebraker
- 1981
(Show Context)
Citation Context ...ssumptions, or updates. Early work in this area was done by Stonebraker, who showed that hypothetical databases can be efficiently implemented by slight extensions to conventional database mechanisms =-=[51, 50]. He -=-pointed out that hypothetical databases are useful for debugging purposes, for generating test data, and for carrying out a variety of simulations. He also argued that "there are advantages to ma... |

3 |
N-Prolog and Equivalence of Logic Programs (Part 1
- Olivetti, Terracini
- 1992
(Show Context)
Citation Context |

3 |
Financial Planning in Action
- Olson, Jr
- 1989
(Show Context)
Citation Context ...eed for computer systems that reason hypothetically. Decision support systems (DSS) are a good example, especially in domains like financial planning where many "what if" scenarios must be c=-=onsidered [34, 44]-=-. A typical example is an analyst who must predict a company's deficit for the upcoming year assuming that employee salaries are increased by a given percentage. Or he might want a table of deficit pr... |

1 | The complexity of reusing and modifying rulebases - Bonner - 1992 |

1 |
Clause Queries and Generlizations
- Horn
- 1985
(Show Context)
Citation Context ...imb one level in the polynomial-time hierarchy. As a corollary, we conclude that stratified linear rulebases express exactly the second-order definable queries. Unlike other results in the literature =-=[32, 53, 19]-=-, the results in this section do not assume that the data domain is linearly ordered. The assumption of ordered domains is a technical device that is often used to achieve expressibility results, but ... |

1 |
Computation with Run Time Skolemization
- Gabbay, Reyle
- 1993
(Show Context)
Citation Context ... OE can be derived. In databases and logic programming, it is common to ask queries that return a set of tuples as answers. Such queries can be defined using embedded implications with free variables =-=[10, 15, 9, 26, 27, 39]-=-. We address this issue in Section 7, where we discuss database queries in general. Until then, however, we shall assume that a query is a closed formula. Indeed, as far as the above inference system ... |